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UNITED STATES DEPARTMENT OF LABOR
F rances P erk in s,

S
ecretary

B U R E A U O F L A B O R S T A T IS T IC S
Isador L u b in ,

C m
om issioner

T he M aking and U sing o f
Index Numbers
By
WESLEY C. MITCHELL

This Bulletin is a Reprint of Part I
From Bulletin No. 284 of the
Bureau of Labor Statistics

B u lletin 7s£o. 656
March 1938

U N IT E D S T A T E S
G O V E R N M E N T P R I N T I N G O FFIC E
W A S H I N G T O N : 1938

F o r sale b y th e S u p e rin te n d e n t o f D o cu m e n ts, W a sh in g ton , D . C . -




-

-

P rice 20 cen ts




PREFACE

This study o f the making and using of index numbers, by Wesley
C. Mitchell, was originally published in 1915 as part of Bulletin 173
o f the Bureau of Labor Statistics, dealing with wholesale-price index
numbers in the United States and foreign countries. A revision of
this bulletin, including a revision of Dr. Mitchell’s section, was issued
as Bulletin 284 in 1921, following the world-wide revolution in prices
caused by the war.
Insofar as these bulletins dealt with current price-reporting
methods they are, of course, long since obsolete. However, the sec­
tion by Dr. Mitchell on the making and use of index numbers has
been in continuing demand, particularly in colleges and universities,
and, to meet this demand, is now being reprinted, without change,
from the original plates.
I sador L

u b in ,

C om m ission er o f L a b o r S ta tistics .

M ay 18, 1938.




in




CO N TEN TS.
Part I .— T h e m aking and using of index n u m bers:

I. The history of index num bers.. . ...................................... ...................................... 7-10
II. The difficulties of measuring changes in the level of prices.......................... 10,11
III. The characteristics of price fluctuations............................................................... 11-23
IY . Varieties of methods used in making index numbers........................................ 23-93
1. The relations between methods and uses................................................ 23-25
2. Collecting and publishing the original quotations.................. ............25-27
3. Market prices, contract prices, institution prices, and importexport values . . . , ........................................................................................ 27-31
4. Relative versus actual prices....................................................................... 31-33
5. The numbers and kinds of commodities included............................... 33-59
6. Problems of weighting.................................................................................... 59-68
7. Averages and aggregates................................................................................ 68-81
8. Base periods, chain index numbers, and fixed base series'............. 81-91
9. The *‘ ideal ” formula.................................................................................... 91-93

V. A comparison of the leading American index numbers for the years 1890
94-112
to 1918.................................................................
1. Analysis of the similarities and differences, by years, 1890 to 1918 94-105
2. Comparison of four leading American index numbers, by months,
July, 1914, toNovember,1918.......................................................... 105-108
3. Critical evaluation of the Bureau of Labor Statistics’ , Bradstreet’s,
and Dun’sindex numbers..................................................................108-112
YI. Conclusions............................................................................................................ 112-114
L IS T OF C H A R T S .

Chart 1.— Conspectus of yearly changes in prices, 1891-1918................. (Facing)
Chart 2.— Distribution of 5,578 price variations (percentages of rise or fall
from prices of preceding year)..........................................................................................
Chart 3.— Distribution of the price variations of 241 commodities in 1913
(percentages of rise or fall in price)...................................................................... ..
Chart 4.— Index numbers made from the market prices and from the import
and export values of identical lists of commodities.
England, 1871-1902.
(Based on Table 5 .) ........................................................... ............ ......................................
Chart 5.— General-purpose index numbers including 25, 50, and 242 com­
modities. (Based on Table 6 .).........................................................................................
Chart 6.— Index numbers of the prices of 20 raw materials and of 20 products
manufactured from them. (Based dn Table 7 . ) ......................................................
Chart s .— I ndex numbers of the prices of wool, cotton, hides, wheat, and pig
iron in their raw, partially manufactured, and finished forms.
(Based on
Table 7 .) ................................................. ..................................................................................
Chart 8.— Index numbers of the prices of 19 mineral products and of 18 farm
crops. (Based on Table 8 .) ...............................................................................................
Chart 9.— Index numbers of the prices of manufactured goods used for family
consumption and for industrial purposes. (Based on Table 9 .) ............... ........
Chart 10.— Index numbers of the prices of 25 food products and of 25 miscel­
laneous commodities. (Based on Table 13.)..............................................................
Chart 11.— A comparison of medians and arithmetic means of 145 commod­
ities. (Based on Table 16.)..................... ............................. ............................................
Chart 12.— Index numbers of the Bureau of Labor Statistics, Dun, and Bradstreet, 1890 to 1918. (Based on Table 1 9 .) ................................................ (Facing)
Chart 13.— Secular trends of index numbers of Bureau of Labor Statistics,
Dun, and Bradstreet, 1896 to 1914. (Based on Table 21.)....................................
Chart 14.— Index numbers of Bradstreet, compared with their secular trend,
1896 to 1914. (Based on Table 2 1 .) ...............................................................................
Chart 15.— Index numbers of Bureau of Labor Statistics compared with their
secular trend, 1896 to 1914. (Based on Table 2 1 .)...................................................
C hart 16.— Index numbers of Dun, compared with their secular trend, 1896
to 1914. (Based on Table 2 1 .).........................................................................................
Chart 17.— Yearly deviations from secular trend of index numbers of the
Bureau of Labor Statistics, Dun, and Bradstreet, 1896 to 1914. (Based on
Table 2 1 . ) . . . .......................................................................................................................
C hart 18.— Index numbers of the Bureau of Labor Statistics, Dun, Bradstreet,
and the War Industries Board, July, 1914, to December, 1918. (Average
prices July, 1913, to June, 1 9 1 4 = 1 0 0 .)..........................................................................




3

15
19
20
30
38
44
45
47
49
55
77
96
98
99
100
101
103
10T




PAR T I.— T H E M A K IN G A N D U SIN G O F IN D E X
NUM BERS.
BY

W E S L E Y C. M IT C H E L L .1

I.—THE HISTORY OF INDEX NUMBERS.
The honor of inventing the device now commonly used to measure
changes in the level of prices probably belongs to an Italian, G. R,
Carli. In an investigation into tlxe effect of the discovery of America
upon the purchasing power, of money, he reduced the prices paid for
grain, wine, and oil in 1750 to percentages of change from their prices
m 1500, added the percentages together, and divided the sum by
three, thus making an exceedingly simple index number. Since his
book was first published in 1764, index numbers are over 150 years
old.2
It was in England, however, where practically the same device had
been hit upon by Sir George Schuckburg-Evelyn in 1798,8 that the
theory and practice of index numbers were chiefly developed. The
generation that created the classical political economy was deeply
interested in the violent price fluctuations that accompanied the
Napoleonic wars and the use of an irredeemable paper currency from
1797 to 1821. Several attempts were made, to measure these fluc­
tuations, and in 1833 G. Poulett Scrope suggested the establishment
of a “ tabular standard of value.” 4
Interest in the study of price fluctuations lagged somewhat in the
forties; but the great rise of prices after the Californian and Aus­
tralian gold discoveries started fresh investigations. W. S. Jevons
in England and Adolf Soetbeer in Germany gave a powerful impetus
to the theoretical discussion and the practical computation of index
numbers. The problem changed somewhat in form but received even
more attention after 1873, when a prolonged fall of prices began. In the
sixties the chief aim of investigation had been to discover the relations
between the rise of prices and the increased production of gold; in the
seventies and eighties the chief aim was to find the relations between
the fall of prices and the restrictions placed upon the free coinage of

1

The writer has received generous help from Prof. Irving Fisher, Prof. A llyn A . Young, D r. R oyal
Meeker, and Mr. C. H . Vorrill, all of whom read the first draft of this paper and m ade m any effective criti­
cisms. In revising the paper the writer has m ade free use of the criticisms of the first edition published
b y Prof. F. Y . Edgeworth, Econom ic Journal, June, 1818, V ol. X X V I I I , p p . 176-197, anji b y Prof. Fred­
rick R . Macaulay, American E conom ic R eview , March, 1916, V o l. V I , p p . 203-209. H e is indebted once
more to Dr. R oyal Meeker for critical and constructive suggestions, and t o Prof. W . F. Ogburn for super­
vising certain com putations and for reading the manuscript. Prof. Macaulay has considered the theoretical
sections with care and suggested numerous im provem ents in b oth text and tables.
Del Valore e della Proporzione de’ Metalli Monetati con i generiin Italiaprim a delle Scoperte deir Indie
col confronto del Valore e della Proporzione de’ Tem pi nostri. Republished b y Custodi in his Scrittori
Italiani de Econom ia Politics. Parte Moderna, V ol. X I I I , p p . 297-363, especially pp. 335-354.
* “ A n account of som e endeavors to ascertain a standard of weight and measure, ” Philosophical Trans­
actions of the R oya l Society of L ondon, 1798, P art I, A rt V I II , p p . 133-182, especially p p . 175 and 176.
Principles o f Political E conom y, L ondon , 1833, p p . 405-408. It is interesting to note, however,
that neither D avid Ricardo, who wrote several pamphlets on currency and prices during the “ bank re­
striction, ” nor Thom as Tooke, w ho published an elaborate H istory of Prices in 1793-1847, m ade use of
index numbers.

2

4




7

THE

8

M A K I N G A N D U S IN G OF IN D E X N U M B E K S .

silver. The weightiest theoretical contributions of this period were
made by Prof. F. Y. Edgeworth, who served as secretary of a com­
mittee appointed by the British Association for the Advancement of
Science ‘ ‘for the purpose of investigating the best methods of ascer­
taining and measuring variations in the value of the monetarv stand­
ard.” 5
The problem of price fluctuations entered upon another phase when
the world-wide rise of prices which began in 1896-97 had oeen under
way for several years. After 1900, and more insistently after 1910,
complaints about the rising cost of living became common in all
civilized countries. Efforts to measure this increase as well as
efforts to explain it multiplied.
Index numbers are both troublesome and expensive to compile,
yet now in the United States not less than seven wholesale-price
series are currently maintained, four of them by financial papers.
In England there are four important series; in France one; m Ger­
many, before the beginning of the World War, there were three;
while the Governments of Canada, Australia, South Africa) India,
Netherlands, and New Zealand now publish official index numbers,
and private investigators have made series for Italy, Japan, Belgium,
Denmark, Norway, Austria, Spain, and Sweden, although not all of
these were kept up during the war period. This list may well be
incomplete at present, and is almost certain to require additions
within a short time.
Most of the series just mentioned have been established but recently.
The oldest— that of the London Economist— was begun in 1869.6
Sauerbeck’s English series dates from 1886, Conrad’s German series
from 1887 (though in a sense it continues investigations made by
Laspeyres in 1864), and Bradstreet’s American series from 1897.
Of the remaining index numbers regularly published at present, all
date from years since 1899, and the majority from years since 1909.7
With this increase in numbers there has come an improvement in
quality. The early index numbers were made by private investi­
gators, at irregular intervals, from such price quotations as chance
had preserved. As public appreciation of the importance of meas­
uring changes in price levels has developed, the work has more and
more been assumed by financial journals and Government bureaus.
This shift has produced a greater measure of continuity in the series,
as well as greater frequency, regularity, and promptness in the pub­
lication of the results. Even more important is the improvement
in the character and the scope of the price quotations from which
the index numbers are made Whereas the individual investigator
had to take what he could get in the way of data, financial journals
and Government bureaus can collect those current prices that are
best adapted for statistical treatment, and can give better assurance
of the representative value of their quotations and the uniform
quality oi the commodities included in successive years.

6

For the reports of this com m ittee, see the R eports of the British Association, 1887, p p . 247-254; 1888,
p p. 181-188; 1889, p . 133; 1890, pp . 485-488, See particularly the m emoranda by.Prof. Edgeworth subjoined
to these reports.
e From 1864 to 1869 the Econom ist published the relative prices of com m odities, b ut such separate figures
without a sum or an average do not constitute an index number proper.
i The years m entioned are the dates of first publication, not the earliest dates for which relative prices
are shown. In m ost cases the computers carried their investigations back into the past, frequently for a
decade or more.




H IS T O R Y OF IN D E X

NUM BERS,

9

This improvement in the quantity and* quality of index numbers
is as marked in the United States as elsewhere. Price quotations
had been published with more or less care and system by various
newspapers and periodicals for many years before the first effort to
compile an average of price variations was made. In 1881, Mr. H. C.
Burchard, Director of the Mint, made an index number covering the
years 1825 to 1880 from quotations that had been printed in certain
reports of the Secretary of the Treasury, supplemented by quotations
from a New York newspaper. But his data were of uncertain quality
and his series was allowed to lapse after 1884.8 After an interval
of eight years, the Senate Committee on Finance authorized a more
ambitious effort. Under the direction of Dr. Roland P. Falkner,
the statistician of this committee, the (then) Department of Labor
made a huge collection of price quotations, running back as far as
1840, and compiled'an index number including more than 200 com­
modities for the years 1860 to 1891, and 85 commodities for 1840 to
1891J But this also was a single investigation, and the United States
9
did not have an index number regularly maintained year after year
until the establishment of Bradstreet’s series in 1897. A quasi
continuation of the Senate Finance Committee’s work, covering
the years 1890-1899, was prepared by Dr. R . P. Falkner, and pub­
lished by the Department of Labor in March, 1900.1 Another short­
0
lived series was begun by Prof. John R. Commons and’Dr. N. I. Stone
in the Quarterly Bulletin of the Bureau of Economic Research later
in the same year.1 In January, 1901, the second continuous Ameri­
1
can series was started by Dui^s Review and gradually carried back
to 1860; the third, covering the years 1890 to date, was added by
the Federal Department of Labor in March, 1902. Other series of
this type were begun by Thomas. Gibson’s weekly market letters in
1910, b y the New York Times Annalist in 1913, and by the Federal
Reserve Board in 1918.
This activity in the making of index numbers was accompanied by
a rapid growth of the literature of the subject. Among the later
contributions dealing with theoretical issues, the first place belongs to
the work of an American scholar, Mr. C. M. Walsh. His great
treatise upon the Measurement of General Exchange-Value, published
in 1901, is still the most comprehensive book upon the subject. But
the bibliographies that aim to cover the field now include hundreds
of items, and to them must go the student who wishes a guide to
further reading.1
2
Some of the more important new series known to have been estab­
lished since the war are the series compiled by the Price Section of
the War Industries Board and published in its “ History of Prices
s See Finance Reports, 1881, pp t 312-321; 1882, p p . 252-254; 1883, p p. 316-318; R eport of the Director of
the Mint on the Production of the Precious Metals, 1884, p p. 497-502. Compare the criticism of this series
b y Prof. J. Laurence Laughlin, Quarterly Journal of Econom ics, A pril, 1887, pp. 397 and 398.
» See the description given on p p . 149-159.
See Bulletin No. 27 of the Department of Labor, March, 1900.
See the issues for July and October, 1900.
For such bibliographies see W alsh, The Measurement of General Exchange-Value, pp . 553-574, and
J. L . Laughlin, Principles of Money, p p . 221-224. The m ost im portant contributions of later date than
Laughlin’s entries are Prof. Irving Fisher’s Purchasing Power o f Money, pp. 385-429, Mr. C. M. W alsh’s
“ The Problem o f E stim ation,” Prof. Irving Fisher’s paper on “ The Best F orm o f In d ex Numbers,”
with discussion, in the’ Quarterly Publication o f the American Statistical Association, March, 1921,and
Mr. A . W . F lux’s paper on “ The Measurement o f Price Changes,” with discussion, in the Journal o f the
R oyal Statistical Society, March, 1921.

1
0
1
1




10

THE

M A K IN G

AND

U S IN G

OF IN D E X N U M B E R S .

During the War,” the series compiled by the Federal Reserve Board
from data gathered by the United States Bureau of Labor Statistics,
the series designed by the same board for making international com­
parisons, the series published by the United States Food Administra­
tion in 1918 in a pamphlet entitled “ General Index Numbers of Food
Prices on a Nutritive Value Base,” the series established by the
London Times for Great Britain and by the Handelstidning for
Sweden, the series for Italy compiled by rrof. Riccardo Bachi, the
series compiled by the Bank of Japan, and those published by the
Governments of South Africa and New Zealand.
II.—THE DIFFICULTIES OF MEASURING CHANGES IN THE
LEVEL OF PRICES.
It is a curious fact that men did not attempt to measure changes
in the level of prices until after they had learned to measure such
subtle things as the weight of the atmosphere, the velocity of
sound, fluctuations of temperature, and the precession of the equi­
noxes. Their tardiness in attacking that problem is the more strange
because price changes had frequently been a subject of acrimonious
debate among publicists and a cause of popular agitation. Long
before the high development of the credit system and a class of
permanent wage earners practical issues of grave importance were
raised by the instability of prices, as the disturbances created in
sixteenth-century Europe by the inflow of American silver and gold
abundantly show. Perhaps disinclination on the part of “ natural
philosophers ” to soil their hands with such vulgar subjects as the
prices of provisions was partly responsible for the dela y;1 but after
3
all a number of eminently “ respectable” men wrote upon economic
topics in every generation after the days of Columbus— to go no
further back. Nor can the technical difficulties of the problem
explain this tardiness; for the mathematical intricacy of index num­
bers, and even the necessity of allowing for changes in the pure silver
content of coins, are obstacles far less formidable than those sur­
mounted long before in other fields of research.
Probably the chief cause of delay was that averages of price fluctu­
ations did not promise to command much confidence after they had
been made. The quotations available for use by the early investi­
gators were few in number and often of doubtful accuracy. Carli,
for example, dealt with only three commodities; Shuckburg-Evelyn
with 12. About the vastly greater number of unrecorded price
fluctuations the one firmly established fact was that they exhibited
bewildering diversity. Under these circumstances, could an average
made from a few samples be accepted as a reliable measure of changes
in the general level oi prices ? And if averages could not be trusted,
why trouble to devise a plan of making them? So writers upon

1
3

One of the early British writers on prices, Bishop Fleetwood, remarked: “ * * * as the W orld now
goes, the greatest (tho’ I w ill not think the best) Part of Readers will be rather apt to despise than to com ­
m end the Pains that are taken in making Collections of so m ean Things as the price of W heat & Oats, of
Poultry, and such like Provisions * *
— Chroiiicon Preciosum, 1707, 2d ed., 1745, p. . Sir G. Shuck­
burg-Evelyn, in the paper referred to above, also felt himself on the defensive in presenting the first English
index num ber: “ * * * * H owever, I m ay appear to descend below the dignity o f philosophy, in such
economical researches, I trust I shall find favour with the historian, at least, and the antiquary.
Shuckburg-Evelyn’ s discussion of index numbers, indeed, was merely a minor appendix to his discussion of
standards of weights and measures. B ut it has become his chief claim to remembrance.




6

DIFFICULTIES

OF

MEASURING

CHANGES

IN

PRICES.

11

prices long contented themselves with statements about the fluc­
tuations o f particular commodities, and with indefinite assertions that
the purchasing power of money had changed little or changed much.
So, also, when certain bold investigators did finally venture to make
index numbers, no one was particularly impressed by the significance
of their achievement.
This lack of faith in the validity of averages of price variations was
overcome rather slowly, partly in consequence of improvements in
business organization. The multiplication of commercial news­
papers and the more systematic keeping of private and public records
provided a larger and more accurate body of quotations. Improved
means of transportation made wholesale prices in the larger cities
basic for many local markets. The grading and standardizing of
commodities increased the number of articles which could be ac­
cepted as substantially uniform in quality from one year to the
next. More important still was the discovery by statisticians that
social phenomena of most kinds, though seeming to result from the
uncontrolled choice of individuals, yet reveal a striking regularity
when studied in large numbers.1 The demonstration that a formerly
4
unsuspected regularity lay hidden in one set of numerical data after
another encouraged economists to believe that the known price varia­
tions might after all be fair samples of the more numerous unknown
variations. The general similarity of the results reached by different
investigators using dissimilar data confirmed this faith. Thus em­
boldened, economic statisticians devoted much time to extending the
scope and improving the technique of index numbers. And their
growing confidence in the trustworthiness of their series was gradually
imparted to the public.
To-day few, if any, competent judges doubt the validity of index
numbers or the substantial accuracy of the results they show when
properly constructed from carefully collected data. Indeed the
danger at present is rather that the figures published will be taken
too absolutely as a complete representation of the facts about price
fluctuations. It is therefore well to begin a study of index numbers,
not by analyzing the finished series, but by inspecting the actual
changes in prices from which they are made, and which they purport
to summarize. In no other way, indeed, can the value and the
limitations of index numbers be learned.
IIL—THE CHARACTERISTICS OF PRICE FLUCTUATIONS.
An excellent collection of materials for the study of changes in
wholesale prices is found in Bulletin No. 149 of the Bureau of Labor
Statistics, published in 1914. Here are given the average annual
prices at wholesale of more than 230 commodities for a period of al­
most a quarter of a century. Comparison of the changes in these
actual prices* is facilitated by the publication of two series of relative
rices lor each commodity. One series reduces the quotations in
ollars and cents to percentages of the average actual prices in the
decade 1890-1899. The second series, which may be called u chain
relatives/’ shows the percentage by which each article rose or fell in

§

1
4

The Belgian statistician, Adolphe Quetelet, and Thomas H enry Buckle, author of the H istory of
Civilization in England, 1857 and 1861, were perhaps the most effective demonstrators o f this fact.




THE M A K IN G AND USING OF INDEX NUMBERS.

12

price each year as compared with the year before.1 Since this sec­
5
tion is concerned wholly with problems of method which have no
connection with, any given period of time, there is no reason for
bringing all the illustrative materials down to date.
A survey of these relative figures for the 230 commodities thrown
the diversity of price fluctuations into high relief. (1) During the
24 years 1890-1913 no two of the commodities quoted underwent the
same changes in price. Some articles rose rather steadily in price and
fluctuated on a much higher level in 1913 than in 1890; for example,
rosin and crude petroleum. Other articles fell much more than they
rose and fluctuated on a much lower level at the end than at the be­
ginning; for example, soda and wood alcohol. Some articles were
steady in price, seldom changing from one year to the next; for ex­
ample, bread and certain kinds of tools. Other articles changed in
price every year; for example, cotton and pig iron. (2) In every
year a considerable proportion of the commodities rose in price, a
considerable proportion fell, and a somewhat smaller proportion
remained unchanged. (3) The range covered even by the fluctua­
tions from one year to the next was very wide. For example, in .1896
potatoes fell 54.6 per cent, while coke rose 41.5 per cent; in 1899
wheat flour fell 20.2 per cent, while steel billets rose 103.3 per cent;
in 1913 onions fell 38.5 per cent, while cabbage rose 58.5 per cent.1
6
Such extreme diversities as have been cited, however, give a mis­
leading impression of chaos among the fluctuations. A just impres­
sion can be had only from some scheme of presentation which takes
account of all the commodities at once. Table 1 is a first rough
approximation toward this end.1 It shows for each year how many
7
ot the commodities quoted rose, remained unchanged, or fell in price,
and divides those which rose and those which fell into six groups,
according to the magnitude of their fluctuations.
1
5
The reader m ay follow the discussion more easily if he runs over the following sample of the figures
referred to.
Cotton, upland, middling.

Year.

Average
price per
pound.

A verage, 1890-1899. $0.07762
1890............................
.11089
1891............................
.08606
1892............................
.07686
.08319
1893............................
.07002
1894............................

Relative
price.

100.0
142.9
110.8
99.0
107.2
90.2

Per cent
of in­
cr e a s e ^ )
or d e­
crease ( —)
compared
with pre­
ceding
year.

—22.4
-1 0 .7
+
-1 5 .8

8.2

1895 ...............
1896............
1897...........................
1898............................
1899...........................

.07298
.07918
.07153
.05972
.06578

102.0
92.2
76.9
84.7

+ 4.2
+ 8.5
— 9.7
-1 6 .5
+ 10.1

1900...........................
1901...........................
1902...........................
1903..........................
1904...........................

.09609
.08627
.08932
.11235
.12100

123.8
111.1
115.1
144.7
155.9

Year.

Per cent
o f in ­
crease ( + )
Average
or de­
price per R elative crease ( —)
price.
pound.
com pared
w ith pre­
ceding
year.

21.0

1905........................... $0.09553
1906...........................
.11025
1107...... ....................
. 11879
1908...........................
. 10463
1909...........................
.12107

123.1
142.0
153. 0
134.
156.0

—
+15.4
+ 7.7
—11.9
+15.7

1910...........................
1911..........................
1912...........................
1913...........................

194.8
168.
148. 2
164.

+24.9
—13.8
—
+

.15118
.13037
. 11503
.12792

8
0
8

+46.1
-1 0 .2
+ 3.5
+25.8
+ 7.7

94.0

11.8
11.2

i® A ll of these figures show percentages of rise or fall from the average prices of the com m odities in question
in the preceding year.
n The figures in this table have been brought down to 1918 to harmonize with the material in Section V ,
on “ A comparison of the leading American index numbers for the years 1890 to 1918.”




T a b l e 1 . -C O N SPE C TU S OF T H E CHANGES IN W H O L E S A L E PRICES IN TH E U N IT E D STATES., B Y Y E A R S , 1891 TO 1918.

[Based upon the percentages of increase or decrease in price from one year to the next, computed from Table 9 of Bulletin of the United States Bureau of Labor Statistics, No. 269,
M ay, 1920.]

1

Number of commodities that fell in price by—

20.0

10.0

to
to 5.0 to
50.0 per
cent or 49 9 * w 19.9 per 9.9 per
cent. 'ce n t.
more.
cent.

1891....................................................
1892...................................................
1 8 9 3 -...............................................
1814...................................................
189,'...................................................

232
232
234
236
237

106
140
114
192
138

18%...................................................
1897...................................................
1899...................................................
1899....................................................
1900 .................................................

240
241
242
242
242

133
118
73
46
38

1901..................................................
1902...................................................
1903...................................................
1904 ..................................................
1906........................................ . .........

242
242
242
242
242

128
61
92
106
89

1906...................................................
If07 .................................................
1908..................................................
K 0 9 ....................................................
1910...................................................

242

25

233
253

47
48
155
98
81

253
253
252
329
342

147
80
84
152
137

0
12
12

342
337
348

12

1911...................................................
1912...................................................
1913...................................................
1914...................................................
1916...................................................
1916...................................................
1917...................................................
1918...................................................




2'2
2i2

1

9
56

13
)

1
1

1
6
29
10
22
9
2
1
3

10
6
9
12
3

5

2
2

2.0

to
4.9 per
cent.

21

22
35
21
12

21
16
9

10
9

40
13
23
24
13

32
14

35

11

10
5

9
9
32

4
7

25
70
35

55
22
16
7
4

13

2
1
2
2
26

19
27

34
40
30
27

12
22

28
24

13
18
30
28

50
15

21
2
0

18

1

10
31

43
18
16
‘A
29

30
25
23
28
27

1
2

5

3
3

16

14

9

2
11

2,0

30
39
44
44
41

26

3

2

Less
than
per
cent.

12

14
35
35

22

12

Number
o f com ­
modities
that
d id n o t
ch an ge
in price.

Number of com m odities that rose in price b y—

2.0

18
16
15
15 1
!

44
37
42
25

23
24
13

34
31
34
27 1

12

16
17

2
0
23

10

14
18
32
26
25

22

Number
of com ­
m odities
5.0 to
to 50.0 per
to
to
cent or that rose
4.9 per 9.9 per 19.9 per 49.9 per
in price.
cent.
cent.
cent.
cent.
more.

17
9
17

15

2
0
2
2

12

2
0

25
38

2
0

19

20.0

10.0

2.0

17
19
15
4
15

15
17
7

6

18
16

28
25

2
2

15
13

2
1
3
17

18
30
34
45
59
16
35
44
28
37

16

12
10
3
12
18

11

40
39
57

2
1

29
29

22
23
22

22

31
29
32
26

28
32
32
31
26

13
27
14
24
24

31
?5
14
24
33

52
43
40
42

52
45
H
17
25

19

16
27
39
38

14
34
30
35
39

13
35
27
23
38

27
7
18

37
16
27

19
33
35

31
36
35
59
36

3

10

2
2

2
1

19

Less
than
per
cent.

16
27

21

28
30

22
11
2

9

21

12

10

31

8
8
30
73

14
3
13

1
1
2

6
17
2
2

2
2

16

2

26
33

14
3

9
27

2
1

10
16
15

19

2
0

2

2
2
10
2
0
7
8

3

28

4

115
172
136

42

4
16

3

2
1

100
28

82
55
78
19
77
73
92
135
109
184
89
143
128
113
131
167
162
55
124
146
75
137
133
118
169

CHARACTERISTICS OF PRICE FLUCTUATIONS,

Year.

Total
Number
number
of com ­
of com ­
modities
modities
that fell
quoted
each year. in price.

320
326
291

C
O

14

TH E M AK IN G AND USING OF INDEX NUMBERS.

A more significant presentation of the same set of price fluctuations
is given by Table 2. To make this table a tally sheet was drawn
up for each year from 1891 to 1918, on which the changes from
prices in the preceding year were entered in the order of their
magnitude, beginning with the greatest percentage of fall and run­
ning up through "n o change” to the greatest percentage of rise.
Then the whole number of recorded fluctuations for each year was
divided into 10 numerically equal groups, again beginning with the
case of greatest fall and counting upward. Finally the nine dividing
points between these 10 equal groups were marked off in the percent­
age scale of fall, "n o change,” or rise. For example, the tally sheet
for 1913 showed how the average prices of 252 commodities in that
year differed from their average prices in 1912. One-tenth of these
252 commodities showed a fall of prices ranging between 38.5 per
cent and 10.4 per cent, the second tenth ranged between a fall of
10.4 per cent and one of 3.7 per cent; the third tenth ranged between
a fall of 3.7 per cent and one of 1 per cent; the fourth tenth between
a fall of 1 per cent and "n o change;” the fifth tenth between "n o
change” and a rise of 0.5 per cent, and so on. These dividing points
( —10.4 per cent, —3.7 per cent, —1 per cent, ± 0 per cent, +0.5
per cent, etc.) between the successive tenths into which the data
were split are called "deeds.” The midmost deed, which of course
divides the whole number of observations into two equal groups, is
called the "m edian.” Table 2 presents the results drawn from the
tally sheets— that is, the nine deeds for .each year, together with the
percentages of greatest fall and of greatest rise from prices in the
year before.
T able 2 .—C H A IN IN D E X N U M B E R S O F PR IC E S A T W H O L E S A L E IN T H E U N IT E D
S T A T E S , B Y Y E A R S , 1891 TO 1918.
[The decils are those points in the percentage scale of rise or fall in price which divide the whole number
of price changes recorded each year into 10 equal groups. Based upon the percentages of increase
or decrease in price from one year to the next, com puted from Table 9 of Bulletin of the United States
Bureau of Labor Statistics, N o. 269, May, 1920. ]

( —indicates a fall; +indicates a rise; ± 0 indicates “ no change.” )
Year.

1891..............
1892..............
1893..............
1894..............
1895..............
1896..............
1897..............
1898..............
1899..............
1900..............
1901..............
1902..............
1903..............
1904..............
1905..............
1906..............
1907..............
1908..............
1909..............
1910.............
1911:............
1912..............
1913..............
1914..............
1915..............
1916..............
1917..............
1918___ . . . .
A verage..

Great­
est
falL

1st
decil.

2d
decil.

3d
decil.

4th
decil.

Me­
dian.

6th
decil.

7th
decil.

8th
decil.

9th
decil.

Great­
est
rise.

Per ct.
-3 0 .5
-4 1 .2
-2 7 .5
-4 4 .3
- 3 8 .0
-5 4 .6
-5 0 .9
-2 1 .9
-2 0 .2
-2 9 .2
-4 2 .6
-4 0 .6
-3 3 . 7
-4 3 .8
-4 4 .9
-3 9 .1
-4 3 .0
-3 9 .5
-2 9 .8
-3 7 .7
-4 7 .4
- 36.1
-3 8 .5
-3 7 .3
-6 0 .4
-1 9 .1
-3 4 .1
—51.0

Per ct.
-1 3 .2
- 1 6 .0
-1 1 .9
-2 1 .4
-1 4 .0
-1 7 .8
—11.5
- 7.0
- 3.8
- 3.6
-1 5 .0
- 7.4
-1 2 .6
-1 5 .0
- 7.6
- 4.8
- 3.2
-2 1 .3
- 7.7
- 6.1
-1 5 .1
- 6.8
-1 0 .4
- 1 2 .0
- 1 2 .0
+ 2.1
+ 8.7
- 6.0

Per ct.
- 4.8
- 8.5
- 5.5
-1 3 .4
— 6.5
- 7.5
- 4.4
.4
± 0
+ 3.2
- 6.1
± 0
- 2.1
- 3.5
- 1.0
± 0 *
± 0
-1 0 .8
- 1.1
.4
- 7.0
.5
- 1.0
- 4.1
- 1.9
+ 10.5
+25.1
+ 8.6

Per ct.
- 1.4
- 5.4
- 2.4
-1 0 .8
- 4.1
- 3.0
- 1.7
± o
+ 2.6
+ 5.1
- 3.7
± o
± 0
.6
± o
+ 2.8
+ 1.2
- 5.8
± o
± 0
- 4.2
± 0
± o
- 1.3
.1
+ 14.4
+ 28 .6
+ 14.8

-1 0 .1

-

+

Per ct.
± 0
- 3.1
± o
- 7.1
- 2.4
- 1.2
± o
+ 1.8
+ 5.5
+ 7.5
- 1.5
+ 2.2
+ 1.3
± 0
+ .7
+ 5.1
+ 3.9
- 3.8
± o
+ 1.5
.9
+ 1.0
+ .5
± o
dt 0
+ 18.6
+ 34 .8
+ 18.5
+ 3.0

P er ct.
± o
- 0.5
± o
- 5.0
± 0
± 0
± 0
+ 5.0
+ 7.6
+ 9.6
± o
+ 4.7
+ 3.7
+ 1.3
+ 3.2
+ 6.4
+ 6.6
.9
+ 1.7
+ 3.6
± 0
+ 3.6
+ 2.4
± o
+ 2.7
+ 24.0
+ 42.1
+ 22.1
+ 5.1

P er ct.
+ 1.5
± o
+ 1.1
- 3.3
+ .7
+ .3
+ 2.9
+ 8.3
+ 10.6
+ 12.7
+ 1.3
+ 7.1
+ 5.3
+ 3.0
+ 5.9
+ 9.7
+ 8.9
± o
+ 5.0
+ 6.3
± o
+ 6.7
+ 4.5
+ 1.5
+ 6.0
+30.1
+ 49.3
+ 28 .6
+ 7.3

Per ct.
+ 5.0
+ 1# 1
+ 4.8
- 1.3
+ 4.2
+ 4.3
+ 6.2
+ 13.3
+ 16.4
+ 17 .4
+ 4.9
+ 12.1
+ 8.3
+ 5.9
+ 9.6
+ 14.5
+ 12.3
+ .8
+ 8.1
+ 9.2
+ 2.9
+ 11.0
+ 7.5
+ 5.0
+ 10.1
+ 38 .7
+57.5
+36.1
+ 11.5

P er ct.
+15.3
+ 5.5
+ 11.0
± 0
+ 12.1
+ 10.2
+ 12.7
+ 19.8
+30.8
+ 25.6
+ 13.2
+ 20.4
+ 14.1
+ 11.7
+ 15.9
+ 18.9
+ 17.6
+ 6.2
+ 16.0
+ 18.6
+ 11.0
+ 17.7
+ 12.7
+ 9.1
+ 18.7
+ 53 .4
+ 69.3
+46.3
+ 19.0

Per. ct.
+ 53.0
+ 28.0
+ 59.1
+ 31.1
+ 61.9
+ 41.5
+ 101.6
+ 60.4
+ 103.3
+ 72.8
+ 53.0
+ 58.9
+ 37.4
+ 39.9
+ 46.0
+ 40.7
+ 67.8
+ 44.9
+ 70.1
+ 49.5
+ 86.1
+ 46.2
+ 58.5
+ 76.4
+ 172.9
+ 155.1
+ 154.2
+ 118.0

-3 1 .9

Per ct.
- 8.0
-1 1 .2
- 8.0
-1 5 .8
- 9.6
-1 1 .3
- 7.2
- 3.3
± o
± o
-1 0 .2
- 1.6
- 5.3
- 7.6
- 3.9
+ 0
± 0
- 1 6 .0
- 3.7
- 2.4
- 9.8
- 2.9
- 3.7
- 7.4
— 5.9
+ 6.7
+ 19.4
+ 2.0
- 5.0




*.9

.9

+ 71.0

Ch

a rt

1 .— CONSPECTUS

O F Y E A R L Y CH AN GES IN P R IC E S, 1891-1918.
(Based on Table 2.)

f311739 0 — 41




( T o face page 15.)

CHARACTERISTICS C)F PRICE FLUCTUATIONS.

15

Chart 1, based upon Table 2 and drawn to a logarithmic scale, gives
a more vivid idea of these price fluctuations. It shows for each ye^r
the whole range covered by the recorded changes from prices in the
preceding year by vertical lines, which connect the points of greatest
rise with the points of greatest fall. These lines differ considerably
in length, which indicates that price changes cover a wider range in
some years than in others. The heavy dots upon the vertical lines
show the positions of the deeds. One-tenth of the commodities
quoted in any given year rose above their prices of the year before by
percentages scattered between the top of the line for that year and the
highest of the dots. Another tenth fell in price by percentages scat­
tered between the bottom of the line and the lowest of the dots. The
fluctuations of the remaining eight-tenths of the commodities were
concentrated within the much narrower range between the lowest
and the highest dots. The dots grow closer together toward the
central dot, which is the median. This concentration indicates, of
course, that the number of commodities showing fluctuations of
relatively slight extent was much larger than the number showing
the wide fluctuations falling outside the highest and lowest deeds,
or even between these deeds and the deeds next inside them.
The middle dots or medians in successive years are connected by a
heavy black line, which represents the general upward or downward
drift of the whole set of fluctuations. To make this drift clear the
median of each year is taken as the starting point from which the
r
upward or downward movements in the following year are meas­
ured. Hence the chart has no fixed base line. But in this respect
it represents faithfully the figures from which it is made; since these
figures are percentages of prices in the preceding year, a price fluc­
tuation in any year establishes a new’ base for computing the percent­
age of change in the following year. The fact that prices in the
preceding year are the units from which all the changes proceed is
further emphasized by connecting the nine deeds, as well as the
points of greatest rise and fall, with the median of the year before by
light diagonal lines. The chart suggests a series of bursting bomb
shells, the bombs being represented by the median dots of the years
before and the scattering of their fragments by the lines which radiate
to the deeds and the points of greatest rise and fall.1
8
Time is well spent in studying this chart, because it is capable of
giving a truer impression of the characteristics of price changes than
can be given by any other device. The marked diversity of the fluctu­
ations of different commodities in the same year— some rising, some
falling, some remaining unchanged— the wide range covered by these
fluctuations, the erratic occurrence of extremely large changes, and
the fact that the greatest percentages of rise far surpass the greatest
percentages of fall are strikingly showm; but so also are the much
greater frequency of rather small variations, the dense concentration
near the center of the field, the existence of a general drift in the whole
complex of changes, and the frequent alterations in the direction and
the degree of this drift. But if the chart is effective in giving these
impressions, it leaves them rather vague. To render certain of them
1 Owing to the constant shifting of the base line, no fixed scale of relative prices can be shown on the
8
margin ofthe chart. Because of its intricacy, the chart had to be reproduced on a larger scale than in the
other cases, but of course that fact does not alter the slant of the lines, and this slant is the matter of
importance.




THE M AK IN G AND USING OF INDEX NUMBERS.

16

more definite, recourse must be had to the figures from which the
chart was drawn.
These figures,, already given in Table 2, enable us to measure the
concentration of the mass of fluctuations about the center of the field.
One way to measure this concentration is to compute the differences
between the successive deeds; that is, to find the range within wdiich
successive tenths of the fluctuations fall. This “ range” is, of course,
a certain number of points in the percentage scale of change from
prices in the year before. When this computation is made for the
whole period covered by the table, we get the results presented in
Table 3. As heretofore, the successive tenths of the fluctuations
represented are reckoned by starting with cases of greatest fall in price
and counting upward to cases of greatest rise. The central division
of the table shows that the average range covered by the fluctuations
diminishes rapidly as we pass from the cases of greatest fall toward
the cases of little change, and then increases still more rapidly as we
go onward to the cases of greatest rise. The right-hand group of
columns shows how the range increases if we start with the two
middle tenths, take in the two-tenths just outside them, then the twotenths outside the latter, and so on until we have included the whole
body of fluctuations. The left-hand group of columns, on the other
hand, combines in succession the two-tenths on the outer boundaries,
then the two-tenths immediately inside them, and so on lentil we get
back again to the two central tenths. Perhaps the most striking sin­
gle result brought out by this table is that eight-tenths of all the fluc­
tuations are concentrated within a range (29.1 per cent) slightly wider
than that covered by the single tenth that represents the heaviest de­
clines (21.8 per cent), and much narrower than that covered b y the
single tenth that represents the greatest advances (52 per cent).
T able 3 .—A V E R A G E C O N C E N T R A T IO N OF P R IC E F L U C T U A T IO N S A R O U N D T H E M E­
D IA N , 1891 T O 1918.
[ Based upon Table 2.

The fluctuations represent percentage changes from average prices in the preceding
year.l
Average range covered b y the—

1st and
10th
tenths
of the
price
fluctu­
ations.

9th
of the
price

3d and 4th and
8th
7th
tenths tenths
of the of the
price
price
fluctu­ fluctu­
ations.
ations.

-

12.6
<

6.3.




6.()j

5th and
6th
tenths
of the
price
fluctu­
ations.

Central
tw o
tenths
Successive tenths
of the price fluctu­ of the
ations.
price
fluctu­
ations.

1st
2d
3d
4th
5th
4.2{ 6th
7th
8th
9th
10th

tenth,
tenth,
tenth,
tenth,
tenth,
tenth,
tenth,
tenth,
tenth,
tenth,

21.8
5.1
2.1
3.8
2.1
2.1 }
2.2
4.2
7.5
52.0

42

Central
four
tenths
o f the
price
fluctu­
ations.

Central
six
tenths
of the
price
fluctu­
ations.

Central
eight
tenths
of the
price
fluctu­
ations.

W hole
num ber
o f th e
price
fluctu­
ations.

1 10.2

> 16.5

■ 29.1

■ 102.9

CHARACTERISTICS OF PRICE FLUCTUATIONS.

17

Such results as these gain greatly in significance by being put
beside corresponding results for other groups of statistical data. The
best comparison to make, however, is one between the actual distri­
bution of our price fluctuations about their average and a “ normal”
distribution of the same data— that is, a distribution according per­
fectly with the so-called “ normal law of error. ” This law shows how
phenomena are distributed about their average when the number of
phenomena observed is very large, and when each phenomenon is the
resultant of numerous independent factors, none of which is of pre­
ponderating importance. It has been found that many kinds of phe­
nomena tend to conform rather closely to this normal distribution; for
example, human heights, errors of observation, shots at a target, wage
rates m different occupations, etc.1 When it can be shown that phe­
9
nomena are distributed approximately in this fashion, their average
can safely be accepted as a significant measure of the whole set of
variations, since even the deviations from the average are then
grouped in a tolerably definite and symmetrical fashion about the
average.
With such a comparison in view we may treat each recorded per­
centage of rise or fall in price as an observation of the degree and
direction in which prices vary from one year to the next. Taking
all the commodities and all the years up to 1913 covered by the bu­
reau^ chain relatives, we have 5,578 observations for analysis. Table
4 shows how these cases are distributed along a percentage scale of
rise or fall in prices which jumps two points at a time. The columns
headed “ number of cases” show how many price variations of the
given magnitude and directions occur, and the columns headed
“ proportion of cases’7 show the same numbers in the rather clearer
form of percentages of their sum (5,578).
Such is the actual distribution of the phenomena under analysis.
To compare it with the “ normal” distribution, we put these figures
on a chart, which presents the facts clearly to the eye. Here the
horizontal scale represents percentages of rise or fall in price, and
the vertical scale represents the number of times each percentage of
change is observed. The dotted line shows how our 5,578 cases
would have been distributed had they followed strictly the normal
law of error. The areas included by the unbroken line and the dotted
line are equal.2
0
1 See, for example, Prof. F. Y . Edgeworth’s article “ Probability,” Part II, Encyclopaedia Britannica,
9
11th ed., and the references there given.
8 Table 4 and Chart 2 m ight be im proved b y a change in form. If the ‘ ‘price variations” in each
9
year were com puted as percentage deviations from their geom etric mean in that year, the d istribution
of their variations would doubtless be more sym m etrical than is the distribution here shown.
1311739 0 - 4 1 ------2




THE M A K IN G AND USING OF INDEX NUMBERS,

18

T able 4 __ D IS T R IB U T IO N OF 5*578 CASES OF CH AN G E IN T H E W H O L E S A L E P R ICE S
OF C OM M ODITIES FR O M ONE Y E A R TO T H E N E X T , ACCORDIN G TO T H E
M A G N I­
TU D E A N D D IR E C T IO N OF T H E C H AN G E S.
[Based upon the chain relatives in Table II of Bulletin No. 149 of the Bureau of Labor Statistics.]

R ising prices.
Per cent of
change from,
the average
price of the
preceding
year.
102-103.9
100-101.9
98- 99.9
96- 97.9
94- 95.9
92- 93.9
90- 91.9
88- 89.9
86- 87.9
84- 85.9
82- 83.9
80- 81.9
78- 79.9
76- 77.9
74- 75.9
72- 73.9
70- 71.9
68- 69.9
66- 67.9
64- 65.9
62- 63- 9
60- 61.9
58- 59.9
56- 57.9
54- 55.9
52- 53.9
50- 51.9
48- 49.9

N um ­
ber of
cases.

Propor­
tion of
cases.

1
1

0.018
.018

1
1
1
1

.018
.018
.018
.018

1
4
1
3
4

.018
.072
.018
.054
.072

4
6
1
3
4
1
5

.072
.108
.018
.054
.072
.018
.090

Falling prices.

Per cent of
change from
the average
price of the
preceding
year.

N um ­
ber of
cases.

46-47.9
44-45.9
42-43.9
40-41.9
38-39.9
36-37.9
34-35.9
32-33.9
30-31.9
28-29.9
26-27.9
24-25.9
22-23.9
20-21.9
18-19.9
16-17.9
14-15.9
12-13-9 >
10-11.9
8 - 9 .9
6 - 7.9
4 - 5.9
2 - 3.9
Under 2.
N o change.

Proportion
of cases.

11
10
6
14
17
11
18
17
22
30
29
47
45
65
73
1 102
106
115
167
i 237
261
1356
355
i 410

0.197
.179
.108
.251
.305
.197
.323
.305
.394
.538
.520
.843
.807
1.165
1.308
1.828
1.900
2.062
2.994
4.249
4.679
6.382
6.364
7.350

1697

12. 494

Per cent of
change from
the average
price of the
preceding
year.
Under 2.
2- 3.9
4- 5.9
6 - 7.9
8- 9.9
10-11.9
12-13.9
14-15.9
16-17.9
18-19.9
20-21.9
22-23.9
24-25.9
26-27.9
28-29.9
30-31.9
32-33.9
34-35.9
36-37.9
38-39.9
40-41.9
42-43.9
44-45.9
46-47.9
48-49.9
50-51.9
52-53-9
54-55.9

Propor­
tion o f
cases.

Num ber
of cases.

i 405
I 375
329
1 238
200
173
i 120
107
76
71
45
39
32
17
27
16
.7
10
.7
5
5
4
2
1
1
1

7.261
6.723
5.898
4.267
3.585
3.101
% 151
1.918
1.362
1.273
.807
.699
.574
.305
.484
.287
.125
.179
.125
.090
.090
.072
.036
,018
.018
.018

1

.018

S u m m a ry .

Num ber
of cases.

Proportion
of cases.

Rising prices...........................................................................................................................
N o change...............................................................................................................................
Falling prices.........................................................................................................................

2,567
697
2,314

46.021
12.494
41. 485

T ota l...................................................: ........................................................................

5,578

100.000

1 Locatlon of thedecils.

There are several points to notice here. While the actual and the
“ normal” distributions look much alike, they are not, strictly speak­
ing, of the same type. The actual distribution is much more pointed
than the other, and has a much higher “ mode,” or point of greatest
density. On the other hand, the actual distribution drops away
rapidly on either side of this mode, so that the curve representing it
falls below the curve representing the “ normal” distribution. The
actual distribution is “ skewed” instead of being perfectly symmetri­
cal. The outlying cases of a “ normal” distribution extend precisely
the same distance from the central tendency in both directions,
whereas in the actual distribution the outlying cases run about twice
as far to the right (in the direction of a rise of prices) as to the left
(in the direction of a fall). This fact suggests that the actual distri-




19

CHARACTERISTICS OF PRICE FLUCTUATIONS.

bution would be more symmetrical if it were plotted on a logarithmic
scale, one which represents the doubling of one price by the same
distance from zero as the halving of another price.. Another aspect
of the difference in symmetry is that the central tendency about
which the variations group themselves is free from ambiguity in one
case but not in the other. In the “ normal” distribution this ten­
dency may be expressed indifferently by the median, the arithmetic
mean, or the mode; for these three averages coincide. In the actual
distribution, on the contrary, these averages differ slightly; the
median and the “ crude” mode stand at ± 0 , while the arithmetic
Ch art 2 —D IS T R IB U T IO N OF 5.578 PRICE V A R IA T IO N S (P E R C E N T A G E S OF
F A L L FR O M PR ICE S OF PR E C E D IN G Y E A R ).

R ISE

OR

mean is +1.36 per cent.2 These departures of the actual distribu­
1
tion from perfect symmetry possess significance; but the fact remains
that year-to-year price fluctuations are highly concentrated about
their central tendency.
This study of the actual distribution of price fluctuations from one
year to the next will be found to throw light upon several problems
presently to be faced in discussing the methods of making index
2 That the arithmetic mean is slightly above zero arises partly from the fact that there are 33 percentages
1
of rise greater than any percentage of fall. But it also arises partly from the fact that our data com e from
a period (1890-1913) when the trend o f year-to-year fluctuations was more often upward than downward;
there were 2,567 cases o f advance in price against 2,314 cases of fall. The median is kept from rising above
zero because the cases of “ no clrange, ” 697 in number, more than offset the difference between the numbers
of advances and of declines in price.




to

o




3$ Case* -

30 C ases '

j& ’ Csges *

r

V a r ia t io n s Jrotn p r i c e s in. /?/£.

J
2 0 Cases, *

/S 'Cases -

/0 C ases

9/p

/Oil*

M f*

t3/fc

W f0

/S/f,

TH E M A K IN G AND USING OF INDEX NUMBERS.

Chart 3.—D IS T R IB U T IO N OF T H E P R IC E V A R IA T IO N S OF 241 C OM M ODITIES IN 1913 (P E R C E N T A G E S OF R IS E OR F A L L IN P R IC E S).

CHARACTERISTICS OF PRICE FLUCTUATIONS*

21

numbers. For the moment we have use primarily for the demonstra­
tion that these fluctuations are highly concentrated about a central
tendency. This conclusion strengthens the hope that we may make
measurements of price fluctuations that fairly represent the net
resultant of all the changes, miscellaneous as they seem to be. For
properly constructed averages have clearly a better chance of being
representative and significant when the phenomena for which they
stand have a strongly marked central tendency about which devia­
tions are grouped than when the phenomena are irregularly scattered
over their range.
But it mustTbe remembered, and with the reminder doubt reenters,
that the variations just analyzed are percentages of increase or de­
crease from the prices of the year before. Most index numbers,
however, attempt to measure price fluctuations, not with reference to
the preceding year, but with reference to a period considerably more
remote. For example, in its old series, here used for illustration, the
Bureau of Labor Statistics measured prices in 1913 in terms of aver­
age prices in the decade 1890-1899. Are price variations computed
in this manner highly concentrated around their central tendency
like the price variations with which we have been dealing ?
Chart 3 answers this question emphatically in the negative. It
represents the distribution of the price variations of 241 commodities
quoted by the Bureau of Labor Statistics for the year 1913.2 These
2
variations are computed in two ways: (1) As percentages of rise or
fall from the prices of 1912; (2) as percentages of rise or fall from
the average prices of 1890-1899. Of course the first set of varia­
tions corresponds in character to the variations represented above in
Chart 2. The distribution of these variations, shown by the area in­
closed by the unbroken line, is similar in type to the actual distri­
bution in Chart 2; although it is less regular— a difference to be
expected, since the number of observations is only 241 here as against
5,578 there. But the distribution of the second set of variations
(percentages of change from the average prices of 1890-1899) as repre­
sented by the area inclosed within the dotted line has no obvious
central tendency; it shows no high degree of concentration around
the arithmetic mean ( + 30.4 per cent) or median ( + 26 per cent)
and it has a range between the greatest fall (52.2 per cent) ana greatest
rise (234.5 per cent) so extreme that two of the cases could not be
represented on the chart.2
3
Price variations, then, become dispersed over a wider range*and less
concentrated about their mean as the time covered by the variations
increases. The cause is simple: With some commodities the trend
of successive price changes continues distinctly upward for years at a
time; with other commodities there is a consistent downward trend;
with still others no definite long-period trend appears. In any large
collection of price quotations covering many years each of these
types, in moderate and extreme form, and all sorts of crossings among
them, are likely to occur. As the years pass by the commodities that
2 The bureau quoted 252 commodities in 1913; but 11 could not be included in the present comparison
8
because h o quotations are given for them in 1890-1899.
23
I n com m enting dn this chart Prof. Edgeworth has shown that, despite appearance^, the distribution
of the price variations from the 1890-1899 base, m ay conform to the normal distribution as closely as the
variation from the preceding year base. For, under the condition presented b y prices, the quantity ob ­
served m ay m ove either up or dow n at each successive interval (here a year), ana w ith a num ber o f obser­
vations such as here used, an ideal distribution would appear m ore or less oblong (as does the dotted line
in Chart 3) rather than bell shaped.—E conom ic Journal, June, 1918, V ol. X X V I I I , pp. 183-185.




22

THE M A K IN G AND USING OF INDEX NUMBERS,

have a consistent trend gradually climb far above or subside far below
their earlier levels, while the other commodities are scattered between
these extremes. Thus the percentages of variation for any given
year gradually get strung out in a long, thin, and irregular line, with­
out a marked degree of concentration about any single point. Another
factor in scattering the percentage variations is probably that the
degree of scatter is a function of the degree of variation, and of course
variations are likely to be larger between 3
^ears far apart than between
years close together.
The consequence is that the measurement of price fluctuations
becomes difficult in proportion to the length of time during which the
variations to be measured have continued. In other words; the
farther apart are the dates for which prices are compared, the wider
is the margin of error to which index numbers are subject, the greater
the discrepancies likely to appear between index numbers made by
different investigators, the wider the divergencies between the aver­
ages and the individual variations from which they are computed, and
the-larger the body of data required to give confidence in the repre­
sentative value of the results.
From this preliminary survey of the characteristics of price fluc­
tuations it appears (1) that year-to-year changes in the price level
can be measured with good prospects of success, because such varia­
tions show a marked degree ot concentration about their central
tendency, but (2) that measurements of variations between years
far apart have a more problematical value. The practical question
whether the index numbers in current use can be trusted, then, may
have two answers. Perhaps they give results that are reliable as
between successive years, and at the same time doubtful for dates
between which 50, 20, or even 10 years have intervened.
The best way to test the reassuring conclusion about index num­
bers for successive years and to resolve the disturbing doubt about
index numbers covering long periods is to compare different series
of index numbers that purport to measure price changes in the same
country during the same time. If the results turn out to be con­
sistent with one another, our faith will be confirmed. If the results
are not consistent, we must find a valid reason for the discrepancies,
or become skeptical about the present methods of measuring changes
in the price level.
When this test is applied, the first impression is unfavorable.
For example, the five currently published American index numbers
show the following results for 1912 and 1913:
Year.

1912.......................................................... .
1913..............................................................
Changes......................................................
Percentage changes .........................^

Bureau of
Labor Sta­ Bra&street’ s
tistics’ index
index
number
number.
(oid series).
*133.6
135.2
+ 1 .6
+ 1 .2 1

$9.1867
9.2076
+.0209
+ .2

Annalist
in d e x
number.
143. 25
139.98
—3. 27
—2.3

Gibson's
ind ex
number.
62.6
58.1
—4.5
—7.2

D u n’s index
num ber.

$124.44
120.89
-3 .5 5
-2 .9

Here no two of the series are as closely consistent with each other
as one could wish. On the contrary, the five series disagree not only
as to the degree but also as to the direction of the change in prices.
And this is a comparison between the same successive years, where
measurements should be especially accurate.




METHODS USED IN M AKIN G INDEX NUMBERS.

23

Such offhand comparisons as the above, however, are not fair,
and the conclusion they suggest as to the unreliability of index num­
bers can not be accepted without further study, for these various
index numbers mean different things. They do not all undertake
to measure the same quantity, hence they do not all employ the same
methods, and hence the discrepancies among their results may reveal
no real inconsistency. No valid comparison of index numbers can
be made, indeed, without a careful examination of what is measured
and how the measurement is made. Such an examination accord­
ingly we must make before we can satisfy our minds upon the question
whether index numbers yield trustworthy results.
IV.—VARIETIES OF METHODS USED IN MAKING INDEX
NUMBERS.
Making an index number involves several distinct operations: (1)
Defining the purpose for which the final results are to be used; (2) de­
ciding the numbers and kinds of commodities to be included; (3) de­
termining whether these commodities shall all be treated alike or
whether they shall be “ weighted ” according to their relative impor­
tance; (4) collecting the actual prices of the commodities chosen,
and, in case a weighted series is to be made, collecting also data
regarding their relative importance; (5) deciding whether the form
of the index number shall be one showing the average variations of
prices or the variations of a sum of actual prices; (6) in case average
variations are to be shown, choosing the base upon which relative
prices shall be computed; and (7) settling upon the form of average
to be struck, if averages are to be used.
A t each one of these successive steps choice must be mad# among
alternatives that range in number from two to thousands. The pos­
sible combinations among the alternatives chosen are indefinitely
numerous. Hence there is no assignable limit to the possible varie­
ties of index numbers, and in practice no two of the known series are
exactly alike in construction. To canvass even the important
variations of method actually in use is not a simple task.
1. TH E RELATIONS BETW EEN M E T H O D S AND USES.

The first step, framing a clear idea of the ultimate use of the results,
is most important, since it affords the clue to guide the compiler
through the labyrinth of subsequent choices. It is, however, the
step most frequently omitted.
Mr. C. M. Walsh and Prof. Irving Fisher, indeed, hold that “ an
index number is itself a purpose.” “ In averaging price variations,”
Mr. Walsh explains, “ the purpose or object is given: It is to measure
variations in the exchange value or purchasing power of money.”
Hence they logically contend that there is one “ best form of index
number.” 2 But this position is untenable. (1) As a statistical
4
device, index numbers have found a wide range of application outside
the field of prices. To deny the term index numbers to series which
show average variations in municipal water supply, rainfall, railroad
traffic, and the like conflicts with established usage. (2) Within the
field of prices index numbers are needed which do not aim to measure
2
4
See Walsh’ s The Problem of Estimation, p . 116, and Fisher’s “ Rejoinder” in Quarterly Publication
of the American Statistical Association, March ,1921, p . 547. T h e merits of the formula which they consider
“ the best?’ are discussed below, in section 9, p p . 91-93.




24

TH E M A K IN G AND USING OF INDEX NUMBERS.

the purchasing power of money. For example, some one should com­
pile a special series for forecasting changes in business conditions.
The compiler might select those commodities whose prices in the
past have given the earliest and most regular indications of changes
that subsequently occurred in the* larger index numbers, he might
weight these series in accordance with their past reliability as price
“ barometers,” and he might use whatever method of averaging the
fluctuations gave the best results for his purpose. Such a series
probably would not be a reliable measure o f variations in “ the pur­
chasing power of m oney/ ’ but it probably would be better adapted
to its special purpose than a series made by the formula which Prof.
Fisher and Mr. Walsh advocate as “ the best.” (3) To “ measure
variations in the exchange value or purchasing power of m oney” is
not a clearly defined aim. For example, in explaining his new form
of the British Board of Trade index number to the Royal Statistical
Society Prof. A. W. Flux pointed out that he might have aimed
either to find the change in the money cost of the things people buy,
or to find the net effect of the general economic situation, and espe­
cially of currency and credit, on prices. In discussing this paper
Prof. G. Udney Yule added a third aim, “ To find the effect of pricechanges on currency and credit.” 2 These three aims, which at first
5
sight seem much the same, turn out on closer scrutiny to differ and
to call for the use of dissimilar formulas, as Prof. Flux and Prof.
Yule argued. Nor is their list of aims in measuring the purchasing
power of money exhaustive. (4) What does “ the purchasing power
of m oney” include? Merely the standardized wares of the wholesale
markets which are sampled with varying thoroughness in the current
index numbers? Or does it include also commodities at retail,
stocks, bonds, labor of all sorts, farm lands and town lots, loans, trans­
portation, insurance, advertising space, and all the other classes of
oods that are bought and sold ? As Mr. W. T. Layton remarked in
iscussing Prof. Flux’s paper, “ The wholesale price index number is
not a measure of the general purchasing power of money, though all
the wholesale price index numbers are constantly quoted as such.” 2
6
In fine, the problem of measuring the purchasing power of money
has not yet been thoroughly explored. To insist that this problem
has but one meaning and therefore one “ best” solution obstructs
progress. It is wiser to exploit all the significant interpretations of
the problem and to consider what solution is appropriate to each.
And in addition to this general problem we should devise “ specialpurpose” index numbers to solve particular problems with a view to
learning all we can about the fluctuations of economic quantities,
physical as well as pecuniary. The making of index numbers is still
in the experimental stage, and it will progress by the differentiation
of many types of series, each with its clearly defined uses.
The most systematic plan of treating the subject, then, would be
to begin with the different uses of index numbers and to consider
the methods appropriate to each. But that plan can not be fol­
lowed in an interpretative study of the currently published series,
because most of the wholesale price index numbers are “ generalpurpose” series designed with no aim more definite than that of
“ measuring changes in the price level.” The only plan feasible

f

25 Journal of the R oyal Statistical Society, March, 1921, p p . 175-179 and 200.




2» Idem , p . 206.

METHODS USED IN M AKIN G INDEX NUMBERS.

25

for such a study at present.is to invert the problem. Instead of
studying methods in the light of uses, we must study uses in the
light of methods. That is, we must analyze the effect of the different
methods followed in practice and so determine what the resulting
figures mean and the uses to which they may properly be put.
The following discussion proceeds upon this plan. It deals prima­
rily with the popular general-purpose series and endeavors to show
how the various methods used in constructing these index numbers
determine the uses to which they are severally adapted.
2.

COLLECTING AND PUBLISHING T H E ORIGINAL QUOTATIONS.

The reliability of an index number obviously depends upon the
judgment and the accuracy with which the original price quotations
were collected. This field work is not only fundamental, it is also
laborious, expensive, and perplexing beyond any other part of the
whole investigation. Only those who have tried to gather from the
original sources quotations for many commodities over a long series
of years appreciate the difficulties besetting the task. The men who
deal with data already published are prone to regard all this prelimiwork as a clerical compilation requiring much industry but
skill. To judge from the literature about index numbers, one
would think that the difficult and important problems concern meth­
ods of weighting and averaging. JBut those who are practically
concerned with the whole process of making an index number from
start to finish rate this office work lightly in comparison with the
field work of getting the original data.
We commonly speak of the wholesale price of articles like pig iron,
cotton, or beef as if there were only one unambiguous price for
any one thing on a given day, however this price may vary from
one day to another. In fact there are many different prices for
every great staple on every day it is dealt in, and most of these differ­
ences are of the sort that tend to maintain themselves even when
markets are highly organized and competition is keen. Of course
varying grades command varying prices, and so as a rule do large
lots and small lots; for the same grade in the same quantities, differ­
ent prices are paid by the manufacturer, jobber, and local buyer; in
different localities the prices paid by these various dealers are not the
same; even in the same locality different dealers of the same class
do not all pay the same price to everyone from whom they buy the
same grade in the same.quantity on the same day. To find what
really was the price of cotton, for example, on February 1, 1920,
would require an elaborate investigation, and would result in show­
ing a multitude of different prices covering a considerable range.
Now the field worker collecting data for an index number must
select from among all these different prices for each of his commodi­
ties the one or the few series of quotations that make the most repre­
sentative sample of the whole. He must find the most reliable
source of information, the most representative market, the most
typical brands or grades, and the class of dealers who stand in the
most influential position. He must have sufficient technical knowl­
edge to be sure that his quotations are for uniform qualities, or to
make the necessary adjustments if changes in quality nave occurred
in the markets and require recognition in the statistical office. He

S




2 6

TH E M A K IN G AND USING OF INDEX NUMBERS.

must be able to recognize anything suspicious in the data offered
him and to get at the facts. He must know how commodities are
made and must seek comparable information concerning the prices of
raw materials and their manufactured products, concerning articles
that are substituted for one another, used in connection with one
another, or turned out as joint products of the same process. He
must guard against the pitfalls of cash discounts, premiums, rebates,
deferred payments, and allowances of all sorts. And he must know
whether his quotations for different articles are all on the same basis,
or whether concealed factors must be allowed for in comparing the
prices of different articles on a given date.
Difficult as it is to secure satisfactory price quotations, it is still
more difficult to secure satisfactory statistics concerning the relative
importance of the various commodities quoted. What is wanted
is an accurate census of the quantities of the important staples,
at least, that are annually produced, exchanged, or consumed. To
take such a census is altogether beyond the powrer of the private
investigators or even of the Government bureaus now engaged in
making index numbers. Hence the compilers are forced to confine
themselves for the most part to extracting such information as they
can from statistics already gathered by other hands and for other
purposes than theirs. •In the United States, for example, estimates
of production, consumption, or exchange come from most miscella­
neous sources: The Department of Agriculture, the Census Office,
the Treasury Department, the Bureau of Mines, the Geological
Survey, the Internal Revenue Office, the Mint, associations of manu­
facturers or dealers, trade papers, produce exchanges, traffic records
of canals and railways, etc. The man who assembles and compares
estimates made by these various organizations finds among them
many glaring discrepancies for which it is difficult to account. Such
conflict of evidence when two or more independent estimates of the
same quantity are available throws doubt also upon the seemingly
plausible figures coming from a single source for other articles. To
extract acceptable results from this mass of heterogeneous data
requires intimate familiarity with the statistical methods by which
they were made, endless patience, and critical judgment oi a high
order, not to speak of tactful diplomacy in dealing with the authori­
ties whose figures are questioned. The keenest investigator, after
long labor, can seldohci attain more than a rough approximation to
the facts. Yet it is only by critical use of the data now available
that current index numbers can be weighted, and the best hope of
improving weights in the future lies in demonstrating not only the
imperfections of our present statistics of production, consumption,
and exchange, but also the importance of making them better.
When all this preliminary work has been done, the original quota­
tions and the weights should be published at length. Unfortunately,
many compilers of index numbers publish only the final results of
their computations, upon the ground of expense or lack of interest
in the detailed information. But much is sacrificed by taking this
easy course. First, the reputation of the index number itself is
compromised, and deservedly. No one can really test whether a
series is accurately compiled from representative quotations unless
the data and their sources are given in full. Second, and more
important, the publication of actual quotations greatly extends the




METHODS USED IN M A K IN G INDEX NUM BERS.

27

usefulness of an investigation into prices. Men with quite other
ends in view than those of the original compilers can make index
numbers of their own adapted to their peculiar purposes if provided
with the original data.
Nor is the importance of such unplanned uses to be rated lightly.
If we are ever to make the money economy under which we live
highly efficient in promoting social welfare we must learn how to
control its workings. What vrares our business enterprises produce
and what goods our families consume are largely determined by
existing prices, and the production and consumption of goods are
altered by every price fluctuation. What we waste and what we
save, how we divide the burden of labor and how we distribute its
rewards, whether business enjoys prosperity or suffers depression,
whether debts of long standing become easier or harder to pay---all
these and many other issues turn in no small measure upon what
things are cheap and what are dear, upon the maintenance of a due
balance within the system of prices, upon the upward or downward
trend of the price changes that are always taking place. But if the
prices of yesterday are powerful factors in determining what we
shall do and how we shall fare to-day, what we do and how we fare
to-day are powerful factors in determining what prices shall be
to-morrow. If prices control us, we also control them. To control
them so that they shall react favorably upon our economic fortunes
we need more insight than we have at present. It is, then, one of
the great tasks of the future to master the complicated system of
prices which we have gradually developed— to find how prices are
interconnected, how and why they change, and what consequences
each change entails. For when men have learned these things they
will be vastly more skillful in mending what they find amiss in
economic life, and in reenforcing what they find good. As yet our
knowledge is fragmentary and uncertain. But of all the efforts
being made to extend it none is more certain to prove fruitful than
the effort to record the actual prices at which large numbers of com­
modities are bought and sold. For such data are the materials with
which all investigators must deal, and without which no bits o f
insight can be tested. Indeed, it is probable that long after the best
index numbers we can make to-day have been superseded, the data
from which they were compiled will be among the sources from
which men will be extracting knowledge which we do not know
enough to find.
3.

M A R K ET PRICES, CONTRACT PRICES, IN STITU TIO N PRICES, AND
IM P O R T -E XP O R T VALUES.

Most American index numbers are made from “ market prices.”
These prices are usually obtained directly from manufacturers,
selling agents, or wholesale merchants; from the records of produce
exchanges and the like; or from trade journals and newspapers
which make a specialty of market reporting in their respective
fields.
Several of the important foreign index numbers are made wholly
or partly from 4 import and export values” ; that is, from the average
‘
prices of important articles o f merchandise as officially declared by
the importing or exporting firms, or as determined by governmental
commissions. For example, Soetbeer's celebrated German series,




28

THE

M A K IN G

AND

U S IN G

OF IN D E X

N UM BERS.

and (until 1921) the British Board of Trade’s official series were made
maiidy from such material, and the official French series was made
wholly from import values until 1911.
A fourth source of quotations often drawn upon in Europe is the
prices paid for supplies by such institutions as hospitals, normal
schools, poorhoufees, army posts, and the like. The official Italian
series, Alberti’s series for Trieste, and Levasseur’s French series are
examples.
These four classes of quotations— market prices, contract prices,
import and export values, and institution prices— usually differ some­
what, not only with respect to the prices prevailing on a given date,
but also with respect to the degree of change from time to time.
Accordingly it is desirable to inquire into the several advantages
possessed by each source of quotations.
Institution prices may be set aside promptly, because index num­
bers made from them have a limited range of usefulness. Though the
institutions vrhose records are drawn upon often make purchases on
a considerable scale, yet the common description of their contract
rates as u semiwholesale ” prices points to the peculiar and there­
fore unrepresentative character of such data. Moreover, there is
often more doubt about the strictly uniform character of the supplies
furnished to these institutions than about the uniformity of the
standardized goods which are usually quoted in the market reports.
If the aim of the investigation is to find the average variations m the
cost of supplies to public institutions, doubtless the prices they pay
are the best data to use. But if the aim is to measure the average
variations in the wholesale prices paid by the business world at
large, then market and contract prices are distinctly the better source.
Indeed, institution prices are seldom used for the latter purpose
except when well-authenticated market quotations can not be had.
So far as is known, the series of index numbers compiled by the
Price Section of the War Industries Board for 1913-1918 is the only
series in which free use has been made of contract prices, and even in
this series contract prices were not obtained for some important articles
handled largely on the contract basis— especially pig iron. Contract
prices, indeed, seem more difficult to ascertain than open-market
prices, and they are really less appropriate data than the latter when
the purpose is primarily to ascertain in what direction prices are tend­
ing from one month to the next. But when it is desired to show the
fluctuations in the prices at which the bulk of business is carried on,
it is clear that the index numbers should be made from both contract
and open-market prices and that the two sets of quotations should be
weighted in accordance with the volume of transactions which each
set represents. In the long run there may be little difference between
the fluctuations in the contract and the open-market prices for the
same commodity; but within short periods the difference is sometimes
wide. In 1915-1918, for example, contract prices taade at the begin­
ning of a year were often far below the level attained by open-market
end of
The
on a
{>rices by theto whichthe year.of their collection of contract pricesprices
arger scale and the analysis
relation with open-market
are matters
the makers of index numbers may profitably
direct greater attention.2
7
2 The best presentation of contract and open-market prices yet m ade is in T he Prices of Coal and Coke,
7
b y Carl E. Lesher, W ar Industries Board Price Bulletin, N o. 35.




M E T H O D S U SED I N

M A K IN G

IN D E X N U M B E R S .

29

The theory on which import and export values are sometimes pre­
ferred to market prices is that the former figures show more nearly
the variations in the prices actually paid or received by a country for
the great staples which it buys and sells than do market quotations
for particular brands or grades of these commodities. For example,
England buys several different kinds of cotton in proportions that
vary from year to year. A price obtained by dividing the total de­
clared values of all the cotton consignments imported by their total
weight will show the average cost per pound actually paid by Eng­
lishmen for cotton with more certainty than will Liverpool market
quotations fora single grade of cotton like u Middling American” —
provided always that the udeclared values’ 7 are trustworthy. Now,
if the aim of the investigation is to find out the variations in the
average prices paid or received for staples— irrespective of minor
changes in their qualities— then the preference for import and export
values is clearly justified, again granted the trustworthiness of the
returns. But if the aim is to measure just one thing— the average
variation in prices— market prices for uniform grades are clearly bet­
ter data. For index numbers made from import and export values
measure the net resultant of two sets of changes, and one can not tell
from the published figures what part of the fluctuations is due to
changes in prices and what part is due to changes in the qualities of
the goods bought and sold.

As might be expected, import and export series generally pursue
a more even course than market-price series. But this difference
may be due less to the sources from which the quotations are obtained
than to differences in the lists of commodities used. Fortunately, we
can arrange a more certain test than any of the common series pro­
vide. In 1903 the British Board of Trade published the average
import or export prices of 25 commodities for which Mr. Sauerbeck
has published market prices.2 Index numbers made from these two
8

2
3

Wholesale and Retail Prices. Return to an Order of the . . . H ouse of Commons . . . for “ Report
on Wholesale and Retail Prices in the United Kingdom in 1902, with Comparative Statistical Tables for a
Series of Y ears.'/ For Sauerbeck’ s figures see his annual articles in the Journal of the R oyal Statistical
Society. The list of commodities in question is as follows:

Comm odity.

B acon..........................
B arley..........................
Coal...............................
C o ffe e ..........................
Copper..........................
C otton ..........................
F la x ..............................
H id es............................
Iron , p ig .......................
Jute...............................
L e a d ............................
L inseed........................
M aize............................
O ats..............................
Oil, olive......................
Oil, palm ...................
Petroleum ................. .
Rice ..........................
Silk
..........................
Sugar, refined.............
T e a ...............................
T in ................................
W h ea t..........................
W ool.............................
D o ..............................




Quotations given b y
Board of Trade.

Brands quoted b y Sauerbeck.

Average im port values
........d o ......................... ..
Average export values.
Average im port values
........d o ........................... .
........d o .............................
........d o .............................
........d o .............................
Average export values.
Average im port values
........d o .............................
........d o .............................
........d o .............................
........d o .............................
....... d o .............................
........d o .............................
........d o .............................
........d o .............................
........d o .............................
........d o .............................
........d o .............................
........d o .............................
........d o .............................
........d o .............................
Average export values.

W aterford.
English Gazette.
W allsend, H etton, in X o n d o n .
R io, good channel.
Chile bars.
Middling A merican.
St. Petersburg.
R iver P lata, dry.
Scotch pig.
Goo<^ m edium .
English pig.
Linseed.
American m ixed.
English Gazette.
Olive oil.
Palm oil.
Petroleum , refined.
Rangoon, cargoes to arrive.
Tsatlee.
Java, floating cargoes.
Congou, com m on.
Straits.
English Gazette.
Merino, Adelaide, average grease.
English, Lincoln, half hogs.

30

THE

M A K IN G A N D

U S IN G OF IN D E X N U M B E R S .

Chart 4 .—I N D E X N U M B E R S M A D E FR O M T H E M A R K E T P R IC E S A N D F R O M T H E IM P O R T
A N D E X P O R T V A L U E S O F ID E N T IC A L LISTS OF C O M M O D ITIE S. E N G L A N D , 1871-1902.
(Based on Table 5 .)1

* This and the succeeding charts h ave been drawn on a logarithmic, instead of an arithmetic, scale in
order that the per cent of change m ay easily be discerned.




M E T H O D S U SED I N

31

M A K I N G IN D E X N U M B E B S .

sets of data for the same commodities for the years 1§71 to 1902 are
given in Table 5. The results confirm the expectation: As compared
with the import and export index number, the market-price index
number starts on a higher level in 1871, falls to a lower point dur­
ing the middle nineties, rises to a higher level in 1900, and again
drops to as low a level in 1902. But the differences are not wide.
T a b l e 5 . — CO M PA R ISO N

O F IN D E X N U M B E R S M A D E F R O M IM P O R T A N D E X P O R T
V A L U E S W IT H I N D E X N U M B E R S M A D E FR O M T H E M A R K E T P R IC E S O F T H E
SAM E COM M O D ITIE S, B Y Y E A R S , 1871 TO 1902.
[Data from the British Board of Trade and from Sauerbeck.]
(A rithm etic means o f relative prices.
Im pott
ana ex­
port
values.

Year.

1871................................................
1872................................................
1873................................................
1874................................................
1875
...................
1870...............................................
1877................................................
1878............. k................................
1 8 7 9 ...,........................................
1880................................................
1881................................................
1882................................................
1833
___ J
1884................................................
1S85................................................
1880................................................

158
169
170
102
152
149
150
139
128
130
133
129
125
118
110
105

A verage prices in 1890-1899=100.

Market
prices.

17t)
185
182
168
155
152
152
138
13i
137
130
125
123
116
112
107

25 com m od ities.)

Year.

Im port
and ex­
port
values.

1887................................................
1888................................................
1889................................................
1890................................................
1891................................................
1892................................................
1893................................................
1894................................................
1895................................................
1896................................................
1897................................................
1898................................................
1899................................................
1900................................................
1901................................................
1902................................................

104
108
108
109
111
105
103
95
93
94
93
95
101
114
107
104

Market
prices.

107
110

no

111
111
103
104
94
94
93
91
95
105
117
106
104

4. RELATIVE VERSUS ACTUAL PRICES.

In February,- 1864, Hunt’s Merchants’ Magazine published the fol­
lowing statement to show how rapidly prices rose after the suspension
of specie payments in December, 1861, and the issue of the irredeem­
able United States notes.2 These figures are the total prices of 55
9
articles quoted by their customary commercial units.
Value of 55- leading articles o f New York commerce.
January, 1862.............................................................................................
April, 1862 ...................................................................................................
January, 1 8 6 3 ..........................................................................................
March, 1863.................................................................................................
July, 1863.............................................................. ......................................
October, 1863..............................................................................................
January, 1864.............................................................................................

$804
844
1,312
1, 524
1,324
1,455
1, 693

For example, in January, 1862, coal, oil is entered as 30 cents per
gallon and pig iron as $24 per ton; molasses is entered as 42^ cents
per gallon and whalebone as $69 per ton ; oats is entered as 38 cents
per bushel and corn as $59.25 per hundred bushels, etc.3
0
Clearly, this simple method of measuring changes in the price level
by casting sums of actual prices is not trustworthy. For a relatively
slight fall in the quotation for whalebone would affect the total, as
Hunt’s Merchants’ Magazine computes it, much more than a rela­
tively enormous increase in the price of molasses. The fact that com




™ V ol. 50, p. 132.

33 See vol. 48, p. 129.

32

THE

M A K IN G

A N D U S IN G OF IN D E X N U M B E R S .

happens to be quoted by the hundred bushels makes a 1 per cent
change from its price in January, 1862, equal to a 43 per cent change in
the price of wheat and to a 156 per cent change in the price of oats,
both of which are quoted by the bushel.
It was to avoid such patent absurdities that Carli threw his actual
prices of grain, wine, and olives in 1750 into the form of percentages
of rise or fall from their prices in 1500, and then struck the average
of the three percentages. When this operation is performed it makes
no difference whether the commodities are quoted by large or by
small units. The obvious common sense of this precedent has caused
it to be followed or reinvented by most makers of index numbers to
this day— with one slight modification. To avoid the awkwardness
of the plus and minus signs necessary to indicate whether prices have
advanced or receded, it is usual to substitute for percentages of rise
or fall relative prices on the scale of 100. For example, a rise of 10
per cent and a fall of 10 per cent are expressed by relatives of 110
and 90, respectively. Occasionally, however, percentages of rise or
fall are still used as by Carli; as, for instance, in the chain relatives
published b y the Bureau of Labor Statistics in Bulletin No. 149 and
averaged in Table 4 of this bulletin. A second unimportant variant,
long practiced by the London Economist, but now seldom used,
is to publish as the final result the sums of relative prices, instead
of their averages.3
1
In recent years a few statisticians have gone back from the use of
relative to the use of actual prices, adopting various devices to avoid
such crude errors as those perpetrated in the figures cited from Hunt’s
Merchants’ Magazine. In 1897 Bradstreet’s began reducing all its
original quotations by the gallon, ton, dozen, square yard, etc., to
prices by the pound, and presenting as its index number the aggregate
prices per pound of 98 articles.3 Four years later, Dun’s Review
2
followed this lead with an important difference. Instead of reducing
actual quotations to quotations by the pound, it multiplied the actual
quotation for each article included by the quantity of that article sup­
posed to be consumed in the course of a year by the average indi­
vidual. These products were then cast up, and the sums, in dollars
and cents, were presented as an index number purporting to show
the changes in the per capita cost of a year’s supplies.3
3
Still later (1912), the method practiced by Dun was adopted by
the Commonwealth statistician of Australia as the basis of his official
series. However, after he had calculated the aggregate expenditure
of Australians upon his bill' of goods in terms of pounds sterling, he
threw these pecuniary sums back into the form of relative numbers
on the scale of 1,000. In 1914 the United States Bureau of Labor
Statistics dropped its former practice of averaging relative prices on
the 1890-1899 base, and began to use aggregates of actual prices,
weighted by quantities entering into exchange and thrown into the
form of relatives to facilitate comparison.

Accordingly, three types of index numbers are now in general use:
(1) Averages of relative prices or average percentages of change in
3 Gibson’s index number is such a sum. See pp. 172 to 175. The difference between sums of relative
1
prices and these sums divided b y the number of articles included is, of course, purely formal. Averages
have displaced sums in current use m ainly because it is easier to make comparisons on the scale of 100
than on the scale of 2,200, or whatever number is given b y the addition o f relative prices.
-32 For a criticism of this m ethod, see p . 110.
3 The confidence merited b y this index num ber is discussed in Section V .
3




M E T H O D S USED I N

M A K IN G

IN D E X N U M B E R S .

33

prices; (2) sums in dollars and cents showing changes in the aggre­
gate cost of certain definite quantities of certain commodities; (3)
relative figures made from series of the second sort. The first type
shows average variations, the second type shows the variations oi an
aggregate, the third type turns these variations of an aggregate into
percentages of the aggregate itself as it stood at some selected time.
The differences between these types, it is true, are differences of
form, not differences of kind. As will later be shown, by using
a certain scheme of weights an aggregate of actual prices can be
made to give precisely the same results when turned into relatives
that will be given by an average of relative prices computed from
the same data. But it will also be shown that the differences of form
are important. The advantages and shortcomings of the several
types will appear as the various problems encountered in making
index numbers are discussed.
5. TH E NUM BERS AND KIN D S OF C O M M O D ITIE S INCLUDED.

Since the earlier makers of index numbers had to use such price
quotations as they could find, the problems how many and what
kinds of commodities to include were practically solved for them.
As Prof. Edgeworth remarks, u Beggars can not be choosers/7
Paucity o f data still hampers contemporary efforts to measure
variations of prices in the past; but the compilers of index numbers
for current years have a wider range of choice. The scope of their
data is limited not by the impossibility but by the expense of col­
lecting quotations. And in the case of governmental bureaus or
financial journals the limits set by expense are neither narrow nor
rigid. Such organizations can choose many commodities if they will
or content themselves with few.
One principle of choice is generally recognized. Those commodities
are preferable that are substantially uniform from market to market
and from year to year. Often the form of quotation makes all the
difference between a substantially uniform and a highly variable com­
modity. For example, prices of cattle and hogs are more significant
than prices of horses and mules, because the prices of cattle and hogs
are quoted per pound, while the prices of horses and mules are quoted
per head.
It is often argued that the application of this common-sense prin­
ciple rules out almost all manufactured goods, because such articles
are continually altered in quality to suit the technical exigencies
of new industrial processes or the varying tastes of consumers. But
minor changes in quality, provided their occurrence is known, do not
necessarily unfit a commodity for inclusion. When the brand for­
merly sold is replaced by a variant it is usually possible to get over­
lapping quotations for the old and new qualities during the time of
transition. Then the new series may be spliced upon the old by
means of the ratio borne by the price of the new grade to the price of
the old grade in the years when the substitution is made. Statis­
ticians'willing to take the extra precautions and trouble involved by
such operations can legitimately include not only a large number of
staple raw materials and their simplest products, but also an even
larger number of manufactured goods.
+311739 0 —41------3




34

THE

M A K IN G

A N D U S IN G OF IN D E X

N U M BERS.

Some of the modem index numbers, accordingly, have long lists of
commodities. Dun’s index number seems to be built up from about
300 series of quotations, the official Canadian index number includes
271, the Bureau of Labor Statistics’ index number for 1919 has 328,
and the index number compiled by the Price Section of the War
Industries Board has 1,366 price series. On the other hand, many
of the best-known index numbers use less than 50 series pf quota­
tions. Forty-five is a favorite number, largely because of the high
reputation early established by Sauerbeck’s English series. The
British Board of Trade’s series to 1921, the official French series, the
New Zealand series, Von Jankovich’s Austrian series, and Atkinson’s
series for British India all have just 45 commodities, while the new
series of the London Economist and the relative prices published by
the former Imperial Statistical Office of Germany include 44 articles.
Even shorter lists are often used. For example, Schmitz’s German
series has only 29 commodities, the New York Annalist series 25,
and Gibson’s series 22. Private investigators working with limited
resources sometimes confine themselves to a bare dozen commodi­
ties, or even less.3
4
These differences of practice raise important questions of theory.
Does it make any substantial difference in the results whether 25
or 50 or 250 commodities be included—^provided always that the
lists be well chosen in the three cases ? If differences do appear in
the results, are they merely haphazar d, or are they significant differ­
ences? If there are significant differences, which set of results is
more valuable, that made from the long or from the short lists?
And what does the proviso that the lists be well chosen mean ? *In
short, do the index numbers including hundreds of commodities pos­
sess advantages over those including 50 or 25 sufficient to compen­
sate for the greater trouble and expense of compiling them?
The best way to answer these questions is to experiment with large
and small index numbers, made on a strictly uniform plan for the
same country and the same years. Table 6 presents six such index
numbers which differ only in respect to the number and kind of
commodities included. The first column includes all the commod­
ities quoted by the Bureau of Labor Statistics in 1913 except the 11
whose prices do not run back of. 1908.3 Many of the commodities
5
in this list are merely different varieties of the same article; for exam­
ple, there are two kinds of corn meal, four kinds of leather, six kinds of
women’s dress goods, eleven kinds of steel tools, etc. The second
column gives an index number in which all such groups are repre­
sented by single averages, so that the number of series which enter
directly into the final results is cut down to 145.3 The third column,
6
which includes 50 commodities, is made up from the list adopted for
3 These statements refer to the number of series of relative prices averaged to get the final results as m m
4
presented. Often tw o or more different varieties of an important article are counted as separate com ­
modities, and, on the other hand, the relative prices of slightly different articles are sometimes averaged
to make one of the series which enters into the final averages. In view of the diversity of practice in this*
respect, a perfectly consistent counting of the num ber of distinct “ com m odities” included in the general
series is impossible. Moreover, the figures are often published with such imperfect explanations as to
make the counting of the com m odities included doubtful or impossible on any interpretation of that term.
In 1921 the num ber of price series used m the British Board o f Trade ind ex was increased to 150.
3 T o facilitate comparison, decimals have been dropped and the index for each year rounded off to the
5
nearest whole number. Regarding the changes in the number of commodities included, see Bulletin
No. 149, p. 11. T h e reader m ay foe rem inded once m ore that this is the Bureau’s ©M index number, m ade
before th6 improved m ethod of com pilation was introduced.
3 This*experimental list of 145 com m odities is given below. W hen the relative prices of closely related
6
articles are averaged to make a single series, the number of these articles quoted b y the Bureau and
included in the group is indicated. Most of the bureau’s series which d o not cover the whole period,




METHODS USED IN M A K IN G INDEX NUM BERS.

35

the Gibson index number in its original form.3 The fourth series is
7
made from the prices of 20 pairs, each commodity being given in two
forms, raw and manufactured, e. g., barley and malt, cattle and beef,
copper Ingots and copper wire, etc.3 The last two columns contain
8
1890-1913, are dropped altogether. As the basis of a general-purpose index number, this revised list is
worse than the bureau’s list in certain respects and better in others. See Section V .

Barley.
Cattle, 2.
Corn.
Cotton.
Flaxseed.
H ay.
Hides.
H ogs, 2.
H ops.
Oats.
R ye.
Sheep, 2.
W heat.
FUEL A N D L IG H T IN G .

1.
2.
3.
4.
5.
6.
7.

Candles.
Coal, anthracite, 4.
Coal, bitu m inou s,3.
Coke.
Matches.
Petroleum , erude.
Petroleum , refined, 2.
FOOD, ETC.

1.
2.
3.
4.
5.
6.
7.
8.

A pples, evaporated
Beans.
Bread, craekers, 2.
Bread, loaf, 3.
B utter, 3.
Cheese.
Coffee.
Currants.

10. Fish, 4.

11. Flour, buckwheat.
12. Flour, rye.
13.
14.
15.
16.
17.
18.
19.

Flour, wheat.
Lard.
Meal, corn, 2.
Meat, beef, 3.
Meat, pork, 4.
Meat, m utton.
M ilk.

21.
22.
23.
24.
25.

Onions.
Potatoes.
Prunes.
Raisins.
R ice.

20. Molasses.

26. Salt.
27. Soda.

28.
29.
30.
31.
32.
33.

Spice, pepper.
Starch, corn.
Sugar, 3.
Tallow .
Tea.
Vinegar.

LU M BE R AND BUILDING
M ATERIALS.

CLOTHS AND CLOTHING.

FARM PRODUCTS.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.

Bags.
B lankets, 3.
B oots and shoes,3.
B roadcloths.
C alico.
Carpets, 3.
C otton flannels, 2.
C otton thread.
Cott on yarns, 2.
D enim s.
D rillings, 2.
Flannels.
Ginghams, 2.
Horse blankets.
H ose.
Leather, 4.
L in en thread.
Overcoatings, 2 .
P rint cloths.
Sheetings, 7.
Shirtings, 5.
S ilk, 2.
Suitings.
Tickings.
Underwear, 2.
W om en ’s dress goods, 6*
W o o l, 2.
W orsted yarns, 2.
METALS AND IM P LEM ENTS.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

Bar iron, 2.
Barb w ife.
B uilders’ hardware, 3.
Copper, ingot.
Copper, wire.
L ead, pig.
Lead pipe.
N ails, 2 .
Pig iron , 4.
Quicksilver.
Silver.
Spelter.
Steel billets.
Steel rails.
T in , pig.
T ools, 11.
W ood screws.
Zinc.
DRUGS AN D

1.
2.
3.
4.
5.
6.
7.

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.

B rick.
Carbonate of lead.
Cement.
Doors.
H em lock.
L im e.
Linseed oil.
Maple.
Oak, 2.
O xide o f zinc.
Pine, white, 2.
Pine, yellow.
Plate glass, 2.
Poplar.
P utty.
R osin.
Shingles, 2.
Spruce.
Tar.
Turpentine.
W indow 'glass, 2.
HOUSE-FURNISHING GOODS.

1.
2.
3.
4.
5.

Ear then w*are, 3.
Furniture, 4.
Glassware, 3
T able cutlery, 2.
Woodenwrare, 2.

1.
2.
3.
4.
5.
6.
7.
8.
9.
ID.
11.

Cottonseed meal.
Cottonseed oil*
Jute.
M alt.
Paper, 2.
P roof spirit s.
R ope.
R ubber.
Soap.
Starch, iaundiy
T obacco, 2.

M ISCELLANEOUS.

CHEMICALS*

A leoh ol, grain.
A lcoh ol, wood.
A lu m .
Brim stone.
Glycerine.
M uriatic acid.
O pium .

8. Quinine.

9. Sulphuric acid.

3
7

The list is as follows: W heat, wheat flour (tw'o kinds), barley, oats, corn, corn meal, potatoes, rye,
sugar 89°, sugar 96°, coffee, tea, steers, fresh beef, salt beef, sheep, m utton, hogs, bacon, hams, butter,
cotton, cotton yarns (tw o kinds), jute, wrool (tw o kinds), worsted yarns, raw silk (tw o kinds), pig iron,
bar iron, cement, copper ingots, copper sheets, lead, anthracite coal, bituminous coal (tw o kinds), hides,
leather, cottonseed oil, linseed oil, petroleum (crude and refined), rubber, spruce lumber, yellow-pine
lumber, and paper. See J. P. Norton, “ A revised index number for measuring the rise in prices,”
Quarterly Journal of Economics, August. 1910, vol. 24, pp. 750-758.
3 The remaining 17 pairs are corn ancl corn meal, cotton and cotton textiles, flaxseed and linseed oil,
3
window glass and glassware, hides and leather, hogs and pork, lead (pig) and lead pipe, m ilk and cheese,
petroleum (crude and refined), pig iron and nails, pine boards and pine doors, rye and rye flour, sheep
and m utton, spelter and zinc, steel billets and steel tools, wheat and wheat flour, wool and woolen textiles.




36

THE M AKING AND USING OF INDEX NUMBERS,

index numbers each made from the prices of 25 important articles
selected at random* the two lists having no items in common.3
9
T able 6 .— S IX IN D E X N U M B E R S F O R TH E U N IT E D S TA T E S M A D E F R O M Q U O T A T IO N S
F O R D IF F E R E N T N U M B E R S O F C OM M ODITIE S, B Y Y E A R S , 1890 TO 1913.
[Data from the Bulletin of the Bureau of Labor Statistics, No. 149.]
(A rith m etic means.

Year.

A verage prices in 1890-1899= 100.)

242 to 261
25 com ­ 25 com ­
145 com ­ 50 com ­
40 com ­
com m od­ modities. modities. modities. modities, modities,
second
ities.
first list.
list.

1890......................................................................
1891......................................................................
1892......................................................................
1893......................................................................
1894......................................................................
1895......................................................................
1896......................................................................
1897......................................................................
1898......................................................................
1899......................................................................
1900......................................................................
1901......................................................................
1902......................................................................
1903......................................................................
1904.....................................................................
1905......................................................................
1906......................................................................
1907......................................................................
1908......................................................................
1909......................................................................
1910......................................................................
1911......................................................................
1912......................................................................
1913......................................................................

113
112
106
106
9?
<
94
90
90
93
102
111
109
113
114
113
116
123
130
122
125
130
126
130
130

114
113
106
105
96
93
89
89
93
103
111
110
114
114
114
116
122
130
121
124
131
130
134
131

Averages 1890-1899...........................................
1900-1909..........................................
1910-1913..........................................
Number of points b y which prices rose ( + )
or fell ( - ) i n 1890-1896.........................................................
1896-1907.........................................................
1907-1908.........................................................
1908-1912.........................................................

100
118
129

Difference between highest and lowest rel­
ative prices....................................................
Average change from year to year.............

3
9

114
114
105
105
94
94
87
89
95
103
112
109
116
115
116
118
123
132
125
132
135
129
138
138

113
114
105
101
93
95
88
89
95
108
115
116
122
118
118
122
128
138
129
135
141
135
142
139

115
112
103
103
92
95
88
90
96
107
113
111
116
11*
122
123
130
132
124
133
133
129
140
142

113
118
112
107
96
93
85
84
90
103
109
107
117
117
110
115
122
132
122
128
134
131
138
133

100
118
132

100
120
135

100
124
139

100
122
136

100
118
134

-2 3
+40
- 8
+ 8

-2 5
+41
- 9
+13

-2 7
+45
- 7
+ 13

-2 5
+50
- 9
+13

-2 7
+44
- 8
+ 16

-2 8
+47
-1 0
+16

40

45

51

54

54

4.0

4.1

,

4.9

5.5

54
6.2

5.0

The first list includes cotton, corn, wheat, hides, cattle, hogs, coffee, wheat flour, salt, sugar, tea,
potatoes, wool, silk, anthracite coal, bituminous coal, crude petroleum, pig iron, steel billets, copper ingots,
lead (pig), brick, average of nine kinds of lumber, jute, and rubber.
The second list includes hay, oats, rye, eggs, sheep, lard, beans, corn meal, butter, rice, milk, prunes,
cotton yarns, worsted yarns, coke, cement (Rosendale 1890-1899, Portland domestic 1900-1913), tallow,
spelter, bar iron, tin (pig), quicksilver, lime, tar, paper, proof spirits.




37

METHODS USED IN M AKING INDEX NUMBERS.

N um ber o f p oin t8 by which the selected index num bers were greater ( + ) or less (—) than the
Bureau o f Labor Statistics* series.
25 com ­ 25 com ­
40 com ­ modities, modities,
145 com ­ 50 com ­
second
modities. modities. modities. first list.
list.

Year.

±
+
+
+
+

4- 1
+ 1
± o
- 1
± o
- 1
- 1
- 1
± 0
+ 1
db o
+ 1
+ 1
± 0
+ 1

4- 4
4- i

4- 3
+ 1
4- 3
4- 2
± 0
4- 2
4- 3
4- 7
+ 5
+ 3
+ 8
4- 9

Arithmetic sums .................................................................................
Algebraic sum s ....................................................................................
Average differences com puted from the—
Arithmetic sums ........................................................................
Algebraic sum s.............................................................

23
4- 9

60
+44

Maximum differences..
M in im um differences. . .

.........................................................
....................................................

±

±
+

±

o

1
0
1
1
1

4- 4

+
+
+

4- 4

+

8

±

db o

o

97
f 73

+

2.5
1.8

1.0
.4

5
6
5
8
7
10
11
9
12
8

2
0
3
3
4
1
2
0
3
5
+ 2
+ 2
+ 3
+ 4
+ 9
+ 7
+ 7
+ 2
+ 2
+ 8
4- 3
+ 3
+ 10
+ 12

+
+
+
+
+
+
+
+
+
+

o

+

4-

0
2
1
5
3
1
2
1
2
6

129
+105

4- 1
+ 2
- 1
- 1
- 2
± o
- 3
- 1
4- 2
4- 1
4- 1

1890......................................................................................... \
1891.........................................................................................
1892.........................................................................................
1893.........................................................................................
1894.........................................................................................
1895.........................................................................................
1896 ................
.....................................................
1897.........................................................................................
1898.........................................................................................
1899.........................................................................................
1900.........................................................................................
1901.............................................................................................................
1902.........................................................................................
1903.............................................................................................................
1904....................... ...............................................................................
1305.............................................................................................................
1906.........................................................................................
1907.............................................................................................................
1908.............................................................................................................
1909............................................................................................................
1910.............................................................................................................
1911 .......
.................................................................
1912.........................................................................................
1913.............................................................................................................

4

7
9

4

5.4
4.4

+
+
±

+
±
+
±
+
+

12

+

+
+
+
+
±
+
+
+
-

4-

±
+

4+
+
+

±

2

4
3
3
1
1
2
0
3
4
5
8
3

70
+22

4.0
3.0

+12

o

0
6
6
1
0
1
5
6
3
1
2

+
1

\

o

+
±

2.9
.9
8
o

N um ber o f p oin ts by which each index num ber rose (4 ) or fell ( —) in each successive year.

Year.

25 com ­
25 com ­
242 to 261
145 com ­ 50 com ­
40 com ­
com m od­ modities. modities. modities. modities, modities,
second
first list.
ities.
list.

1891......................................................................
1892......................................................................
1893......................................................................
1894......................................................................
1895................................... ..................................
1896......................................................................
1897......................................................................
1898..................................................................
1899......................................................................
1900......................................................................
1901......................................................................
1902......................................................................
1903......................................................................
1904......................................................................
1905......................................................................
1906...................................................................... i
1907...................................................................... !
1908............................................................................................ I
1909.............................................................................................
1910............................................................................................ !
1911............................................................................................. 1
1912............................................................................................ !
1913............................................................................................




1

—
—
±
+
+
+

1

4

— 7
— 1
— 9
— 3
— 4

0

± 0

— 6
± 0

10
2

± 0
9

± 0
-11
± 0

3
9
9

+ 4
+ 10
+ 8

—
+
+
+
+

—

—

+ 4
+ 1

+ 4

—

—

2
1

+ 3
+ 7
+ 7

8

+ 3
4- 5
4
+ 4
rb 0

db

±
+
+
+

1

0
0
2
6
8

— 9

+ 3
+ 7
—

1

+ 4
3
“

7

2
6
8
9
3
7

1
+ 1
+ 2
—

4- 5
+ 9
— 7
+ 7
+ 3
— 6
+ 9
± 0

+
+
+
+
+
+
+
+
±
+
+
+

1
9
4
8
2
7
1
6
13
7
1,
6
4
0
4
6
10

-

9

+ 6
+ 6
6
+ 7
3

_ 3
— 9

± 0
-11
+ 3
— 7
+ 2
+ 6
+ 11
+ 6
—

+
+
+
+
+
+
—

2
5

2
4

1
7

2
8

+ 9
± 0
— 4
+ 11
+ 2

+ 5
— 6
— 5

-11

— 3
— 8
— 1
+ 6
+ 13
+ 6
—

2

+ 10
± 0
— 7
+ 5
+ 7
+ 10
_1_

10
6

+ 6
— 3
+ 7
5

38

THE MAKING AND USING OF INDEX NUMBERS.

Now, these six index numbers, large and small, certainly hare a
strong family likeness. The great movements of American prices
from 1890 to 1913 stand out boldly in them all— the heavy fall of
prices in 1890-1896, the distinctly greater rise^ in 1896-1907, the
sharp decline in 1908, the recovery in 1909, and the wavering course
Ch

a rt

5 .— G E N E R A L -P U R P O S E I N D E X N U M B E R S, IN C L U D IN G 25, 50, A N D 242 COMM OD­
IT IE S , B Y Y E A R S , 1890 TO 1913.

in 1910-1913. If index numbers could pretend to nothing more
than to show roughly the trend of price fluctuations, then it would
indeed matter little which of these series were used. Either of the
sets including only 25 commodities would serve that limited purpose
as well as the set containing nearly ten times as many commodities,
though doubtless the longer lists would command more confidence.




METHODS USED IN MAKING INDEX NUMBERS.

39

But the very success with which index numbers, even when made
from scanty and dissimilar data, bring out the broader features of
price movements encourages one to hope, from this device, for more
than an indication of the direction and a rough approximation to the
degree of change. Instead of concluding that an easy compilation,
based on a few series of quotations “ will do,” we may hope that
careful work covering a wide field will enable us to improve upon our
first results and attain measurements that have a narrow margin of
error.

When we make these more exacting demands upon our six index
numbers we attach importance to the fact that their general similarity
does not preclude numerous differences of detail. For example, two
series indicate that prices rose in 1891, one indicates that prices did
not change, and three indicate a fall; three put the lowest point in
1896, one in 1897, and two make the price level the same m these
years; one series shows a rise in 1901, five show a fall; in 1913 again
one series indicates a rise of prices, three indicate a fall, and two indi­
cate no change; the general level of prices in the final year is made
to vary between an average rise of 30 per cent and one of 42 per cent
above the level of 1890-1899; there is also a difference in steadiness,
the small series fluctuating through a wider range than the large
ones, etc.
To what are these discrepancies due? Are they discreditable to
the large series, or to the small ones, or to neither set ? Can they be
accounted for except as the results of random differences in sampling ?
If an index number made from the wholesale prices of 25, or 50,
or 250 commodities can measure approximately the changes in all
wholesale prices, it must be because the known fluctuations in the
prices of these selected commodities are fair samples of the unknown
fluctuations in the prices of the vastly larger number of other com­
modities for which quotations are not collected. Now if (1) the
price fluctuations of each commodity that is bought and sold were
r
strictly independent of the price fluctuations of every other com­
modity, and if (2) each commodity had just the same importance as
an element in the general system of prices as every other commodity,
then any series of price quotations collected at random would be a
fair sample for determining the average changes in the wholesale
prices of commodities in general. Of course, the larger the number
of commodities included, the more trustworthy would be the index
number. In Table 6, for example, the first index number would be
adjudged the best, and the divergencies between it and its fellows
would be held to result from the scantier material from which the
latter are made.
In fact, howr
ever, the situation is by no means so simple, because
neither of the above-mentioned conditions holds true. Commodities
are far from being all of the same importance as elements in the whole
system of prices. With the complications arising from this fact the
section on the problems of weighting will deal. Neither are the price
fluctuations of different commodities independent of each other. On
the contrary, the price changes of practically every commodity in the
markets of the whole country are causally related to the changes in
the prices of a few or of many, perhaps in the last resort of all, other
commodities that are bought and sold. Most of these relations are




40

THE

M A K IN G

AND

U S IN G

OF IN D E X

NUM BEKS.

so slight that they can not be traced by statistical methods. But
certain bonds are so close and so strong that they establish definite
groups of related prices which fluctuate m harmony with one another
and which differ in definable ways from the fluctuations of other such
groups. The present task is to show the existence of these groups
and the effects which they exercise upon index numbers.
First, the price fluctuations of a raw material are usually reflected
in the prices of its manufactured products. Hence to quote in some
cases both the raw material and several of its finished products, and
to quote in other cases the raw material alone, assigns certain groups
of related prices a larger influence upon the results than is assigned
the other groups. When the aim is to secure a set of samples which
fairly represent price fluctuations as a whole, the existence of these
groups must be taken into account. Neglect on this score may give
a misleading twist to the final index numbers. A celebrated case in
point is that of the Economist index number in 1863-1865. Out of
the 22 commodities included in the Economist’s list as then consti­
tuted 4 consisted of cotton and its products. Hence when the
blockade of Southern ports during the Civil War raised the price of
cotton, the Economist index numbers grossly exaggerated the aver­
age rise in the price level, as appears from the following comparison
between the Economist’s results for 1860-1865 and the corresponding
English figures compiled by Sauerbeck: 4
0

Year.

I860....................................................................................................................................
1861....................................................................................................................................
1862....................................................................................................................................
1863....................................................................................................................................
1864....................................................................................................................................
1865....................................................................................................................................

Econom ist
Sauerbeck’s
index number index number
(prices in
(prices in
1860-100).
I860-100).
100
102
109
136
145
136

100
100
106
109
112
106

Directly opposing the relations which unite the prices of finished
goods with the prices of their raw materials is a second set of influences
which make the price fluctuations of manufactured goods considered
as a group characteristically different from the price fluctuations
of their raw materials considered as a separate group. Table 7
presents several sets of index numbers designed to throw these
characteristic differences into high relief. The first two columns
compare the relative prices of the 49 raw materials quoted by the
Bureau of Labor Statistics in 1913 and of the 183 to 193 more or less
manufactured commodities in its list.4 The second pair of columns
1
contains index numbers made from the prices of 20 raw materials and
of 20 products manufactured from these same materials.4 Then
2
*0 T o make the comparison as fair as possible, both series are here given, not in their original form, but
recomputed on a com m on basis. See Wholesale Prices, Wages, and Transportation, report b y Mr. Aldrich
from the Committe on Finance, Mar. 3, 1893, 52d Cong., 2d sess., Senate R eport No. 1394, Part I, p p. 226
and 255.
See Bulletin No. 149, pp. 13 and 14. The differences between the original figures and those given here
are due (1) to the dropping of decimals, (2) to the exclusion of 11 commodities which the Bureau of Labor
Statistics quotes in the years 1908-1913 only, (3) to the com putation of the arithmetic means in these years
b y the method applied in 1890-1907 in place of the Bureau’s roundabout method.
4 The articles included here are those from which the index number of 40 commodities in Table 6 was
2
made. For the list, see p. 35 and note.




M ETHODS

USED

IN

M A K IN G

TNDEX

NUM BERS.

41

come three columns giving index numbers made from the prices of
five great staples at three successive stages of manufacture: Wheat,
flour, and bread; cotton, cotton yarns, and cotton textiles; wool,
worsted yarns, and woolen textiles; pig iron, steel billets, and steel
tools; hides, leather, and shoes.4 The later sections of the table give
3
the data for each of these last-mentioned groups separately. These
several comparisons establish the conclusion that manufactured
goods were steadier in price than raw materials. The manufactured
goods fell less in 1890-1896, rose less in 1896-1907, again fell less in
1907-1908, and rose less in 1908-1913. Further, the manufactured
goods had the narrower extreme range of fluctuations, the smaller
average change from year to year, ana the slighter advance in price
from one decade to the next.4 It follows that index numbers made
4
from the prices of raw materials, or of raw materials and slightly
manufactured products, must be expected to show wider oscillations
than index numbers including a liberal representation of finished
commodities.
« For the list of'textiles and of tools, see Bulletin No. 99 of the Bureau of Labor, March, 1912, pp. 554-556
and 682-683.
44 Like most generalizations about price changes, these statements are strictly valid only in the case of
averages covering several commodities, but the exceptions are not numerous, even in the case of single
com m odities, as detailed study of the wheat, cotton, wool, iron, and leather groups will show.




42

TH E M A K IN G AND USING OF INDEX NUMBEBS.
T able 7 .—I N D E X N U M B E R S M ADE F R O M T H E P R IC E S OF R A W M A T E R IA L S
[Data from Bulletin No. 149 of Ike
(Arithmetic means.

Year.

Tw enty
pairs.
183 to
49
193
raw m an­
m a­ ufac­
Man­
teri­ tured R aw ufac­
m a­
als. prod­ teri­ tured
ucts. als. goods.

Five triplets.

Average

W heat group.

Inter­
R aw m edi­ F in­
ma­
W heat
ished
teri­ ate goods. Wheat. flour. Bread.
als. prod­
ucts.
1

2

2

1890..........................................................
1891..........................................................
1892
..................................
1893..........................................................
1894..........................................................
1895..........................................................
1896..........................................................
1897..........................................................
1898..........................................................
1899..........................................................
1900......................................... ................
1901..........................................................
1902..........................................................
1903...........................„ .............................
1904..........................................................
1905..........................................................
1906..........................................................
1907..........................................................
1908..........................................................
1909..........................................................
1910.........................................................
1911..........................................................
1912..........................................................
1913..........................................................

115
116
108
104
93
92
84
88
94
106
112
111
122
123
120
121
127
133
124
131
135
135
145
139

112
111
106
106
97
94
92
90
93
101
110
108
111
112
111
115
122
129
121
123
129
124
127
128

113
114
104
99
91
94
85
88
98
114
118
120
127
122
123
127
135
146
135
143
149
144
151
149

112
114
105
103
94
96
92
89
92
103
111
113
118
114
113
117
120
131
124
127
132
127
132
128

125
117
103
95
79
89
87
94
101
111
120
110
123
125
128
132
136
145
130
149
149
135
141
143

119
116
109
100
86
89
88
90
95
107
110
102
110
114
115
115
119
126
117
126
125
115
119
122

108
107
106
105
98
95
95
94
95
98
105
102
103
106
110
114
121
125
120
121
124
120
124
127

119
128
105
90
74
80
85
106
L18
95
94
96
99
105
138
135
106
121
132
160
146
131
140
127

121
126
104
89
78
84
91
110
109
88
88
87
90
97
125
122
97
109
119
139
126
112
122
109

101
101
101
101
101
98
97
101
101
101
101
101
101
101
106
110
110
110
113
116
118
118
122
123

Averages, 1890-1899..............................
1900-1909..............................
1910-1913..............................
N um ber of points b y which prices
rose ( + ) or fell ( —) in—

100
122
139

100
!16
127

100
130
148

100
119
130

100
130
142

100
115
120

100
113
124

100
119
136

100
107
117

100
107
120

1890-1896.........................................
1896-1907.........................................
1907-1908.........................................
1908-1913.........................................
Difference between highest and
lowest relative prices.

-3 1
+ 49
- 9
+ 15
61

-2 0
+37
- 8
+ 7
39

-2 8
+61
-1 1
+ 14
66

-2 0
+39
- 7
+ 4
43

-3 8
+ 58
-1 5
+ 13
70

-3 1
+38
- 9
+ 5
40

-1 3
+ 30
- 5
+ 7
33

-3 4
+36
+ 11
- 5

86

-3 0
+ 18
+ 10
-1 0
61

Average change from year to y e a r...

5.5

4.0

6.4

4.9

8.4

5.5

3.1

13.6

11.6




+
+
+

4
13
3
10
26

1.3

43

METHODS USED IN M A K IN G INDEX NUMBERS,
A N D OF M A N U F A C T U R E D GOODS* B Y Y E A R S , 1890 TO 1913.
Bureau of Labor Statistics.]
prices in 1890-1899=100.)

Cotton group.

Iron group.

W ool group.

Leather group.

W ool­
Steel
Raw Cot­ Cot­ Raw W orst­
en
Steel
ton
Pig
Leather. Shoes.
ton
ed
cot­
tex­ iron. bil­ tools. Hides.
tex­ wool.
lets.
yarn.
ton. yams. tiles.
tiles.

16

4

1

11

1

111
112
112
109,
96
88
87
90
98
100
111
105
106
111
112
119
125
124
121
122
124
120
123
123

131
116
106
96
83
91
88
78
77
134
140
112
155
141
104
124
145
175
125
127
124
112
118
122

142
118
110
95
77
86
88
70
71
145
116
112
142
130
103
112
128
136
122
114
118
100
104
120

107
106
105
103
99
95
96
95
94
101
112
110
115
118
118
128
134
138
134
129
131
123
124
126

100
102
93
80
68
110
87
106
123
132
127
132
143
125
124
153
165
155
143
176
165
158
188
196

101
101
97
97
92
108
95
96
104
109
113
111
113
112
109
112
120
124
119
127
125
121
129
139

106
104
103
101
99
100
101
96
94
95
98
96
96
96
98
106
119
120
114
121
118
116
127
137

100 ; 100
122
135
119
111

100
124
126

100
144
177

100
116
129

100
106
125

-4 3
+ 87
-5 0
- 3
98

-5 4
+ 48
-1 4
- 2
75

-1 1
+42
- 4
- 8
44

-1 3
+68
-1 2
+53
128

- 6
+29
- 5
+ 20
47

- 5
+ 19
- 6
+23
43

17.5

16.0

3.7

14.7

5.0

3.7

24

2

143
111
99
107
90
94
102
92
77
85
124
111
115
145
156
123
142
153
135
156
195
168
148
165

112
113
117
111
93
. 92
93
91
91
89
116
98
94
113
120
106
121
134
109
119
133
125
120
132

117
112
111
109
98
94
95
90
85
91
103
90
100
105
114
107
117
133
116
117
127
125
122
126

132
126
113
102
79
70
71
89
108
111
118
97
101
110
116
127
121
122
118
127
116
108
111
105

122
123
117
110
91
74
73
83
101
107
118
102
112
118
117
125
129
128
118
130
124
116
119
113

100
136
169

100
113
128

100
111
125

10Q
116
110

100 ! 100
120
116
118
123

-4 1
+ 51
-1 8
+30
118

+ 41
-2 5
+23
45

-2 2
+ 38
-1 7
+ 10
48

-6 1
+51
- 4
-1 3
62

-4 9
+55
-1 0
- 5
57

-2 4
+37
- 3
+ 2
38

18.1

9.8

6.1

9.1

8.1

3.9

1

2

-r e




2

4

3

Year.

Number of com ­
modities included.
1890.
1891.
1892.
1893.
1894.
1895.
1896.
1897.
1898.
1899.
1900
1901.
1902.
1903.
1904.
1905.
1906.
1907.
1908.
1909.
1910.
1911.
1912.
1913.
Averages, 1890-1899.
1900-1909.
1910-1913.
Number of points
b y which prices
rose ( + ) or fell
( —) in—
1890-1896.
1896-1907.
1907-1908.
1908-1913.
Difference between
highest and low­
est relative prices.
Average
change
from year to year.

44

THE

M A K IN G

AND

U S IN G

OF

IN D E X

NUM BERS.

Third, there are characteristic differences among the price fluctua­
tions of the groups consisting of mineral products, forest products,
animal products, and farm crops. Table 8 presents index numbers
for these four groups. Fifty-seven commodities are \ncluded, all of
them raw materials or slightly manufactured products.4 Here the
5
Ch

a kt

6 .—IN D E X N U M B E R S OF T H E PR ICE S OF 20 R A W M A T E R IA L S A N D 20 P R O D U C T S
M A N U F A C T U R E D FR O M TH E M .
(Based on Table 7.)

4 The lists of com m odities are as follows:
r>
Farm crops': Cotton, flaxseed, barley, corn, oats, rye, wheat, hay, hops, beans, coffee, rice, pepper, tea,
onions, potatoes, cottonseed meal, and jute—18 articles.
A n im a l products: Hides, cattle, hogs, sheep, eggs, lard, m ilk, tallow, silk, and wool— 10 articles.
F o u s t products: Hem lock, maple, oak, white pine, yellow pine, poplar and spruce lumber, together with
turpentine, tar, and rubber—10 articles.
M ineral products: Salt, anthracite coal, bituminous coal, coke, crude petroleum, copperingois, lead (pig),
pig iron, bar iron, steel billets, quicksilver, silver bars, tin (pig), spelter, zinc, brick, cement, lime, and brim­
stone—19 articles.




M ETHODS

USED

IN

M A K IN G

IN D E X

NUM BERS.

45

striking feature is the capricious behavior of the prices of farm crops
under the influence of good and bad harvests. The sudden upward
jump in their prices in 1891, despite the depressed condition of busi­
ness, their advance in the dull year 1904, their fall in the year of
revival 1905, their failure to advance in the midst of the prosperity of
a r t 7 . — I N D E X N U M B E R S OF T H E PR ICE S OF W O O L , CO TTO N , H ID E S, W H E A T , A N D
P IG IR O N IN T H E IR R A W , P A R T I A L L Y M A N U F A C T U R E D , A N D F IN ISH E D FO R M S.

Ch

(Based on Table 7.)

1906, their trifling decline during the great depression of 1908, and
their sharp rise in the face of reaction in 1911 are all opposed to the
Cneral trend of other prices. The prices of animal products are
istinctly less affected by weather than the prices of vegetable crops,
but even they behave queerly at times, for example in 1893. Forestproduct prices* are notable chiefly for maintaining a much higher

f




46

THE M AK IN G AND USING OF INDEX NUMBERS.

level of fluctuation in 1902-1913 than any of the other groups, a level
on which their fluctuations, when computed as percentages of the
much lower prices of 1890-1899, appear extremely violent. Finally,
the prices o f minerals accord better with alternations of prosperity,
crisis, and depression than any of the other groups. And the anom­
alies that do appear— the slight rise in three years (1896, 1903,
and 1913) when the tide of business was receding—would be removed
if the figures were compiled by months. For the trend of mineral
prices was downward in these years, but the fall was not so rapid as
the rise had been in the preceding years, so that the annual aver­
ages were left somewhat higher than before.4 An index number
6
composed largely of quotations for annual crops, then, would be
expected at irregular intervals to contradict capriciously the evidence
of index numbers in which most of the articles were mineral, forest, or
even animal products.
T able 8 .—I N D E X

N U M B E R S M A D E F R O M P R IC E S OF M IN E R A L , F O R E S T , A N IM A L ,
A N D F A R M P R O D U C T S , B Y Y E A R S , 1890 TO 1913.

[Data from the Bulletin of the Bureau of Labor Statistics, No. 1-19.]
(Arithmetic means.

Average prices in 1890-1899=100.)

Year.

Mineral ! Forest
Animal
products. ! products. products.

Farm
crops.

N umber of com m odities in clu d ed ......................................................

19

10

10

18

1890.............................................................................................................
1891.............................................................................................................
1892............................... ............................................................................
1893.......................................................................................................
1894.............................................................................................................
1895.............................................................................................................
1896 .........................................................................................................
1897............................................................................................................
1898........... ..................... ..........................................................................
1899.............................................................................................................
1900............................................................................................................
1901.............................................................................................................
1902.............................................................................................................
1903.................................................................. .........................................
1904..................................... ......................................................................
1905.............................................................................................................
1906.............................................................................................................
1907.............................................................................................................
................................... ...........................................................
1908
1909.............................................................................................................
1910.............................................................................................................
1911.............................................................................................................
1912.............................................................................................................
1913.............................................................................................................

119
111
105
98
87
91
92
88
92
117
120
113
119
124
115
123
135
137
118
121
120
120
132
136

107
105
99
98
95
96
94
95
99
112
121
113
123
137
142
149
163
169
151
164
181
172
168
169

106
108
109
116
94
95
82
88
97
105
111
112
128
117
113
121
128
135
126
144
152
131
146
150

119
126
110
105
101
92
76
83
92
96
105
114
120
116
124
116
116
125
124
130
134
151
158
135

Averages, 1890-1899.................................................................................
1900-1909.................................................................................
1910-1913................................................................................
N um ber of points b y which prices rose ( + ) or fell ( —) in—
1890-1896............................................................................................
1896-1907.................... ......................................................................
1907-1908............................................................................................
1908-1913............................................................................................

100
123
127

100
143
173

100
124
145

100
119
145

-2 7
+45
-1 9
+ 18

-1 3
+75

-1 8
+ 18

-2 4
+53
- 9
+ 24

-4 3
+49
- 1
+ 11

Difference between highest and lowest relative prices...................

50

87

70

82

Average change from year to yea r.............................................. .......

7 .0

7.4

8.9

8.2

Fourth, there are characteristic differences between the price fluc­
tuations of manufactured commodities bought by consumers for
family use and the price fluctuations of manufactured commodities
bought by business men for industrial or commercial use. Such at
Compare the m onthly figures com piled b y the Bureau of Labor Statistics for its group of “ Metals and
im plem ents,” Bulletin No. 149, p. 18. These figures are largely influenced b y the relatively stable prices
of 11 different, kinds of tools. M onthly data for the 19 mineral products of Table 8 would probably show
even more decline between January and December in these years.




METHODS USED IN M A K IN G INDEX NUM BEKS.

47

least is the story told by Table 9. The data employed here are
quotations for 28 articles from the Bureau of Labor Statistics'list
tnat rank distinctly as consumers’ goods and 28 that rank as pro­
ducers’ goods.’* Though consisting more largely of the erratically
7
fluctuating farm products, the consumers' goods are steadier in
Ch

a rt

8 . — IN D E X

N U M B E R S O F T H E PR ICE S O F 19 M IN E R A D P R O D U C T S A N D
FA R M CRO PS.

O F 18

(Based on Table 8.)

4 The consumers* goods are bread, crackers, butter, cheese, salt fish, evaporated apples, prunes, raisins,
7
beef, m utton , pork, molasses, cornstarch, sugar, vinegar, shoes, cotton textiles, woolen textiles, candles,
matches, quinine, furniture, earthenware, glassware, woodenware, table cutlery, soap, and tobacco. The
producers’ goods are hags, cotton yam s, leather, linen shoe thread, worsted yarns, refined petroleum,
barbed wire, builders’ hardware, copper wire, lead pipe, nails, steel rails, tools, wooa screws, p ine doors,
plate glass, w indow glass, carbonate of lead, oxide of seine, putty, rosin, shingles, m uriatic acid, sulphuric
acid, m alt, paper, proof spirit, and rope.
11 will be noticed that a large proportion of the consumers’ goods are subject to very .slight manufacturing
processes, notably the foods. Hence the difference between the tw o index numbers can scarcely lie re
garded as merely a fresh contrast between the fluctuations of finished goods and of intermediate products-




48

THE

M A K IN G

AND

U S IN G

OF

IN D E X

NUM BERS.

price than the producers' goods, because the demand for them is less
influenced by changes in business conditions.
T able 9 .—IN D E X N U M B E R S M A D E FR O M T H E P R IC E S OF C ON SU M ERS 7 GOODS A N D
OF P R O D U C E R S 7 GOODS, B Y Y E A R S , 1890 TO 1918.
[Data from Bulletin of the Bureau of Labor Statistics, N o. 149.]
(A rith m etic means.

A verage prices in 1890-1899=100.)

Year.

1890...................................................................................................................................................
1891...................................................................................................................................................
1892...................................................................................................................................................
1893......................................................................................................................................
1894...................................................................................................................................................
1895 ...........................................................................................................................................
1896..................................................................................................................................................
1S97...................................................................................................................................................
1898...................................................................................................................................................
1899...................................................................................................................................................
1900...................................................................................................................................................
1901...................................................................................................................................................
1902...................................................................................................................................................
1903...................................................................................................................................................
1904...................................................................................................................................................
1905...................................................................................................................................................
1908...................................................................................................................................................
1907...................................................................................................................................................
1908.............................................. ....................................................................................................
1909...................................................................................................................................................
1911...... ..........................................................................................................................................
1912..................................................................................................................................................
1913................................................................ .................................................................................1
Averages, 1890-1899.......................................................................................................................
1900-1909.......................................................................................................................
1910-1913.................................................................................................... .................
Number of points b v which prices rose ( + ) or fell ( —) in—
1890-1897......... ........................................................................................................................
1897-1907..................................................................................................................................
1907-1908....................................................................................................................... ..........
1908-1913..................................................................................................................................
Difference between highest and lowest relative prices.........................................................
Average change from year to yea r............................................................................................

Consum­
ers 7
goods.

Produc­
ers7
goods.

112

115

109
104
108

111

100

107
102

121

92
91
93
89
93
107
117
113
114
114
114
117
124
133
119
118
126
125
125
123

100

100

108
119

118
125

—22
+24

+44

95
91
90
94
98
106
105
108
105
103
106
110

114
112

114
118
119
118

—26

— 2
+ 9

—14
+ 4

31

44

3.4

4.7

Other groups of related prices having specific peculiarities of fluc­
tuation doubtless exist,4 but the analysis has been carried far enough
8
for the present purpose. That purpose is to show how the existence
of groups of prices which fluctuate in harmony with each other and
at variance with other groups affects index numbers in general and
in particular the six index numbers for the United States given in
Table 6. To apply the knowledge gained from the preceding analysis
to the explanation of the differences among these six index numbers
is not difficult when once the commodities included in each index
number have been classified on the basis of the groups which have
been examined.
First, the list of commodities used by the Bureau of Labor Sta­
tistics includes 29 quotations for iron a n d its products, 30 quotations
for cotton and its products, and 18 for w'ool and its products, besides
8 more quotations for fabrics made of wool and cotton together. On
the other hand, it has but 7 series for wheat and its products, 8 for
coal and its products, 3 for copper and its products, etc. The iron,
For example, there is evidence that the products of industries characterized b y a large measure of
concentration in business control are steadier in price than products of industries characterized b y un­
hampered com petition.—See W . C, Mitchell, Business Cycles, pp. 402-404.




METHODS USED IN M AK IN G INDEX NUMBERS.

49

cotton and wool groups together make up 85 series out of 242, or
35 per cent of the whole number. The same three groups furnish 36
(or 25 per cent) of the 145 series in the second index number in Table 8.
CH AR T

9 .—

I N D E X N U M B E R S OF T H E PR ICE S OF M A N U F A C T U R E D GOODS U SED F O R
F A M IL Y CON SU M PTION A N D F O R IN D U S T R IA L P U R P O SE S .
(Based on Table 9.)

Does this large representation of three staples distort these index
numbers— particularly the bureau’s series where the disproportion is
greatest? Perhaps, but if so the distortion does not arise chiefly
from the undue influence assigned to the price fluctuations of raw
cotton, raw wool, and pig iron. For, contrary to the prevailing
impression, the similarity between the price fluctuations of finished
products and their raw materials is less than the similarity between
t311739 0 —41------ 4




50

TH E M A K IN G AND USING OF INDEX NUMBERS.

the price fluctuations of finished products made from different mate­
rials. Such at least is the testimony of Table 7. As babies from
different families are more like one another than they are like their
respective parents, so here the relative prices of cotton textiles,
woolen textiles, steel tools, bread, and shoes differ far less among
themselves than they differ severally from the relative prices of raw
cotton, raw wool, pig iron, wheat, and hides.4 Hence the inclusion
9
of a large number of articles made from iron, cotton, and wool affects
an index number mainly by increasing the representation allotted
to manufactured goods. What materials those manufactured goods
are made from makes less difference in the index number than the
fact that they are manufactured. To replace iron, cotton, and
woolen products by copper, linen, and rubber products would change
the result somewhat, but a much greater change would come from
replacing the manufactured forms of iron, cotton, and wool by new
varieties of their raw forms.5
0
This similarity among the price fluctuations of manufactured goods
arises from the fact demonstrated by Table 7 that such articles are
relatively steady in price. Does knowledge of this steadiness assist
in explaining the differences among the six American index numbers
of Table 6? To answer we must find the proportions of raw and
manufactured commodities included in each index number. Classi­
fication along this line is rather uncertain in many cases, but the
results shown in the following schedule, if not strictly correct, are at
least uniform in their errors.
T able 1 0 .—N U M B E R A N D P E R C E N T OF R A W A N D M A N U F A C T U R E D C O M M O D ITIE S
IN C L U D E D IN T H E S IX I N D E X N U M B E R S O F T A B L E 6.
Number of—
Index number.

F irst.......................................................................................
Second....................................................................................
T h ird ......................................................................................
Fourth...................................................................................
F ifth .......................................................................................
Sixth.......................................................................................

Percentage of—

Total
number
Manu­
Manu­
Raw
of com ­
Raw
modities. com m od­ factured com m od­ factured
com m od­
com m od­
ities.
ities.
ities.
ities.
242
145
50
40
25
25

49
36
26
17
19
10

193
109
24
23
6
15

20
25 ,
52 :
43
76
40

80
75
48
57
24
60

4 A com pilation of the differences among the relative prices in question taken seriatim for each of the 24
9
years 1890-1913 yields the following results:
Average differences between the relative prices of—
R aw cotton and cotton textiles.................................................................................. 20.7 points.
R aw w ool and woolen textiles..................................................................................... 8.9 points.
Pig iron and steel to o ls ................................................................................................. 14.0 points.
W heat and bread............................................................................................................ 15.0 points.
H ides and shoes................................................................................................................ 31.6 points.
Average.................................................................................................................. 18.0 points.
Cotton textiles and woolen textiles...........................................................................
Cotton textiles and steel to o ls .....................................................................................
Cotton textiles and bread.............................................................................................
C otton textiles and shoes..............................................................................................
W o d e n textiles and steel to o ls ...................................................................................
W oolen textiles and bread...........................................................................................
W oolen textiles and shoes............................................................................................
Steel tools and b r e a d ....................................................................................................
Steel tools and shoes......................................................................................................
Bread and shoes..............................................................................................................

5.3
7.8
6.9
6.7
6,1
7.3
8.1
9.4
9.6
4.7

points.
points.
points.
points.
points.
points.
points.
points.
points.
points

Average.................................................................................................................. 7.2 points.
50 W hile the fluctuations in the prices of manufactured goods are generally slighter than those in the
prices of raw materials, they are nevertheless violent at times, as in the case of cotton yarns and cotton
textiles during the Civil W ar. ( See p . 40.)




51

METHODS USED IN M A K IN G INDEX NUMBERS.

On this showing the Bureau of Labor Statistics series ought to be
the steadiest, and the second series the next steadiest— and so they are,
as the summaries at the bottom of the columns in Table 6 show.
With the smaller index numbers, however, the rule does not work
well, for the most variable of all— the sixth—has a larger per cent of
manufactured goods than the other three. Moreover, number four
is more variable than number three, though it has relatively more
manufactured goods. But the preceding studies of different groups
throw further light upon the matter.
It has been Found that among manufactured commodities those
bought for family consumption are steadier in price than those bought
for business use. To take account of this factor the manufactured
goods in the several series are classified as primarily consumers ’
goods, primarily producers ' goods, or as bought in large measure by
both classes of purchasers.
T able 1 1 .—C LA S S IF IC A T IO N OF T H E M A N U F A C T U R E D CO M M O D ITIE S IN C L U D E D IN
T H E S IX I N D E X N U M B E R S O F T A B L E 6.

Number of—

Index number.

First..............................
Second..........................
T h ird ............................
Fourth..........................
F ifth.............................
S ix th ...........................

Manu­
factured
articles.

193
109
24
23
6
15

Per cent of—

B oth con­
Con­
Pro­
sumers’
Manu­
sumers’
ducers’ and pro­ factured
com ­
com ­
ducers’
articles.
modities. modities.
com ­
modities.
108
51
11
10
3
4

73
47
12
12
3
11

12
11
1
1

80
75
48
57
24
60

Both con­
Con­
Pro­
sumers’
sumers’
ducers’ and pro­
com ­
com ­
ducers'
modities. modities.
com ­
modities.
45
35
22
25
12
16

30
32
24
30
12
44

5
8
2
2

Here it does turn out that the two series (numbers four and six)
which are highly variable despite the inclusion of many manufactured
goods have relatively more of those manufactured goods which as
a group are most variable. So far as this factor counts, then, it
counts toward clearing up the contradiction pointed out in the
preceding paragraph. It also brings out a further reason for the
comparative stability of the first two series.
The one remaining form of analysis suggested above seems-easy
to apply. In the schedule below, raw and slightly manufactured
commodities like those used in Table 8 are distributed among four
groups according as their constituents come chiefly from mines,
forests, animal sources, or cultivated fields. There is little doubt
about the classification here, but there is much doubt about the
significance of the results as applied to our six index numbers. The
figures in the schedule are either such small percentages of the whole
number of series that they can not exercise much influence upon the
results, or such small numbers that they can not claim to be typical
of their groups. Further, the second part of the schedule shows that
there is less difference among the six index numbers than appears at
first sight in the proportions of the raw and slightly manufactured
commodities which consist of mineral, forest, animal, and farm prod­
ucts. Hence it is not surprising that efforts to account for the
divergences in Table 6 by appealing to this schedule and to Table 8




52

THE

M A K IN G

AND

U S IN G

OF IN D E X

NUM BERS.

accomplish little, especially for the smaller index numbers. This
much does appear regarding the first two series: Whenever mineral
products and farm crops move sharply in opposite directions the
Bureau of Labor Statistics’ index diverges from its mate in harmony
with mineral products, while the series of 145 commodities bends
toward the agricultural products— which is what should happen
according to the schedule.
T a b l e 1 2 . —F A R M ,

A N IM A L , F O R E S T , A N D M IN E R A L P R O D U C T S IN R A W O R S L IG H T L Y
M A N U F A C T U R E D FO R M , IN C L U D E D IN T H E S IX I N D E X N U M B E R S OF T A B L E 6.

Per cent of the whole number consist­
ing of—

Number of—

Index
number.

F irst............
Second........
T h ird ..........
F ou rth ........
F ifth ...........
Sixth...........

Total
num ­
ber
Raw
and
of
com ­ slightly
m odi­ manu­
ties.
fac­
tured
goods.
242
145
50
40
25
25

74
57
30
19
23
18

Farm
crops.

A ni­
mal
prod­
ucts.

Forest
prod­
ucts.

18
18
10
6
7
5

15
10
8
6
5
5

12
10
3
1
2
1

Raw
and
Min­
slightly
eral
manu­
prod­
fac­
ucts.
tured
goods.

Farm
crops.

A ni­
mal
prod­
ucts.

Forest
prod­
ucts.

30
39
60
48
92
72

7
12
20
15
28
20

6
7
16
15
20
20

5
7
6
3
8
4

29
19
9
6
9
7

Min­
eral
prod­
ucts.

12
13
18
15
36
28

Per cent of the raw and slightly
manufactured com m odities consistmg of—
Index number.
Farm
crops.

Second.

................................................................................................

Fourth
..............................................................................................
Fifth..................................................................................................................

25
31
33
32
30
28

Animal
prod­
ucts.

Forest
prod­
ucts.

20
18
27
32
22
28

16
18
10
4
9
5

Mineral
prod­
ucts.
39
33
30
32
39
89

Two practical conclusions of moment to both the makers and the
users of index numbers are established by this section. (1) To make
an index number that measures the changes in wholesale prices at
large, samples must be drawn from all the various groups that behave
in peculiar ways. (2) In using an index number made by others, one
must study the list of commodities included critically with these
groups in mind to know what it really does measure.
The first conclusion seems to contradict a rule often practiced and
sometimes preached. Most of the middle-sized index numbers are
confined to raw materials and slightly manufactured goods. Most
of the small index numbers are confined to foods alone. The makers
of both sets argue that their series are more “ sensitive” and therefore
better measures of price changes than the larger series, which are
“ loaded down” wuth a mass of miscellaneous manufactured goods.
And many users of index numbers seem to prefer a series like Sauer­
beck’s with only 45 commodities, or even one like the Annalist’s
with only 25 commodities, to one like that of the Bureau of Labor
Statistics with five or ten times the number.




M E T H O D S U SED I N

M A K I N G IN D E X N U M B E R S .

53

Critics who take this stand usually assume tacitly that the purpose
of an index number is to serve as a “ business barometer,7 or to
7
measure changes in “ the cost of living.7 If these aims were always
7
clearly realized by the critics and clearly stated for their readers the
room left for differences of opinion would be narrow. In Table 6
the index number with 145 commodities shows itself a more sensi­
tive and on the whole more faithful barometer of changing business
conditions during the 24-year period from 1890 to 1913 than the
official series with 242 commodities,5 and the preceding analysis shows
1
that the sluggishness of the larger index number is due chiefly to its
proportion of manufactured goods. For this particular purpose, then,
a series modeled after Sauerbeck7 has strong claims to preference
s
over one including a larger number of commodities. Indeed, in the
light of the preceding discussion one might carry the process of exclu­
sion much further and throw out of the business barometer not only
manufactured goods but also all farm crops, on the ground that their
prices depend on the eccentricities of the weather, and most forest
products, on the ground that their prices in the period covered by
Table 6 were rising so fast as to obscure the effects of bad times, etc.
But clearly such exclusions, while they might make the resulting fig­
ures more responsive to changes in business conditions, would also
make the figures less acceptable as a measure of changes in prices as a
whole. The sluggish movements of manufactured goods and of con­
sumers7 commodities in particular, the capricious jumping of farm
products, etc., are all part and parcel of the fluctuations which the
price level is actually undergoing. Consequently, an index number
which pretends to measure changes in the general level of prices can
not logically reject authentic quotations from any of these groups.
Every restriction in the scope of the data implies a limitation in the
significance of the results.
As for the small series made from the prices of foods alone or from
the prices of any single group of commodities, it is clear that however
good for special uses they may be, the}7 are untrustworthy as generalpurpose index numbers. Table 13 shows what differences are likely
to appear at any time between series confined to foods and series
covering a wider field. The general-purpose indexes are taken from
Table 6, two of the food indexes include the commodities quoted by
the Annalist index number and by the Gibson index number as now
constituted; the third food index is the bureau’s old series for foods,
with decimals dropped and new arithmetic means for 1908-1913.
si Compare p. 3G.




54
T

a ble

THE

M A K IN G AN D

U S IN G OF IN D E X N U M B E R S .

1 3 .—I N D E X N U M B E R S OF T H E PR IC E S OF FOO DS, A N D G E N E R A L -P U R P O S E
I N D E X N U M B E R S, B Y Y E A R S , 1890 T O 1913.
(Data from Bulletin of the Bureau of Labor Statistics, N o. 149.J
(A rith m etic means.

A verage prices in 1890-1899=100.)

General-purpose index
number from Table 6.

Index numbers of the prices of
foods.
25 com ­
22 com ­
m odities,
modities,
Annalist, j Gibson
list.
list.

Year.
242 to 261
com ­
modities.

1890.......................................................................
1891.......................................................................
1892.......................................................................
1893.......................................................................
1894........ ... ..........................................................
1895.......................................................................
1898.....................................................................
1897.......... ...........................................................
1898.......................................................................
1899......................................................................
1900.......................................................................
1901.......................................................................
1902.......................................................................
1903.......................................................................
1904 ..................................................................
1905.......... ............................................................
1906.......................................................................
1907 ....................................................................
1908.....................................................................
1909........................................................*............
1910.......................................................................
1911.......................................................................
1912.......................................................................
1913.......................................................................

113
112
106
106
96
94
90
90
93
102
111
109
113
114
113
116
123
130
122
125
130
126
130
130

Averages, 1890-1899..........................................
1900-1909...........................................
1910-1913...........................................
N um ber of points b y which prices rose (4- )
or fell ( —) in—
1890-1896......................................................
1896-1907......................................................
1907-1908......................................................
1908-1912................................................
1912-1913......................................................

100
118
129
- 23
+ 40
8
H- 8
± o

Difference between highest and lowest
relative prices.................................................

40

25 com ­
modities,
first list.

Average change from j^ear to year................

4.0

115
112
103
103
92
95
80
90
96
107
113
111
116
118
122
123
130
132
124
133
133
129
140
142

j
i
i
i

,

.
;
i

109
119
108
116
102
95
81
84
92
93
99
105
117
107
109
110
115
120
126
134
137
131
143
139

,

:

109
121
108
110
98
94
81
87
96
96
100
106
118
107
115
114
111
121
128
127
137
134
147
139

100
122
136
+
+
-4-

100
114
138
- 29
+ 40
+ 6
+ 17
4

- 28
+ 40
+ 7
+ 19
8

54

63

66

;
,

100
115
139

27
44
8
16
2

48 com ­
modities,
Bureau of
Labor Sta­
tistics list.

5.0

7.1

7.3

112
116
104
110
100
95
84
88
94
98
104
106
111
107
107
109
113
118
122
125
129
127
135
131
100
112
131

.

- 28
4- 34
+ 4
+ 17
4
51
5.0

The three index numbers for foods agree better than might have
been expected in view of the dissimilarity of the lists of commodities
which they quote and the brevity of two of the lists.5 The bureau
2
5 Of the 56 articles included altogether, only 11 are com m on to all three lists. The Gibson list has 8
2
commodities and the Annalist list has 4 commodities classified b y the bureau with farm products instead
of with foods, while the bureau has 34 foods not quoted b y Gibson and 27 not quoted b y the Annalist.
E ven the tw o .short lists have only 15 articles in com m on, while Gibson has 7 articles not quoted b y the
Annalist, and the Annalist has 10 articles not quoted b y Gibson.
For the Bureau’s list see Bulletin N o. 149, pp. 90-107.
The Annalist list runs—oats, cattle, fresh beef, salt beef, hogs, bacon, salt pork, lard, sheep, m utton,
butter (tw o kinds), cheese, coffee, sugar, wheat flour (tw o lands), rye flour, corn meal, rice, beans, potatoes,
prunes, evaporated apples, and codfish.
The Gibson list is—barley, corn, oats, rye, wheat, cattle, hogs, sheep, butter, coffee, wheat flour (tw o
kinds), corn meal, bacon, fresh beef, salt beef, hams, m utton, sugar (tw o kinds), tea, and potatoes.




M E T H O D S U SED I N

M A K IN G IN D E X N U M B E R S .

55

series is rather steadier than the others, because of the larger propor­
tion of manufactured products included in it; but this series and that
of the Annalist invariably agree about the direction in which prices
Ch

a rt

1 0.—I N D E X N U M B E R S OF T H E PR ICE S OF 25 FO O D P R O D U C T S A N D O F 25 MIS­
C E L L A N E O U S COM M ODITIES.
(Based on Table 13.)

are moving,5 and the Gibson figures agree with the other two series
3
in 19 years out of the 24. On the other hand, the three food indexes
& Even in 1903-4 the bureau’s figures record a slight advance of prices in harm ony with the Annalist
3
figures, though this advance is confined to the decimal colum ns and disappears when the decimals are
rounded off.




56

THE

M A K IN G AN D

U S IN G

OF IN D E X

N U M BERS.

often contradict the evidence of the two general-purpose index num­
bers in a striking fashion. Such contradictions occur in 1890-1891.
1892-1893,1900-1901,1902-1903, 1907-1908, and 1912-1913. These
differences are due chiefly to a contrast in the years mentioned between
business conditions and harvest conditions. They parallel the differ­
ences in Table 8 between the index numbers of mineral products and
those of farm crops, or farm crops and animal products taken together;
for the food indexes are made up almost wholly from the pieces of vege­
table crops, food animals, and their derivatives.5 A food index num­
4
ber, then, is likely at any time to give a wrong impression regarding
the shifting of prices in general and is especially treacherous as a busi­
ness barometer. Nor can such an index when made from wholesale
prices be trusted to show changes in the “ cost of l i v i n g f o r living
expenses are made up of retail prices, and fluctuations in retail prices
do not always follow closely those in the wholesale markets.
But while it is clear that an index number intended to measure
fluctuations in “ the general level of prices” should grant due repre­
sentation to the various groups of prices that behave in specific ways,
it is not possible to give a definitive list of these groups. For our
knowledge of the interrelations among prices even in the recent past
is very limited. Moreover, a change, in the social conditions under
which business is done may at any time produce new groupings of
commodities important to the maker of index numbers, or may cause
old groups to fluctuate in novel ways. For example, the distinction
between commodities over which the Government assumed some
form of price control and commodities whose prices were left unre­
stricted became of first importance in the summer of 1917. After
July the controlled prices dropped, and while they advanced again in
the latter part of 1918, they did not again attain the high level at
which they stood when the price control began. Uncontrolled prices,
on the contrary, which stood lower than the other group in July, 1917,
advanced month by month until the armistice was signed.5 Forest
5
products in 1915-1918 illustrate the way in which a group may change
its characteristic price behavior. The demand for lumber has been
declining jerkily in the United States since 1909, primarily because of
the increased use of cement for building. Further, the terms on
which many large lumber holdings are financed compel the owners to
cut and market their timber as fast as possible. Finally, in 1917-18
the War Industries Board discouraged the construction of buildings
that were not called for by the military program. Under these cir­
cumstances, the price of forest products lagged behind most classes
0 The exceptions are salt and soda, and of these articles the Annalist and Gibson quote neither.
4
See the tables in Government Control Over Prices, b y Paul W . Garrett, W ar Industries Board Price
Bulletin, No. 3.
The following index numbers, while not covering the whole ground, bring out the m ain point. One
scries shows the fluctuations of 5S(> commodities that were subjected to price control at some tim e during
American participation in the war; the second series shows the fluctuations of 780 commodities that were
left uncontrolled in price. Since the practice of “ fixing” prices did not begin until several m onths after
the declaration of war (April, 1917), and was extended gradually month by m onth until the signing of the
armistice (Novem ber, 1918), the “ controlled” list contains m any articles that remained uncontrolled until
late in 1918. The t wo series therefore minimize rather tl\an exaggerate the differences between the behavior
of prices that were controlled earlier in the war and prices that were left to find their own levels. That
this understatement is not more serious arises from the fact that the Government naturally took the m ost
important (and therefore most heavily weighted) commodities under control at an early date. It m ay be




M E T H O D S USED I N

M A K IN G IN D E X

57

N UM BERS.

of commodities in the wartime rise.5 To give another illustration,
6
rubber is rapidly passing from the group of forest products to the
group of cultivated crops. These cases give force to the warning that
the groupings with which the economic statistician deals do not
always rest on permanent foundations. It would be as unwarranted
to draw up a list of groups that should be represented in index num­
bers for all periods as to draw up a list of groups to be represented for
all purposes. In every case in which an investigator plans to measure
changes in the general level of prices he should canvass his particular
field to see whether there are not hitherto unrecognized groups of
commodities that fluctuate in similar ways, and then try to represent
each group in the due measure of its importance. Such investigations
may add much, not only to the accuracy of index numbers but also
to our knowledge of the interrelations among price fluctuations.
In most large index numbers the commodities quoted are divided
into several classes; but these classes seldom have economic signifi­
cance or even logical consistency. Among the nine groups recognized
by the Bureau of Labor Statistics, for example, one group, “ Farm
products,” emphasizes the place of production; four groups, “ Food,
etc.,” “ Fuel and lighting,” “ Lumber and building materials,” and
“ House-furnishing goods,” emphasize the use to which commodities
are put; three groups apply a double criterion, use and physical
character of the goods, namely, “ Cloths and clothing,” “ Metals and
implements,” and “ Drugs and chemicals” ; the remaining group is
frankly styled “ Miscellaneous.” Such a classification is not without
usefulness, for there doubtless are readers especially interested in the
prices of, say, all things that are raised on farms, and others who care
especially about the prices of things used to furnish houses, or things
that can be classed together as drugs and chemicals whether they are
used chiefly as medicines or to make farm fertilizers. But if a classi­
fication of this empirical character is maintained, it might with
advantage be accompanied by a classification that throws more light
upon the workings of the complex system of prices.
pointed out also that the commodities early brought under control were articles that, as a group, had risen
more than the average in price before we entered the war.
Index numbers o f commodities that were and 'of commodities that were not subjected to 'price control by the
Governm ent during the war with G erm any.

[From W ar Industries Board Price Bulletin No. 3.]
(Relatives made from weighted aggregates of actual prices.

Year and month.

1917
.Tannary
............................
Fe.hrnary _
.............
March
_ . _
...............
April .
....................
M ay........................................
June.......................................
J u ly .......................................
August .................................
Septem ber............................
October.................................
N o vem ber............................
December..............................

Uncon­
Controlled
com m od­ trolled com ­
ities.
modities.

183
192
201
209
204
205
198
200
193

i46
149
152
160
162
163
167
172
174

Average prices in July, 1913, to June, 1911— 100.)

Year and month.

1918
January.............................
February..............................
March.................................
A p ril...................................
M a y.......................................
June.......................................
J u ly ........................................
A ugust...................................
Septem ber................. ..........
O ctober.................................
N ovem ber............................
December..............................

Controlled
U ncon­
com m od­ trolled com­
ities.
modities.

195
198
197
196
192
189
195
199
204
201
200
204

178
180
182
187
189
191
194
195
199
201
200
197

^ See R . B . Bryant, The Prices of Lumber, W ar Industries Board Price Bulletin, N o. 43, and H om ef
H oyt, The Prices'of Building Materials, W ar Industries Board Price Bulletin, No. 6.




58

THE

M A K IN G A N D U S IN G OF IN D E X N U M B E R S .

Another interesting experiment has recently been made by the
Price Section of the War Industries Board. This section was able
to collect quotations for so large a number of price series (1366 in
form to be used in the index number) that it attempted to classify
its commodities according to industries by which they are manu­
factured. The advantage of this arrangement is that many users
of index numbers desire to follow the fluctuations of the prices that
are paid for materials and received for products in different lines of
business and to compare fluctuations in one line with those in others.
There are many industries in which the plan works well, because the
demarcation between industries follows, at least roughly, commodity
lines; for example, in the cotton, woolen, silk and leather trades.
But many commodities are used in such a variety of industries, and
many industries use such a variety of commodities, that the classifier
is forced to resort at times to other criteria, such as the physical
characteristics of commodities, their uses, or their sources of supply.
Probably the most illuminating way of presenting an index num­
ber that aspires to cover the whole field of prices at wholesale would
be to publish separate results for the groups that have characteristic
differences of price fluctuations, and then to publish also a grand total
including all the groups. The groups to be recognized and the distri­
bution of commodities among them is a difficult matter to decide.
But, as matters stand, the most significant arrangement seems to be
(1) a division of all commodities into raw and manufactured prod­
ucts; (2) the subdivision of raw commodities into farm crops and
animal, forest, and mineral products; (3) the subdivision of manu­
factured products according as they are bought mainly for personal
consumption, mainly for business use, or largely for both purposes.5
1
This classification is based upon differences among the factors affect­
ing the supply of and the demand for commodities that belong to the
several groups— that is, upon differences among the factors which
determine prices. If we wish our index numbers to help toward an
understanding of changes in the price level, a classification along
such causal lines seems to be the most promising line of progress.
Where means permit, it is desirable to supplement this general
scheme by a series of special indexes for classes of commodities that
possess interest for whatever reason. These supplementary indexes
would not rest on classifications which include all the commodities,
and they might, therefore, employ many different criteria and employ
each one only in those cases in which it was significant. Some
commodities might appear in several of the special indexes, and others
might appear in none. There need, then, be no artificial forcing of a
criterion upon facts which it does not fit, and no hesitation about
presenting any classes that merit separate attention.
Large index numbers are more trustworthy for general purposes
than small ones, not only in so far as they include more groups of
related prices, but also in so far as they contain more numerous
samples from each group. What is characteristic in the behavior of
the prices of farm crops, of mineral products, of manufactured wares,
of consumers’ goods, etc.— what is characteristic in the behavior of
any group of prices— is more likely to be brought out and to exercise
its due effects upon the final results when the group is represented by
w Since the first edition of this bulletin appeared, the Federal Reserve Board has adopted this suggestion
with interesting results. In its m onthly bulletin the board publishes the index number com piled by the
Bureau of Labor Statistics recast into the six groups mentioned.




M E T H O D S U SED I N

M A K IN G IN D E X N U M B E R S

59

10 or 20 sets of quotations than when it is represented by only one or
two sets. The basis of this contention is simple: In every group that
has been studied there are certain commodities whose prices seldom
behave in the typical way, and no commodities whose prices can be
trusted always to behave typically. Consequently, no care to in­
clude commodities belonging to all the important groups can
guarantee accurate results, unless care is also taken to get numerous
representatives of each group.
Even here the matter does not end. The different groups that have
been discussed, the other groups that might have been discussed,
and the commodities that are included within the several groups
differ widely in importance as elements in the system of prices. To
these differences, and to the methods of making them count in index
numbers, we must now turn.
6. PROBLEM S OF W E IG H TIN G .

It is customary to distinguish sharply between “ simple” and
“ weighted” index numbers. When an effort is made to ascertain
the relative importance of the various commodities included, and to
apply some plan by which each commodity shall exercise an influence
upon the final results proportionate to its relative importance, the
index number is said to be weighted. When, on the contrary, no such
effort is made, but every commodity is supposedly allowed just the
same chance to influence the result as every other commodity, the
index number is said to be unweighted, or simple.
This expression, however, that “ every commodity has just the
same chance to influence the result as every other commodity”
conveys no clear meaning. It is better to think of all index numbers
as weighted, for so they are whether their maker knows it or not, and
to ask whether the scheme of weights is good or bad. For example,
in Bradstreet’s index the influence of every article upon the result
varies as its price per pound happens to be large or small.5 Again,
8
the decisive objection to making index numbers by merely adding
the ordinary commercial quotations for different articles is that these
nominally simple series are in fact viciously weighted series.5 Nor
9
does the substitution of relative prices for actual prices assure an
“ equal chance” to every article. For instance, in its famous report
of 1893, the Senate Committee on Finance presented three wholesaleprice index numbers— one simple and two weighted; but in the simple
series it included relative prices for 25 different kinds of pocketknives,
giving this trifling article more than eight times as many chances to
influence the results as they gave wheat, corn, and coal put together.
Finally, even if one series of relative prices, and only one, be accorded
each commodity, it does not follow that equal percentages of change
in the price of every article will always exercise equal influence upon
the results. For when relative prices are computed upon a fixed base
and averaged by the use of arithmetic means, those commodities that
have a long period upward trend in price will presently for outweigh
in influence those commodities whose prices are declining.
Lack of attention to weighting, then, does not automatically secure
a fair field and no favor to every commodity; on the contrary, it




5 For details, see p p. 161-168.
8

w See p. 31.

60

THE

M A K I N G A N D U S IN G OF IN D E X N U M B E R S .

results in what Walsh happily termed haphazard weighting.6 Per­
0
haps “ unconscious weighting7 would be an even better expression.
1
The real problem for the maker of index numbers is whether he shall
have weighting to chance or seek to rationalize it.
There are two excuses for neglect of weighting. First, as has been
shown in another connection, to collect satisfactory statistics showing
the relative importance of different commodities is extremely labori­
ous and extremely difficult.6 Second, there are high authorities who
1
hold that the results turn out much the same whether or not formal
weights are used.6 Certainly u tEe weights are of * * * less impor­
2
tance in determining an index number of prices than the prices
themselves.” 6 But whether their importance is negligible is a ques­
3
tion best answered by a study of actual cases such as are shown in the
next table.6
1
The discrepancies here revealed between the averages with hap­
hazard and with systematic weights seldom amount to 10 per cent of
the results, except under the chaotic price conditions created by the
greenback standard in 1862-1873. In many kinds of statistics a 10
per cent margin of error is not accounted large. But in making wholesale-price index numbers for current years we may reasonably try to
get not two, but three, significant figures; and the third figure is
6 C- M. Walsh, The Measurement of General Exchange-Value, pp. 81 and 82. Haphazard weighting is
0
not necessarily the worst weighting; indeed, it m ay be better than.the weighting which results from some
systematic calculations. For example, Bradstreet’s plan of using actual prices per pound is certainly
systematic, but the weighting which this system involves is probably less defensible than the haphazard
weighting involved in most averages of the relative prices of com m odities selected at random . See p. 78.
C See p . 26. W hen the (then) Departm ent of Labor started its former index number it canvassed the
1
subject of weighting, but decided to use a simple average, because of the “ im possibility of securing even
approxim ately accurate figures for annual consum ption in the United States of the com m odities included.”
(Bulletin No. 39, of the Department of Labor, p. 234, March, 1902.) It did, however, allot tw o or more series
to certain com m odities, and thus introduced a rough system of weights. U nfortunately the number of
series allotted to each com m odity seems to have been determined quite as m uch b v the' ease of securing
quotations as b y the im portance of the articles. For criticism of the weighting whicn resulted, see pp. 48
and 49.
6 Compare A . L . B ow ley, Elem ents of Statistics, 2d ed., pp. 113 and 220-224.
2
0 Irving Fisher, The Purchasing Power of Money, revised edition, p . 406. For further details see the
3
papers b y Edgeworth, to which Fisher refers in his footnote.
ci Details concerning the first three sets of sim ple and weighted averages can be found in the documents
referred to in the table. B ut the fourth set of com parisons is based upon hitherto unpublished data and
requires description.
The “ unweighted” index numbers in this set are arithmetic means of the relative prices given in the
bulletins of the Bureau of Labor Statistics for the commodities listed below. B u t where two or m ore series
of relative prices are shown in the bulletins for different grades of the same articles, as in the case of cattle,
hogs, bacon, butter, corn meal, pig iron, etc., they were replaced b y a single average series for the article in
question before the arithmetic means of the group were com puted.
The “ weighted” index num bers were made from these same relative prices in the following way: (1)
For each com m odity included the Bureau of Labor Statistics made a careful estimate, based upon a critical
stu dy of the best available sources of information, of the physical quantity of it entering into exchange in
the year 1909. B y “ quantity entering into exchange” is meant the quantity bought and sold, irrespective
of the number of times it changed hands. ( See p p. 63 and 64.) (2) These physical quantities were m ulti­
plied b y the average prices in 1909 of the respective com m odities. (3) The resulting sums of m oney
were used as weights to m ultiply the relative prices of the respective commodities on the 1890-1899 base.
(4) The sums of the products were cast up for each year, and finally these sums were divided b y the sums
of the weights, i. e., the value in exchange for 1909.
The average prices of the commodities in 1909 m ay be found in any of the recent wholesale-price bulletins,
e. g., No. 149. The com m odities included, and the estimated physical quantity of each entering into
exchange in 1909, are as follows:
Farm products: Barley, 75,300,538 bu.; cattle, 124,346,349 cw t.; corn, 460,778,251 bu.; cotton, 5,409,760,011
lbs.; flaxseed, 20,106,433 b u.: hay, 10,685,804 tons; hides, 922,243,894 lbs.; hogs, 76,438,923 cw t.; hops.
48,076,921 lbs.; oats, 267,859,660 bu.; rye, 29,520,508 bu.; sheep, 11,498,090 cw t.; wheat, 683,416,528 bu.
Food, etc.: Beans, 8,468,385 cw t.; butter, 1,042,709,708 lbs.; cheese, 353,641,892 lbs.; coffee, 1,038,439,285
lbs.; eggs, 926,690,112 doz.; codfish, 684,692 cw t.; herring, 428,804 bbls.; mackerel, 190,565,bbls.; salmon,
18,431,003 doz. cans; buckwheat flour, 2,009,599 cw t.; rye flour, 1,594,346 bbls.; wheat flour, 107,306,408 bbls.;
currants, 32,163,998 lbs.; prunes, 138,795,607 lbs.; raisins, 12,438.044 boxes; glucose, 7,701,223 cw t.; lard
1,243,572,129 lbs.; corn meal, 53,353,466 cw t.; bacon, 741,354,500 lbs.; beef, fresh, 4,209,196,748 lbs.; beef,
salt,'632,388 bbls.; hams, 789,861,744 lbs.; m utton, 495,458,067 lbs.; pork, salt, 4,760,690 bbls.; m ilk,
7,749,070,256 qts.; molasses, 55,689,983 gals.; rice, 1,042,538,693 lbs.; salt, 22,136,489 bbls.; soda, bicarbonate,
165,600,000 lbs.; pepper, 36,241,462 lbs.; sugar, raw, 6,316,033,669 lbs.: sugar, granulated, 7,366,818,210 lbs.;
fallow, 203,209,103 lbs.; vinegar, 98,403,927 gals.; potatoes, 397,491,062 bu.; onions, 4,972.947 cw t.; tea,
113,547,647 lbs.
Metals and implements: Bar iron, 2,166,529.067 lbs.; barbed wire, 6,471,300 cwt.; copper, ingot, lj312,437,919
lbs.; copper wire, 278,964,000 lbs.; lead, p ig,'732,152,538 lbs.; lead pipe, 1,058,280 cw t.; nails, wire, 13,910,097
kegs: pig iron, 9,896,248 tons: tin (pig), 94,248,471 lbs.; silver, 151,969,144 ozs.; spelter, 464,903,059 lbs.; steel
billets, 4,972,179 tons; steel rails, 3,025,009 tons; tin plate, 12,968,174 cwt.




M E T H O D S U SED I N

M A K IN G

61

IN D E X N U M B E R S ,

usually altered in appreciable degree by the substitution of systematic
for haphazard weights. Even the large Canadian series, with its 272
commodities, is shifted 9.5 points, or more than 7 per cent, in 1912 by
weighting.
T able

1 4 . —C O M PA R ISO N S

OF W E IG H T E D A N D U N W E IG H T E D IN D E X N U M B E R S.

[1. From the report of the Senate Committee on Finance, Mar. 3, 1893.
(A rith m etic means.

A ll articles
averaged
according
Simple
to im por­
arithmetic
cer­
means, all tance,ex­
tain
articles.
penditures
being
uniform.

Year.

1860...............................................
1861................................................
1862................................................
1863................................................
1864................................................
1865................................................
1866................................................
1867................................................
1868................................................
1869................................................
1870................................................
1871................................................
1872................................................
1873................................................
1874................................................
1875................................................
1876................................................
1877................................................
1878................................................
1879................................................
1880................................................
1881................................................
1882................................................
1883................................................
1884................................................
1885................................................
1886................................................
1887................................................
1888................................................
1889................................................
1890.......................................... .
1891................................................

100.0
100.6
117.8
148.6
190.5
216.8
191.0
172.2
160.5
153.5
142.3
136.0
138.8
137.5
133.0
127.6
118.2
110.9
101.3
96.6
106.9
105.7
108.5
106.0
99.4
93.0
91.9
92.6
94.2
94.2
92.3
92.2

100.0
95.9
102.8
122.1
149.4
190.7
160.2
145.2
150.7
135.9
130.4
124.8
122.2
119.9
120.5
319.8
115.5
109.4
103.1
96.6
103.4
105.8
106.3
104.5
101.8
95.4
95.5
96.2
97.4
99.0
95.7
96.2

A ll articles
Difference Difference
averaged
between
according
between
to im por­ sim ple and simple and
tance: 68.6
first
second
per cent of weighted
weighted
total ex­
averages.
averages.
penditure.

100.0
94.1
104.1
132.2
172.1
232.2
187.7
165.8
173.9
152.3
144.4
136.1
132.4
129.0
129.9
128.9
122.6
113.6
104.6
95.0
104.9
108.4
109.1
106.6
102.6
93.3
93.4
94.5
96.2
98.5
93.7
94.4

[2. From Bulletin of the Departm ent of Labor, No. 27, March, 1900.
(A rith m etic means.

4.7
15.0
26.5
41.1
26.1
30.8
27.0
9.8
17.6
11.9
11.2
16.6
17.6
12.5
7.8
2.7
1.5
1.8
3.5
.1
2.2
1.5
2.4
2.4
3.6
3.6
3.2
4.8
3.4
4.0

6.5
13.7
16.4
18.4
15.4
3.3
6.4
13.4
1.2
2.1
.1
6.4
8.5
3.1
1.3
4.4
2.7
3.3
1.6
2.0
2.7
.6
.6
3.2
.3
1.5
1.9
2.0
4.3
1.4
2.2

Difference
between
first and
second
weighted
averages.

1

1.8
1.3
10.1
22.7
41.5
27.5
20.6
23.2
16.4
14.0
11.3
10.2
9.1
9.4
9.1
7.1
4.2
1.5
1.6
1.5
2.6
2.8
2.1
.8
2.1
2.1
1.7
1.2
.5
2.0
1.8

January of the years, 1890 to 1899.]

A verages o f 9 quarterly q u otation s, January, 1890, t o January, 1892=100.)

Year and m onth.

1890, Januarv..............................
1891, January..............................
1892, Januarv..............................
1893, January..............................
1894, January..............................
1895, January..............................
1898, January..............................
1897, January..............................
1898, January..............................
1899, January..............................




B y years, 1860 to 1891.]

Prices in 1860= 100.)

A ll articles A ll articles
averaged
averaged
according according Difference Difference
to im por­
to im por­
between
A ll articles tance, cer­ tance, com ­ between
sim ple
sim ple
sim ply
prising
tain ex­
and first and second
averaged. penditures
weighted
68.6 pet
weighted
being con­
cent of
averages.
averages.
total ex­
sidered
uniform. penditure.
102.0
100.6
96.5
97.2
89.6
84.7
85.2
82.0
83.3
86.5

100.1
102.2
100.0
103.4
97.5
93.5
92.8
90.3
91.0
91.0

100.2
103.2
100.1
105.0
96.4
90.5
89.5
85.9
86.8
86.8

1.9
1.6
3.5
6.2
7.9
8.8
7.6
8.3
7.7
4.5

1.8
2.6
3.6
7.8
6.8
5.8
4.3
3.9
3.5
.3

Difference
between
first and
second
weighted
averages.

0.1
1.0
.1
1.6
1.1
3.0
3.3
4.4
4.2
4 .2

62
T

TH E M A K IN G AND USING OF INDEX NUM BERS.

a b l e

^ .^ C O M P A R I S O N S O F W E IG H T E D A N D U N W E IG H T E D I N D E X N U M B E R S —Con.

J3. F rom W holesale Prices, Canada, 1913.
(A rith m etic means.

U n­
Weighted weighted
index
index
number. number.

Year.

1890............................
1891............................
1892............................
1893............................
1894............................
1895............................
1896............................
1897............................
1898 ........... ...........
1899............................
1900............................
1901............................

112.0
111.3
104.9
103.9
97.2
95.6
90.6
89.9
95. 5
99.0
105. 8
106.0

R eport b y R . H . Coats.

Differ­
ences.

110.3
108. 5
102.8
102.5
97.2
95.6
92.5
92.2
96.1
100.1
108.2
107.0

B y years, 1890 to 1913.1

A verage p rices in 1890-1899=100.)

1.7
2. 8
2.1
1.4
1.9
2.3
.6
1.1
2.4
1.0

Year.

U n­
W eighted
weighted
index
index
number.
number.

1902..........................
1903..........................
1904..........................
1905..........................
1906..........................
1907..........................
1908..........................
1909..........................
1910.,......................
1911..........................
1912..........................
1913..........................

109.6
109. 7
110.6
113.8
120.1
129.2
125.1
126.3
128.0
131.1
143.9
139.6

Differ­
ences.

109.0
110. 5
111. 4
113. 8
120.0
126.2
120. 8
121.2
124.2
127.4
134.4
135. 5

0.6
.8
.8
.1
3.0
4. 3
5.1
3. 8
3! 7
9. 5
4.1

[4. From com putations b y the Bureau of Labor Statistics.!]
(A rith m etic means.

13 farm product;s.

Year.

A verage prices in 1890-1899=100.)

37 food product;s.

14 metallic products.

Weighted
Weighted
Weighted
b y esti­
b y esti­
b y esti­
m ated ex­ Dif­
m ated ex­ Dif­
mated ex­
Dif­
pendi­
U n­
U n­
pendi­
U n­
pendi­
fer­
fer­
fer­
tures
tures
weighted.
weighted.
tures
ences. weighted.
ences.
upon each
upon each ences.
upon each
article in
article in
article in
1909.
1909.
1909.

1890.......................
1891.......................
1892............. ....
1893.......................
1894.......................
1895.......................
1896.......................
1897.......................
1898.......................
1899.......................
1900.......................
1901.......................
1902.......................
1903.......................
1904.......................
1905.......................
1906.......................
1907.......................
1 9 0 8 ...................
1 9 0 9 ....................
1 9 1 0 ................ .
1911................... ..
1912.......................
1913.......................

113
124
112
106
96
93
78
84
97
99109
117
130
120
130
125
122
139
135
150
161
166
173
152

109
117
105
107
94
95
86
93
97
98
109
115
129
120
128
123
124
136
135
154
165
150
164
161

4
7
7
1
2
2
8
9
0
1
0
2
1
0
2
2
2
3
0
4
4
16
9
9

114
116
105
112
99
95
83
87
93
98
108
110
114
110
113
110
115
120
122
124
129
128
137
133

114
114
103
111
97
94
86
90
96
96
100
102
108
104
110
109
108
112
119
126
127
125
137
127

0
2
2
1
2
1
3
3
3
2
8
8
6
6
3
1
9
8
3
2
2
3
0
6

128
118
110
102
88
88
93
82
83
124
124
114
114
114
105
116
131
138
103
109
111
111
120
119

131
116
107
98
84
88
91
80
81
124
123
113
114
113
192
113
130
140
108
107
108
103
114
115

3
2
3
4
4
0
2
2
2
0
1
1
0
1
3
3
1
2
5
2
3
S
6
4

1 See explanations in footnote, p. 60.

If rational weighting is worth striving after, then by what method
shall the weights of the different commodities be arrived at? That
depends upon the object of the investigation. If, for example, the
aim be to measure changes in the cost of living, and the data be
retail quotations of consumers’ commodities, then the proportionate
expenditures upon the different articles as represented by collections
of family budgets make appropriate weights. If the aim be to study
changes in the money incomes of farmers, then the data should be




METHODS USED IN M A K IN G INDEX NUMBERS*

63

“ farm prices/’ the list of commodities should be limited to farm
products, and the weights should be proportionate to the total money
receipts from the several products. If the aim be to construct a
“ business barometer/’ the data should be prices from the most repre­
sentative wholesale markets, the list should be confined to com­
modities whose prices are most sensitive to changes in business pros­
pects and least liable to change from other causes, and the weights
may logically be adjusted to the relative faithfulness with which the
quotations included reflect business conditions. If the aim be merely
to find the differences of price fluctuation characteristic of dissimilar
groups of commodities, or to study the influence of gold production
or the issue of irredeemable paper money upon the way in which
prices change, it may be appropriate to strike a simple arithmetic
average of relative prices. If, on the other hand, the aim be to make
a general-purpose index number of wholesale prices, the question is
less easy to answer.
One proposition, however, is clear. The practice of weighting
wholesale-price index numbers by figures drawn from family budgets
is to be deprecated; for family budgets do not show the importance
of wheat and cotton, of petroleum and spelter, of tar and lime, of pig
iron and hides, of brick and lumber; indeed, to apply budget weights
to half or more of the articles in any wholesale list is nonsensical.
And to pretend that wholesale-price index numbers when weighted
on the basis of family expenditures show fluctuations in the cost of
living is to overtax the credulity of those who know and to abuse the
confidence of those who do not.
Allied to the family-budget method of weighting and yet vastly
better for wholesale-price index numbers is the “ aggregate expend­
iture” method.6 Here an attempt is made to ascertain the aggre­
5
gate sums of money laid out by the people of a whole country upon
the articles quoted and to adjust the weights upon this basis. Of
course the country as a whole buys raw materials, as single families
do notr and of course consumers’ commodities can be taken at their
aggregate values in wholesale markets. Similar in net effect is the
weighting on the basis of consumption practiced by the British
Board of Trade. For “ consumption is taken to mean any process
by which the commodity is substantially changed in character. In
other words, consumption in manufacture is recognized as well as
consumption by an individual.” 6 Somewhat different weights would
6
result i f quantities or values produced were taken in place of quanti­
ties or values consumed. Mr. Walsh thinks it best to combine these
two criteria— that is, to take “ either the total product or the total
consumption according as the one or the other is the greater.” 6
7
Prof. Irving Fisher prefers “ an index number in which every article
or service is weighted according to the value of it exchanged at base
prices in the year whose level of prices it is desired to find.” 6 On
8
this system the weight assigned to each article would be affected by
the number of times it changed hands on its way from producer to
• See G. H . Knibbs, Prices, Price Indexes, and Cost of Living in Australia. Commonwealth Bureau ol
Census and Statistics, Labour and Industrial Branch, Report No. 1, p p . 11-14.
6 Report on Wholesale and Retail Prices in the United Kingdom in 1902. London, 1903, p. 411. The
6
accuracy of the statistics upon which the Australian and British index numbers are based m ay be open
to question. N ot the data, but the m ethod is of interest here.
67 C. M. W alsh, The Measurement of General Exchange-Value. New York, 1901, p. 95.
« Irving Fisher, The Purchasing Power of M oney, revised edition. New Y ork, 1913, p p. 217 and 218.




64

TH E M AKIN G AND USING OF INDEX NUMBERS.

final consumer. A variation of his plan is therefore represented by
the proposal to weight each article according to the quantity of it
which enters into the country’s commerce, irrespective of the fre­
quency with which it changes hands.
The practical consequences of adopting these different systems of
weighting may be illustrated by considering their application to
cotton, corn, and coffee in the United States. Production weights
would give cotton much greater importance than consumption or
aggregate-expenditure weights, because so large a part of the Ameri­
can crop is exported and consumed abroad. Exchange weights would
be practically equivalent to production weights, because practically all
the cotton grown is sold b y the planters and enters into the commerce
of the country, and relatively little cotton is imported. On Prof.
Fisher’s plan, however, the exchange weights would be some multiple
of the production weights, depending upon the average number of
American hands through which the cotton passed. In the case of
corn, production and consumption weights would substantially agree,
for we import little corn and export but a small percentage of the
)rOduction. On the other hand, exchange weights would be much
ess than either production or consumption weights, because a large
part of the corn crop is never sold, but is consumed on the farms
where it is grown. In the case of coffee, production weights would
be zero, while consumption and exchange weights would correspond
closely.
We are helped toward a choice among these rivals by common
agreement upon a slightly different point. In arranging any system
of weights except Prof. Fisher’s, double counting is to be avoided so
far as possible. For example, if cotton is counted at its full impor­
tance as a raw material, then cotton yarns and later cotton fabrics
made of the yarn can not be counted at their full importance with­
out assigning triple weight to the raw cotton which is represented at
these two successive stages of manufacture. Now, if this sensible
observation be applied to cases like those of corn, hay, etc., it casts
the die in favor of exchange weights. For if these articles, which
are used largely by the original producers in making things quite
different from corn and hay (for instance, pork and beef) are counted
at the full amount produced or consumed, and if their products (the
pork and beef) are also counted at the full amount produced or con­
sumed, there will be a great deal of double counting. Not all but
much of this duplication can be eliminated by counting only the
amount of corn and hay sold by the producers and letting the rest of
these articles produced and consumed get their proper representation
under the caption of pork, beef, etc.6
9
If for this reason exchange appears a rather better criterion of
importance than production, consumption, or a combination of the
two, it remains only to decide whether the number of times a thing
is exchanged should be recognized. Prof. Irving Fisher had good
cause to propose multiple counting, for he wanted an index number
of prices for constructing the “ equation of exchange,” a mathematical
expression of the necessary equivalence between the total volume of

J

6
9
6 f course, this same end might be attained without surrendering the production or consum ption
basis if the rule against double counting of raw materials and products were made broad enough to include
corn, for exam ple, as the raw material of pork; but needless to say there is little liklihood that the com m on
meaning of terms will be stretched to such an extent.




METHODS USED IN M AKIN G INDEX NUM BEKS.

65

business done in a country and the total volume of payments effected
by means of money and credit instruments. Of course the oftener
an article is sold and paid for the more important it is as a factor in
this equation. 'But it does not follow that the economic importance
of an article is greatly changed by reorganizing the chain of business
enterprises that deal in it. “ Integration of industry,” as expressed
in our trusts, does not make pig iron less significant as an item in the
country’s economic life, except in the sense that it reduces the
average number of transfers of ownership. The quantity of the
ar.ticle that enters into exchange, then, irrespective of the number
of turnovers, is probably the most satisfactory gauge of importance
to apply in making general-purpose index numbers. Anyone experi­
enced in the search for statistical information will need no warning
that in the working out of weights .along this line many puzzling
cases will arise in which consistency will be difficult to maintain, to
say nothing of the wide gaps and weak places that will;be revealed
among the available data. That* this system of weighting is feasible
in practice as well as desirable in theory, however, was proved b y the
Bureau of Labor Statistics in 1914, when it gave up averaging
relative prices and began multiplying actual prices by the quantities
of commodities that entered into trade in the base year 1909.7
°
Three interesting questions remain: Should the weights be sums
of money or physical quantities? Should the weights be changed
from year to year or kept constant ? Should the weights be adjusted
to the importance of the commodities as such, or should there be
taken into account also the importance of the commodities as repre^
sen ting certain types of price fluctuations ?
When relative prices are being used the weights should be reduced
to a common denominator. As multipliers, of course, weights may
be regarded as merely abstract numbers; but in studying the weights
themselves it is necessary to have some common standard by which
the relative importance assigned to various commodities can be
accurately compared. The only common denominator for all com­
modities that is significant for economic ends and capable of quanti­
tative expression is money value. But it is ill advised to weight by
money values when actual prices are being used, for the common
denominator is already present in the quotations themselves. These
price quotations are best multiplied by the physical quantities of the
goods produced, exchanged, or consumed, as the case may be.
Like most of the issues on which authorities differ, the question
whether it is desirable to change weights at frequent intervals depends
upon the precise end in view. Most makers of index numbers have
wished to isolate the price factor from other changes in the economic
complex. Hence they have preferred to keep their weights as nearly
constant as possible. For when the weights are altered the index
number becomes a measure of two sets of changes, and no one can
tell what part of the net results is due to variations in prices and what
to variations in weights.7 Yet it is clear that a system of fixed
1
weights applied over a long period is certain to become inaccurate
for most o f the years, however carefully it is adjusted to conditions
prevailing at some base period. Practically, then, a compiler who
wishes to ascertain how prices have changed must choose between
7 For details see Bulletin N o. J8i of the Bureau of Labor Statistics.
0
7 See the criticism of index imbibers made from import-export values, p p . 29-31.
1
f311739 0 —41------5




66

THE

M A K IT O A
3STD USIHG OF ItfDEX NUM BEBS.

two eyils— inaccurate weights and ambiguous price measures. Some­
times he can minimize the first evil by collecting data showing the
average importance of his commodities over a period of years, for
these averages are less likely to go awry than figures for a single
year. In other cases the least objectionable compromise is probably
to revise the scheme of weights, say, once a decade, and to show the
effect of this change by computing overlapping results for a few years
with both the old and new weights.7 A further practical reason in
2
favor of this compromise is found in the heavy expense in time and
labor required for frequent revisions of the weights.
Writers like Mr. Walsh, Prof. Pigou, and Prof. Fisher, who urge
the adoption of a formula in which the weights are changed every
year, put another aim in the foreground. Their primary purpose is
to secure the utmost possible nicety in measuring the rise or fall of
prices in each pair of years treated. Of course an index number
made with .these changing weights “ measures neither the varying
cost of a constant amount of goods nor the varying amount of goods
which a dollar will buy.” 7 But, since the importance of price fluc­
3
tuations depends largely upon the accompanying changes in the
quantities of goods bought, there is use for index numbers that do
not attempt to measure the price factor in isolation. By changing
weights each year it is possible to make these constantly occurring
changes in quantities bought influence the price index, and therefore
to secure results better fitted for certain uses than the results of an
unambiguous measure of fluctuations in prices.7
4
To the third question, whether weights should be adjusted to the
importance of the commodities as such, or whether there should also
be taken into account the importance of these commodities as repre­
sentatives of certain types of price fluctuations, little attention has
been paid. But the preceding section shows that this neglected
problem is both important and difficult. The prices of raw materials
behave differently from the prices of manufactured goods; among
the raw materials the prices of farm crops, of forest, animal, and
mineral products behave differently; there are also differences of
behavior between the prices of manufactured goods bought by pro­
ducers and by consumers, etc. Is an accurate measure of changes
in the level of all wholesale prices obtained unless all of the different
types of fluctuation, doubtless including types not vet definitely rec­
ognized, are represented in accordance with the relative importance
of the commodities exhibiting each type ?
How can such representation be attained ? If all the commodities
bought and sold could be included on a strictly uniform basis in the
index number, it would suffice to weight each by the criterion of its
own individual importance. Since that is out of the question, it is
desirable to draw from each part of the whole system of prices
samples sufficient to determine its characteristic fluctuations, and then
to make sure that each part of the whole system counts for the proper
amount in determining the final result. On this plan commodities
would be weighted simply as commodities in making the subtotals

7
2
7
3

Compare G. H . K nibbs, Prices, Price Indexes, and Cost of Living in Australia. Commonwealth
Bureau of Census and Statistics, Labour and Industrial Branch, Report N o. 1, p p. x x iv and xlix.
Prof. Warren M. Persons: "F ish er’s formula for index n u m b ers/’ R eview o f E conom ic Statistics,
May, 1921, p. 115, note.
7 Seethe discussion of the " id e a l'/ formula in section-9, p. 91.
<




METHODS USED IN M A K IN G INDEX NUMBERS

67

f©r each recognized group, and these subtotals would be weighted
again in making up the grand totals.
Such a plan was adopted by the Price Section of the War Industries
Board in making their index number of prices in 1913-1918. As
noted above, the subdivisions used by the Price Section were 50
classes of commodities based, so far as possible, on the organization
o f industries. Within each class, raw materials were weighted
according to quantities used by the industry represented, and
products were weighted according to the quantities produced. A
separate index number was made in this way for each of the 50
classes. These indexes and the materials from which they were
made, both price quotations and weights, seemed fairly satisfactory
as such matters go; but before the aggregates of the commodity
prices times commodity weights for these 50 classes were assembled
to make aggregates for “ all commodities,” it was clear that there
would be wide differences in the fullness with which prices in the
various industries had been covered. In some industries 75 to 90
per cent of the value of the transactions was represented by the
prices multiplied by the weights; in other industries the percentage
sank below 25. Again, there were industries in which it had been
possible to quote commodities at three stages—raw materials, partly
manufactured goods, and finished products— while in other industries
the available data represented only raw materials or only finished
products. That is, while the weights within each class had been
systematized, and while the plan of systematizing the weights was
uniform in all classes systematized, the weights as between different
classes were haphazard to a degree. To overcome the difficulty, the
Price Section .prepared a second set of weights. It estimated the
value of the products sold by each industry represented, divided these
estimates by the aggregate of commodity prices times commodity
weights, and so obtained a set of factors which when applied to the
class aggregates give each class an influence upon the index for “ all
commodities ” proportioned to its estimated importance.7
5
Professor Edgeworth has pointed out a yet further desideratum in
weighting. Most index numbers are made from samples of the data
which logically fall within the field investigated; and the task is to
make from these samples the best approximation to a measure of the
unknown whole. Now “ the theory of errors-of-observation shows
that in the combination of the given observations, 1less weight should
be attached to observations belonging to a class which are subject to
a wider deviation from the mean. Such would be prices of articles
which, exclusive of the common price movement of all the selected
articles, are liable to peculiarly large proper fluctuations.’ ” 7
6
Perhaps it is a counsel of perfection to urge such refinements in
systems of weighting. Certainly the difficulties to be encountered
are very great. Statistical knowledge is not complete enough to sup­
ply accurate data for weighting all the different parts of the system of
prices that are known to have characteristic peculiarities of fluctua­
tion. Nor have these different types and the commodities exhibiting

7
5

See H istory of Prices During tlie .War, Summary, W ar Industries Board Price Bulletin, No. 1.
s Econom ic Journal, June, 1918 (V ol. X X V I I I , p. 188). The quotation within the quotation is from
tho British Association M em orandum , 1887 (p. 36). To.m ake his point clearer, Prof. Edgeworth adds in a
footnote this remark from the corresponding memorandum of 1889 (p. 157): “ I f m ore weight attaches to a
change of price in the article rather than another, it is not on account of the importance of tnat article to the
consumer or the shopkeeper, but on account of theim portance to the calculators of probabilities as affording
an observation which is peculiarly likely to be correct.”




68

TH E M A K IN G AND USING OF INDEX NUMBERS.

eacD been adequately studied. And puzzling difficulties are raised
by overlapping among the type^— there are commodities that belong
in two places at once. But here is certainly a promising lead for
future efforts to improve present measurements of changes in the
price level. Even now it might be feasible by taking pains to secure
rough justice as between raw and manufactured commodities, and as
between raw vegetable, animal, forest, and mineral products. One
modest step in the right direction can readily be taken by any com­
piler of index numbers: He can make clear that his results do not
measure changes in the general level of wholesale prices accurately
when they are obtained without an effort to represent each part of the
field according to its due importance.
7. AVERAGES AND AGGREGATES.

Among aJJ the problems involved in the making of index numbers
the one that has been most discussed is the best form of average
to strike. Most of these discussions have come from men inter­
ested ip the mathematical side of statistics rather than in the
problem of ascertaining what changes have actually occurred in
rices. The practical makers of index numbers, on the contrary,
ave seldom troubled themselves greatly about theoretical con­
siderations. Indeed, the two problems of finding out how prices
have actually changed, and finding the best method of measuring
changes, appeal to two types of interest, which are seldom
strongly developed in the same mind. The mathematical statistician
is likely to know little and care less about the field work of collecting
price quotations. To the practical statistician this field work is of
overshadowing importance, and the subsequent manipulation of his
data is a matter of secondary interest. Hence, a study of index
numbers as they are made need not carry One into long mathematical
flights.7
7
First, it should be recalled that certain compilers of index numbers
do not strike averages at all. The old form of the Economist index
and Gibson's present index, for example, are sums of relative prices.
More important are the series which dispense with the use of relative
prices for each commodity, and give results in the form of sums of
actual prices, or such sums thrown back into a series of relative num­
bers. These cases are still exceptional, however, and most index
numbers are made by finding some sort of average from the relative
prices of the commodities included.
The sort of average struck is usually th,e arithmetic mean— that
is, the sum of the relative prices divided by their number. Occasion­
ally medians are used— that is, the midmost relative prices which
divide the whole number of cases into two equal groups, half above
the median and half below. In one famous investigation,7 geometric
8
means were employed— that is, all the relative prices for a given date
were multiplied together and the nth roots of the products were
extracted, n standing for the number of commodities included. But
Jevons has had few imitators, though Mr. A. W. Flux has just adopted

E

w The best systematic discussions o f averaging for the purpose in hand are to be found in Prof. Edgeworth’ s papers referred to in the footnote on p. 8; Irving Fisher’s The Purchasing Power o f Money,
revised edition, 1913, pn. 385-429; and C. M. W alsh's The Measurement o f General Exchange-Value, 1901,
and his new treatise, Tne Problem of Estimation, 1921.
7 W . 8. Jevons, “ A serious fall in the value o f gold ascertained,” 1863. Reprinted in his Investigation
8
in Currency and Finance, 1884, p p. 13-118




METHODS USED IN M A K IN G INDEX NUMBERS.

69

the geometric mean for the new form of the British Board of Trade
index number. The other standard forms of averages-—the mode
and the harmonic mean— have been discussed frequently, but so
far as is known they have never been consciously used in making
index numbers.7
9
For the geometric mean two great merits are claimed. First, unlike
the arithmetic mean, it is not m danger of distortion from the asym­
metrical distribution of price fluctuations. Chart 2 shows that in a
large collection ol percentage variations from the prices of the pre­
ceding year, the extreme cases of rise run about twice as far up the
scale as the extreme cases of fall run down. Such a distribution is
characteristic of relative prices in general. Indeed, the case cited is
distinctly moderate; most collections of variations covering many
years would show a greater difference. There is indeed no limit to
the possible percentage of rise in prices, while the possible percentage
of fall can not exceed 100.8 The cases of extraordinary advance,
0
accordingly, tend to raise the arithmetic mean more than the cases
of extraordinary decline tend to depress it. If, for example, one
commodity rose tenfold in price and another commodity fell to onetenth of the old price, the arithmetic mean would show an average
rise of 505 per cent (1,000 + 10^2), while the geometric mean would
show no change in the average, since ^IjOOO x 10 = 100.
This favorite imaginary case of 10 and 1,000 seems extreme, but
contrasts approximately as violent as that actually occurred in the
recent war. The Price Section of the War Industries Board has
computed the relative prices of 1,437 commodities in 1918 on the
basis of their average prices in the twelve months, July, 1913, to
June, 1914. These figures are reproduced in somewhat condensed
form in Table 15. Here the array of relative prices is far more elon­
gated in one direction than in the other, and the highest relative
price is upwards of 100 times as great as the lowest m ative price.8
1
Accordingly, the arithmetic mean (217) stands high above the geo­
metric mean (194) and median (191).8
2
Concerning the properties of these averages see, for example, F. Zifcek, Statistical Averages (translated
b y W . M. Persons), and G. U. Yule, Introduction to the Theory o f Statistics, pp. 120-123, 128-129. The
“ crude m od e” is that relative price which occurs m ost frequently in the data under examination, e. g., in
Chart 2 i t i s “ no change.” The true m ode is “ the value oith e variable corresponding to the m axim um of
the ideal frequency-curve which gives the closest possible fit to the actual distribution. ” “ The harmonic
mean o f a series o f quantities is the reciprocal of the arithmetic mean o f their reciprocals.”
80 Negative prices are conceivable o f course; but do they ever occur in the sources which the maker of
index numbers uses? Suppose that some kind o f factory waste, which usually commands a low price,
should fail o fits m arket, and accumulate so as to becom e a nuisance. The factory manager m ight logically
set it dow n at a negative price; but he is m uch more likely to offer a positive price for another com m odity—
the rem oval o f the waste.
« This ratio o f 100 to 1 was indeed surpassed in some months. The highest relative price found was
5,081 (acetiphenetidin, N ovem ber, 1916).— See History of Prices during the W ar (W ar Industries Board
Prirc Bulletin, N o. 54, p. 18).
sa From the skewed distribution characteristic of relative prices when arranged on the ordinary arith­
metic scale, Prof. Frederick R . Macaulay has developed an ingenious argument in favor of the geometric
mean.
He puts the matter in this way: “ W hat is the most probable value for the general percentage m ovem ent?
If the f errors* (variation due to the influence of particular com m odity factors) were distributed arithmetically
according to the normal law, the arithmetic mean—least mean square deviation—would certainly seem
indicated. B ut if the logarithms of the percentages and not the percentages themselves follow more closely
the curve of error, is not the geometric mean indicated? From that point the curve of the squares of the
logarithms of the percentage deviations will be a minimum; and is not this what sound theory should de­
m an d?” American Econom ic Review, March, 1916, V ol. V I, p . 207.
The answer to Prof. Macaulay’ s final question is that what sound theory demands depends upon the
precise magnitude one desires to measure. It is argued hereafter in the text that if the purpose be to measure
the average ratio of change in prices, the geometric m ean is in strictnessthe only proper average to em ploy.
Those who can utilize measures of average change for their purposes will be gratified to know that the arrays
from which their averages are m ade usually conform better to the normal law of distribution than the arrays
from which arithmetic means of relative prices are derived. As Prof. Edgeworth humorously admits, “ it
is a merit in a statistical group to conform to the normal law. ” (Econom ic Journal, June, 191S V ol. X X V I I I ,
p. 182). B ut, of course, the sym m etry of the distribution of data from which different averages are derived
is but one, and generally a subordinate, consideration in the choice of averages.




70

TH E M A K IN G AND USING GE INDEX NUMBERS.

T able 1 5 — D IS T R IB U T IO N OP T H E R E L A T IV E P R IC E S OP 1,137 COM M ODITIE S IN 1918.
(A verage prices ip. .July, 1913, t o June, .1914=100.)

Relative prices.

N um ­
ber of
cases.

36............................
49............................
50*69......................
70-89......................
90-109....................
110-129...................
130-149..................
150-169..................
170-189...................
190-209...................
210-229.................
230-249..................

1
1
4
17
61
64
130
212
219
164
135
104

Relative prices.

250-269.................,
270-289...................
299-309...................
310-329...................
330-349...................
350-369..................
370-389...................
390-409..................
410-429..................
430-449...................
450-469..................
470-489..................

Num­
ber of
cases.
76
54
42
30
31
16
13
7
7
3
4
4

Relative pricks.

490-509.................
510-529.................
530-549............... j
550-569.................
587........................'
-627...................... ^
727........................
730......... •
........... J
7 4 3 ..:.................i
761...................... ;
784........................
826..........................

N um ­
ber of
cases.
4
5
3
4
1
1
1
1
1

1
1
1

Relative prices.

•848.
900
1,165..
1,356..
1. 585
1, 764
2, 049
2,863.. .
3,009...

Num­
ber of
cases.

. .

1
1
1
1
1
1
1
1
1

The second merit claimed for geometric means is that they can be
shifted from one base period to another without producing results
that seem to be inconsistent. 'Suppose, for example, that the price
of wheat falls from SI per bushel in 1913 to 50 cents in 1914, while
the price of corn remains unchanged at 40 cents. Then the relative
prices are—
(1) On the basis, prices in 1913 = 100:
1913
W h ea t...........................................................
C orn...............................................................

100
100

1914
50
100

(2) On the basis, prices in 1914 = 100:
1913
W h ea t...........................................................
•
Corn............................................................ .

200
.100

1914
100
100

The arithmetic and geometric means of these relative prices are—
(1) On the basis of prices in 1913:
Arithmetic means.

1913....................
1914.................. ;

Geometric means.

(4004-400)-7-2 = 100
(504-100)-4-2= 75

V^OOX 100 = 1 0 0 .0 0
V 50X100 = 70.714-

(2) On the basis of prices in 1914:




Arithmetic means.

4913......................
1914......................

Geometric means.

(2004-100) -r- 2 = 150
(1 0 0 4 -1 0 0 )-2 = 100

V200X.100= 1 41 .42VlO 0XiOO = 100.00

METHODS USED IN M A K IN G INDEX NUMBERS,

11

Here the arithmetic means can not, but the geometric means can,
be shifted from the 1913 base to the 1914 base or vice versa by
simply dividing the index number for one year by that for the other.
That is, 100^75 = 133|, not 150; but 1004-70.71 =141.42.8 By
3
shifting the base in this simple fashion geometric means can be made
to give direct comparisons between the price levels at any two dates
covered by the investigation, whereas with arithmetic means com ­
parisons not made in terms of prices at the original base period give
results that may present formal inconsistencies and results whose
meaning is difficult to grasp and put into words.
A third advantage of geometric means is that they are likely to
be nearer the modes of the distributions which they represent than
are arithmetic means. The importance of this point will be more
generally appreciated as statisticians come to study the whole array
of the price fluctuations with which they deal, instead of concen­
trating their attention merely upon averages.
The chief objection to geometric means in an index number
intended for general use is that this form of average is unfamiliar and
therefore more likely to be misinterpreted than arithmetic means.
Further, geometric means do not have any direct bearing upon
changes in the purchasing power of money as do arithmetic means
and weighted aggregates of actual prices.8 Finally, geometric means
4
are somewhat more laborious to compute than arithmetic means or
medians. Instead of adding the relative prices just as they stand and
dividing the sums by their number, the computer must convert the
relative prices of every commodity into their logarithms, add these
logarithms, divide the sum by the number of logarithms, and look up
the natural numbers corresponding to the quotients.8 Statisticians
5
are the more loath to incur the extra labor of this process since the
special merits of the geometric mean are shared in part by certain
ra See the discussion of shifting bases, pp. 83-90.
8 This point is more fully explained on pp. 76 and 77.
4
8 I f relative prices are n ot needed for any other purpose, it is quicker to com pute the geometric mean from
^
the logarithms of the successive actual prices and then to find the ratios between the results. But even
that is a somewhat longer process than calculating relative prices, casting them up, and dividing b y their
number.
That geometric means can he computed either with or without the use of relative prices is readily shown.
Let po, Vx\
P 'o , P ' x i

st

an(3f o r the actual prices of n

commodities in the two years

o

and x.

* V Pj
Then the relative prices of these articles in the year x on the basis of actual prioes in the year o are
,

P x,

P X,

P o'

p'o

p n
X

n

The geometric mean of these relatives is

But this expression is equal to

V (px) (p'J • • • (4 )
•V(P o o
o)

• ■ • (4 )

And the latter expression, of course, is the ratio between the geometric means of the actual prices in the
tw o years.




72

THE M A K IN G AND USING OF INDEX NUMBERS.

other forms of index numbers. Like geometric means, aggregates
of actual prices, or relatives made directly from them, can be
shifted to any base desired without raising difficult problems of
interpretation. Like geometric means, again, medians are not
more affected b y cases of exceptionally great advances in price than
by cases of exceptionally great declines. Hence in practice most
makers of index numbers who distrust arithmetic means abandon the
practice of averaging relative priees or use medians instead of geo­
metric means.
Medians, indeed, have several distinguished champions among
theoretical statisticians.8 It is generally claimed that of all averages
8
medians are the easiest to compute, for a quick ordering of the data,
followed by a counting of the items, takes the place of casting sums
and dividing by the proper number. But in this day of adding
machines the palm for ease of computation has shifted to the arith­
metic mean a*pd the aggregate of actual prices. More important
is the fact demonstrated by Prof. Edgeworth that the median is
safer than the arithmetic mean when, as m the case of index numbers,
the items to be averaged are samples drawn from a larger field.
For, according to the theory of probabilities, the probable error
of the median can not in any case be much greater than that of the
arithmetic mean and in other cases it may be very much less.8
7
But medians have their drawbacks. (1) They are not perfectly
reversible; that is, they can not be shifted from one base to another by
simple division without ambiguity. (2) Medians of different groups
can not be combined, averaged, or otherwise manipulated with ease as
can arithmetic means. For example, in making up its old form of
index number the Bureau of Labor Statistics could add the sums used
for making arithmetic means of the relative prices of farm products,
foods, .cloths and clothing, etc., and from the sum of these sums strike
the grand average for all commodities. It could not handle medians
in this convenient fashion; instead of combining the sums from the
groups it would have to reexamine and rearrange the relative prices
of those commodities which fell near the respective medians. Simi­
larly, a reader who finds arithmetic means of two groups in different
sources can compute the arithmetic mean of these means, provided
the number of items in each group be stated, with no greater error
than that arising from the dropping of fractions in the published
data; but he can not approximate except in the vaguest way the
median of two medians.8 (3) When the number of items to be
8
averaged is small, medians are erratic in their behavior. For in
such groups there is often a considerable interval between the mid­
most relative price and the relative prices standing next above it
and. next below. No change in any of the items, large or small, can
alter the position of the median unless it shifts an'item from the
8 Compare, for example, F. Y . E dgew orth ,“ Index num bers,” Dictionary of Political E conom y, V ol. II,
6
p. 380; Irving Fisher, The Purchasing Power of Money, revised edition, p. 425: A. L. Bowley, Elements of
Statistics, second edition, p. 22-1. Walsh, however, docs not recognize the m edian as a mean. See Quar­
terly Publication of the American Statistical Association, March, 1921, p. 542, and the numerous
references to medians in his Problem of Estimation.
8 See his paper “ On the use of analytical geometry to represent certain kinds of statistics, ” Journal of
7
the R oyal Statistical Society, June, 1914, V ol. L X X V I I , p. 733.
8 It is a convenient feature of arithmetic means com puted from relatives based on average prices over a
3
period of years that the mean of these means for the base period must be 100—again barring discrepancies
caused b y dropping fractions. For exam ple, the arithmetic means of the Bureau c f Labor Statistics oldstyle index numbers for the 10-year period 1890-1899 would always add up to 1,000.0, had all the fractions
been kept and had all commodities been quoted in every year of the decade. I f medians made from these
figures add up to 1,000.0 in 1890-1899, it is accidental.




METHODS USED IN M AKIN G INDEX NUMBERS.

73

upper half of the list to the lower half, or vice versa. But any
change of this character, large or small, will make the median jump
over the whole interval between its former position and that of the
next highest or next lowest relative price, unless the change happens
to place a new item within these limits. In large groups such erratic
jumps are less likely to occur, because the intervals between the
median and its nearest neighbors are usually slight.8 (4) If the num­
9
ber of commodities included in an index number is even, the position
of the median may be indeterminate, though within a determinate
range.
Most of the advantages and defects of arithmetic means have been
mentioned incidentally, but it is well to list them all together: (1)
Arithmetic means (and aggregates of actual prices) stand first in
ease of computation, when an adding machine is available, especially
when the items are to be averaged first in small and later in large
groups. (2) Their familiarity to all readers is supposed to be an
advantage in work intended for wide reading though perhaps this
familiarity means a dangerous lack of curiosity rather than clear
understanding of the figures. (3) They can themselves be averaged
and manipulated algebraically in various other ways.9 On the other
0
side of the score it must be said (4) that arithmetic means are liable to
distortion from the occurrence of a few extremely high relative prices,
(5) that arithmetic means of relative prices can not, consistently be
shifted from one base to another witnout recomputation in full,9
1
and (6) that they may conceivably give contradictory results con­
cerning the direction in which prices are moving, according as relative
prices are computed on one base or on another.9
2
Concerning the numerical value of the three averages under dis­
cussion, it can be proved that the geometric mean is always less than
the arithmetic. On the other hand, the median may be either above
or below the arithmetic mean, and likewise either above or below.the
9 “ This objection is m et,” says Prof. Edgeworth, “ b y denying that the interval between tw o adjacent
9
observations at the m iddle of the group is likely to be ‘ considerable ’ ; large relatively to the magnitude
with which it is proper to compare that interval—that is, the minimum mensurable, as we m ay say—that
interval which is equal to (or of the same order as) the smallest degree which the com pared method of
measurement is capable of distinguishing with accuracy. For this m inim um we m ay take at the least
the ■probable error’ incident to the arithmetic mean. That the interval between adjacent ebservations is
likely to be small compared with this m inimum is sufficiently evidenced b y the following proposition:
W hen the number of observations (n ) is large the interval at the m iddle of the group, which is as likely as
not vacant, within which it is an even chance that no observation falls, is m ost probably very small com ­
pared with the probable range of the arithmetic mean (in the ratio of about 1: yjn). W hen the number of
observation is not large the proposition is less accurate. B ut it remains roughly true, as the num ber can
not be supposed very small consistent with the applicability of the theory of probabilities.” Econom ic
Journal, June, 1918, V ol. X X V I I I , p. 193.
Granting the justness of these general remarks, the writer has found several cases in his own work where
the medians of groups numbering 25 or more items m oved in a w ay not representative of the whole array.
For examples see “ A critique of index numbers of the prices of stocks,” Journal of Political E conom y, July,
1916, V ol. X X I V , pp. 674, 675. It m ay, indeed, be set dow n as an advantage of medians that working with
them m ay bring the full array of fluctuations under the eye and lead to the detection of peculiarities which
would have escaped notice had arithmetic means been employed. W hen medians are used in averaging
small groups the practice of scrutinizing the whole set of data is strongly recommended as a means of guard­
ing against the occasional cases of erratic movem ent.
90 See, for example, G. U. Vale, Introduction to the Theory of Statistics, pp. 114-116.
9 See section 8 below.
1
9 Take, for example, the following data:
2




1913
Wheat, per bushel.........................................
Corn, per bushel.............................................

1914

$0. 50
.48

$1.00
.24

74

THE M AK IN G AND USING OF INDEX NUMBERS.

geometric mean. For example, if the relative prices of the 145 com­
modities represented in the second index number of Table 6 be aver­
aged in these three ways, the results are as follows for 1913:
Geometric mean, 125.7; median, 126.9; arithmetic mean, 131.3.
A more striking example of differences among the averages ’was
incidentally remarked abovei The relative prices of Table 15 yield
the following figures:
Geometric mean, 194; median, 191; arithmetic mean, 217.
A fuller study of the relations between medians and arithmetic
means is provided for by the following table.9 In the chain index
3
the two forms of average never quite coincide; the median is smaller
in 20 cases and larger in 3.; it is also steadier than the arithmetic mean
in the sense that it indicates an average annual change of 2.22 per
cent from prices in the preceding year, as against 3.64 per cent for
the arithmetic mean. In the fixed-base series for 1890-1913, in­
cluding 145 commodities, the median is likewise steadier than the
arithmetic mean, showing a smaller percentage of change, except dur­
ing the middle nineties, when the price level was at its lowest. The
second series for these years illustrates the behavior of medians and
arithmetic means' when used to average small groups. Here the
median is greater than the arithmetic mean in 13 years, the same
in 1 year, and less in 10 years. Moreover, it shows a greater aver­
age change from one year to the next than the arithmetic mean.
Finally, the median drops 9 points in 1913 while the arithmetic mean
rises 2 points. Scrutiny of the full array of relative prices in this year
as compared with 1912 shows that this violent drop is not an apt
Then compute index numbers on the basis 1913= 100:

1913
Wheat, relative prices...................................
Corn, relative prices.......................................
Index num bers............ .......................

1914

J00
100

200
50

200
100 .

250
125

Al30, com pute index numbers on the basis 1914=100:
1913

1914

W heat, relative prices...................................
Corn, relative priees.......................................

50
200

100
100

Index num bers....................................

250
125

200
100

Thus it appears that pfiqes were 25 per cent higher in 1913 than in 1914 and also that they were 25 per cent
higher in 1914 than in 1913.
Much stress is often laid upon illustrations of this sort, but they are not seriously damaging to the good
repute of arithmetic means when properly interpreted. W hat they really say is: The»arithmetic mean
variation of prices from 1913 to 1914 m ay conceivably be upward in percentages of priees in 1913, and at the
same time be downward in percentages of prices in 1914. N o real inconsistency is involved in that statement
to one who can keep the meanings of’the tw o results in mind. It should be added that cases in which such
apparent inconsistency occurs, while cbm m on in theoretical discussions, seldom if ever occur in the practi­
cal computation of wholesale-price index numbers. In retail-price indexes they are not unknown. A n
example has been pointed out in the British Board of Trade’s reports upon cost of living of the working
classes. See the reviews by J. M. Keynes in the Econom ic Journal, September and December, 1908.
9 Irnr mimorionl uvorrmloc nfof nf\T fiu’n AM
3 For numerical examples rYgeometric and arithmetic means ~_.__4
rm
rl
__3 r___ A
i__
I11G dOftE S9'6 F
com puted from the same data', see F. Y
A defense of index numbers,” Econom ic Journal, V ol. V I (1896), p. 137,, and A . W . F lux,
w ~
Modes of constructing index numbers,” Quarterly Journal of Economics, Vol. X X I (1907), p. 627.
On the character of chain indexes, see the following section (pp. 81 to 91).

9
3




M E T H O D S U SED I N

M A K IN U

75

IN D E X N U M B E R S .

summary of the combined movements.9 The figures for prices dur­
5
ing the period of irredeemable paper money (1862-1878, inclusive)
show how far arithmetic means may depart from the medians when
a few commodities attain very high relative prices. The maximum
difference occurs in July, 1864, when the arithmetic mean exceeds the
median by 42 points, or more than 20 per cent. This excessive dif­
ference is due to the high prices of cotton, tar, and other southern
products. It is precisely in cases such as this that the median is
distinctly safer to trust than the arithmetic mean.
T able 1 6 .—C O M PARISON S OF M E D IA N S A N D A R IT H M E T IC M E A N S AS A V E R A G E S O F
R E L A T IV E PR IC E S.
[Data from Bulletin N o. 149 of-the Bureau of Labor Statistics.)

Chain index number
(prices in preceding
year= 100).«
Year.
Medians.

1890................................................
1891............... .................................
1892................................................
1893................................................
1891................................................
1895................................................
1896......................................... ......
1897................................................
1898................................................
1899................................................
1900................................................
1901................................................
1902................................................
1903................................................
1904................................................
1005................................................
1:908................................................
1907...............................................
1908................................................
1909................................................
1910................................................
1911................................................
1912................................................
1913..............................................

Arithmetic
means.

a Compare Tables 2 and 17.

Medians.

112

Arithmetic
means.

Relative prices of 25
com m odities
(aver­
age prices in 18901899= 100). c

Medians.

Arithmetic
means.

- 0.2
- 4. 4
.2
- 8.7
- 1. 5
- 2.8
+ .2
+ 4.8
+10. .4
+ 9.4
- 1.1
+ 4 .6
+ 1.2
.1
+ 2.9
+ 5.8
+ 0.0
— 6. 6
+ 3.2
+ 4. 1
- 1.9
+ 3.4
+ 1.2

2. 22

3.64

111
107
104
96
94
90
91
94
100
109
107
110
111
112
114
119
129
1,19
121
124
125
127
127

Ill
113
10G
105
96
93
89
89
93
103
111
110
114
114
114
116
122
130
121
124
131
130
134
131

116
109
106
102
90
94
89
92
99
108
117
112
115
112
124
126
131
133
125
130
126
131
136
127

115
112
103
103
92
95
88
90
96
107
113
111
116
118
122
123
130
133
124
133
133
129
140
142

100
115
126

±0
- 3 .1
+.0
- 7 .1
-2 . 4
—1. 2
±0
+ 1.8
+ 5. 5
+ 7 .5
-1 . 5
+2. 2
+ 1.3
±0
+ .7
+ 5 .1
+ 3 .9
- 3 .8
±0
+ 1.5
- .9
+ 1.0
+ .5

Averages, 1890-1899...................
1900-1909...................
1910-1913...................
Average change from one year
to the n ex t..............................

Relative prices of 145
com m odities (aver­
age prices in 18901899=100) b

100
118
132

101
123
130

100
122
136

3.61

&Compare Table 6, second series.

4.13

5.70

5.09

< Compare Table 6, fifth series.
•

Of the 25 commodities 13 rose In price and 12 fell; the median percentage of change from prices in tho
year before is +1.0.




76
T able

TH E M A K IN G AND USING OF INDEX NUMBERS.
1 6 .— C O M P A R IS O N S O F M E D IA N S A N D A R IT H M E T IC M E A N S AS A V E R A G E S O F
R E L A T I V E P R IC E S —Concluded.
[From W . C. Mitchell, Gold Prices and Wages under the Greenback Standard, pp. 59, 60.]

92 com m odities at wholesale (prices in 1860=100).

Year.

Me­ Arith­
metic
dians. means.

1860, January............
April. .1.............
J u ly ...................
October.............
1861, January............
A p ril................
J u ly ...................
October.............
1862, January............
A p ril.................
J u ly ...................
October.............
1863, January............
A p ril.................
J u ly ...................
October.............
1864, January............
A p r il.................
J u ly...................
October.............
1865, January............
A p ril.................
J u ly............... ..
October.............
1866, January............
A p ril.................
J u ly ...................
October.............

100
100
100
100
100
96
96
97
100
100
100
111
125
137
134
135
156
169
194
200
216
190
158
175
182
173
181
173

102
102
100
102
100
98
95
103
115
112
120
126
142
160
155
1.55
179
197
236
239
248
206
183
205
199
186
191
188

Year.

1867, January..........
A p ril...............
J u ly .................
October...........
1868, January..........
A p ril................
J u ly.................
October...........
1869, January..........
A p ril...............
J u ly .................
October...........
1870, January..........
A p ril................
J u ly .................
October...........
1871, January..........
A p ril...............
J u ly .................
October...........
1872, January..........
A p ril...............
J u ly .................
October...........
1873, January..........
A p ril..............
J u l y . . . ...........
October...........

Arith­
Me­ metic
dians. means.
169
166
150
162
158
162
154
159
159
159
158
153
147
140
132
135
133
131
130
129
133
140
130
133
135
137
130
131

179
175
170
172
171
176
165
166
165
165
158
157
152
146
145
143
142
140
137
139
141
145
139
143
142
144
140
140

Year.

1874, January........
A p ril.............
J u ly ...............
October.........
1875, January........
A p ril..............
J u ly ...............
October.........
1876, January........
A p ril..............
J u ly ...............
October.........
1877, January........
A p ril.............
J u ly ...............
October.........
1878, January........
A p r il.............
J u ly...............
October.........
1879, January........
A p ril..............
J u ly...............
October.........
1880, January........
A p ril..............
J u ly ...............
October.........

Me­ Arith­
dians. m etic
means.
130
129
130
130
127
125
121
120
117
115
110
108
114
108
100
102
99
98
90
94
88
84
85
95
108
107
102
101

140
141
138
138
138
132
129
127
122
122
118
117
121
118
114
110
107
105
99
102
100
99
98
103
114
116
110
111

Average change from one quarter to the next: Medians, 5.66 points; arithmetic means, 5.65 points.

Wise choice of the average to use in making an index number,
then, involves careful consideration of the materials to be dealt with
and of the purpose in view. (1) If that purpose be to measure the
average ratio o f change in prices, the geometric mean is the best;
indeed, in strictness, it is the only proper average to employ— on one
interpretation of that somewhat indefinite problem. For, alone
among our averages, the geometric mean always allows equal in­
fluence to equal .ratios of change in price, quite irrespective of the
previous levels of the prices in question, the amounts of money rep­
resented by the changes themselves, or any other factor. As has
been said already, in a geometric mean the "doubling of one price is
precisely offset by the halving of another price— though if the two
prices were originally the same the rise amounts in money to twice
the fall. And further changes of 10 per cent from the two new prices
will again be precisely equal in their influence upon a geometric
mean, although 10 per cent of the price that has doubled represents
a sum of money four times as great as 10 per cent of the price that
has been halved. (2) But these same examples show that geometric
means are not proper averages for measuring alterations in the
amount of money that a given bill of goods costs. And as a rule our
interest does center in the money cost of goods rather than in the
average ratio of changes in price. For example, when we are inves­
tigating the increased cost of living, the doubling of one item in
the family budget may well be twice as important as its halving;
and when we are studying the “ relation of prices to the currency, a




METHODS USED IN M AK IN G INDEX NUMBERS,

Ch

a rt

77

11.—A C O M PA R ISO N O F M E D IA N S A N D A R IT H M E T IC M EAN S OF T H E R E L A T IV E
PR IC E S O F 145 COM M ODITIES.




(Based on Table 16.)

78

THE M A K IN G A
JsTD U SI¥G OF ISTDEX J5TUMBEBS.

large upward variation should count for more than a small down­
ward variation, for it requires more currency;” 9 provided always
6
that the changes in prices are not offset or more than offset by con­
trary changes in quantities bought. For such purposes the arith­
metic mean is the logical average to use. (3) Frequently, however,
the very fact that an article has advanced greatly in price cuts down
its market, so that the increase in money cost represented b y the
arithmetic mean exists on paper rather than in fact.9 When such
7
cases of extreme advance are numerous among the relative prices to be
averaged, the median may give more significant results than the
arithmetic mean. (4) When the number of commodities included
in the index number is small, however, medians may occasionally
prove erratic, representing less the general trend of prices than the
peculiarities of the data from which they are made. (5) If the index
number is designed for the public at large, the familiarity of arith­
metic means is an argument in their favor; but it counts for nothing
in the case of figures intended for specialists. (6) Often the useful­
ness of a new index number may be enhanced without detriment to
its special purpose by throwing it into a form directly comparable
with that of index numbers already in existence. Then, of course,
not only the form of average but also the base period employed in
making" the existing series has special claims for imitation. (7) Fi­
nally, the desirability of making index numbers that can be shifted
from one base to another without raising difficult problems of inter­
pretation, deserves more consideration than is commonly accorded
it. On this count the score is in favor of the geometric mean. If
geometric means were invariably used, all index numbers could
readily be compared with one another, whatever the bases on which
they were originally computed. And that would be a great gain to
all students of prices.
No single form of average made from relative prices, then, is with­
out its merits and its defects. Can we not escape the necessity of
relying upon any one of them by giving up the use of relative
prices and falling back upon aggregates of actual prices ?
Index numbers made on this latter plan practically compel the
compiler to adopt a systematic scheme of weighting. This should
constitute a great safeguard against crude work, though in view of
Bradstreet’s method of weighting one can not claim that it always
is effective. For the haphazard weighting involved in merely adding
up the raw quotations for different commodities in terms of their
ordinary commercial units is far more dangerous than the haphazard
weighting involved in using the same materials after reduction to relalative prices.9 It is true that sums in dollars and cents are likely to
8
run in amounts awkward for comparison; but these sums can quickly be
turned into a series of relatives on the scale of 100. The same device
96 Irving Fisher, The Purchasing Power of Money, revised edition, p. 426, note 2. Mr. Flux and Mr. Yule
hold that to measure changes “ in the m oney cost of the things we b u y ” is “ the retail-prices problem ,”
and is not the appropriate aim of a wholesale-price index; but they do not consider the arguments which
Prof. Fisher advances. Journal of the R oyal Statistical Society, March, 1921, pp. 175-9, and 200, 201.
Such cases m ight be m et b y reducing the weight allowed the article in question; but w e have seen thal
revising weights blurs the m eaning of the index number, b y making it im possible to say how far the finat
results measure the change i n prices and how far they measure the change in weights. (See p. G5.)
9 See the exam ple from H un t’s Merchant’s Magazine given on p p . 31 and 32. However, a very rough
8
system of weights based upon guesswork m ay give quite as good results as the haphazard weighting of
relative prices. Prof. Irving Fisher suggests to the writer a “ lazy m an’s index number,” m ade b y adding
actual prices for ordinary commercial units, with their decimal points shifted forward or backward, or
left unchanged, according to the estimated importance of each article.

9
7




METHODS USED IN M A K IN G INDEX NUMBERS.

79

meets the objection that the introduction of new commodities, neces­
sary at intervals in any large index number that is'kept up to date, dis­
turbs a sum of actual prices more than it disturbs an average of rela­
tive prices. This statement is valid because the quotations for new
commodities, however adjusted, are just so much added to the old
sum; while the relative prices of new commodities may be either
above or below the old average, and often exercise but a trifling net
effect upon its value. But by noting the ratios between the sums of
actual money which include and which exclude the new commodities,
and by using these ratios to adjust successive aggregates, the compiler
meets this difficulty quite as well as if he were averaging relatives
from the start.
The technical difficulties attending the construction of index num­
bers made of actual prices, then, can be surmounted. Offsetting these
difficulties are numerous and subst antial .advantages. Aggregates of
money prices weighted according to the importance of the several arti­
cles are even easier to understand than arithmetic means of relative
prices. They are less laborious to compute than any other form of
weighted series, for no relative prices are used; the original quotations
are multiplied directly by the physical quantities used as weights, -and
the products added together. They are not tied to a single base
period; but from them relative prices can quickly be made upon
the chain system or any fixed base that is desired, and these relative
prices themselves can be shifted about at will as readily as geometric
means." Hence they are capable of giving direct comparisons
between prices on any two dates in which an investigator happens
to be interested. Hence, also, they can be compared with any index
numbers covering the same years, on whatever base the latter are
computed. They can not be made to give apparently inconsistent
results like arithmetic means. When published as sums of money,
they can be added, subtracted, multiplied, divided, or averaged
in any way that is convenient. When comprehensive in scope and
weighted on a sound system, they are not likely to be unduly dis­
torted by a very great advance in the price of a few articles, and yet,
unlike medians, tney allow every change in the price of every article
9 The legitimacy of shifting these relatives b y the “ short” m ethod is best shown b y the use of symbols.
9
Let
^represent the m oney prices of the two commodities p and p' in three years o, x, and y.
Then the sums of these actual prices will be—
P o + p 'o in the year o.
Px+ p 'x in the year x.
Py+P'y in the year y.
Relative prices in the year x com puted from these sums will be—
- — ^ r o n the basis of prices in the year o, and
Po+P o
r
J
Py~+P'y0D
kasis
Prices in the year y.
Relative prices in the year y will be—.
^ -^ jP /on the basis of prices in the year o.
N ow the relative price in the year x, com puted on the basis of prices in the year o, can be turned into
the relative price for the year x on the basis of prices in the year y, b y dividing the relative for the year
x on the basis of prices in the year o b y the relative for the year y oh the basis of prices in the year o. For
P x+ p 'x . P y+p'y _ P x+ p 'z
Po+ P 'o Po+P'o P v+ P ’y
The reason why ordinary arithmetic means of relative prices can not be consistently shifted to another
base b y this simple method is explained on p. 83.




80

THE M A K IN G AND USING OF INDEX NUMBERS.

to influence the result. In fact, they combine most of the merits
and few of the defects characteristic of the various methods of averag­
ing relative prices.
But the main issue has still to be faced. Do we wish to know
how certain sample prices have changed on the average, or do we
wish to know how the total cost of a sample bill of goods has changed ?
This is practically the same question we considered on pages 76 to
78 in discussing how best to average relative prices. Ana the answer
given there is valid here. If our interest really lies in measuring
average ratios of change, then geometric means are best. But (l)
the unfamiliarity of this average outside technical circles is itself an
objection to measuring average changes in an index number designed
for wide use, and (2) a measure of change in the money cost of goods
probably serves more uses than a measure of average ratios of change
m prices. Now, the weighted aggregate of prices is the best measure
of change in the money cost o f goods; it is better in several ways
than the simple arithmetic mean of relative prices, and in addition
it has all the merits of the latter form of average. For the relatives
which can be computed from these aggregates with little trouble are
identical with arithmetic means of relative prices, when the latter are
weighted by the money value of the physical quantities used as
weights for the corresponding actual prices.
This identity of the variations of a weighted aggregate of actual
prices and the arithmetic-mean variations of similarly weighted rela­
tive prices can readily be demonstrated. Suppose that we have
collected the price quotations and the quantities to be used as weights
in an index number, and have decided what period to make the base
for comparisons. Then if we want an aggregate of actual prices, we
merely multiply the quotations of each commodity at each date by the
physical quantities used as weights, and add these products. To
measure the variations of these aggregates in terms of prices at the
base period, we have only to divide the aggregate for each period by
the aggregate for the base period. But if we plan to make a weighted
arithmetic mean of price variations, we begin by turning the quota­
tions into relative prices. That is, we divide the actual price of each
commodity at each date by its price in the base period. Then we
weight these relatives, not by physical quantities as in the first case,
but by the money values of the physical quantities at the prices of
the base year. But in this step the prices of the base year, which
were just used as divisors to get relative prices, are used again as
factors by which the relative prices are multiplied. Hence our results
are the same as if we had neither multiplied nor divided by the prices
of the base year; in other words, the same as if we had multiplied the
quotations of each commodity in each year by the physical quantities
used as weights. But that is just what we did when we set out to
make an aggregate of actual prices. So far, then, the two processes
are identical in their outcome. And the remaining steps are also the
same. The products must be added, and the sums divided by the
physical quantities used as weights times the actual prices of the base
year. Therefore, to make relative prices from aggregates of actual




METHODS USED IN M AKIN G INDEX NUMBERS.

81

prices is a shorter way of getting the same results as are obtained by
making similarly weighted arithmetic means of relative prices.1
But while an arithmetic mean of relative prices is always equiva­
lent to some aggregate of actual prices turned into relatives, this fact
does not mean that the arithmetic mean of relatives is as desirable
a form of general-purpose index number as its rival. For the par­
ticular aggregate of actual prices to which a given arithmetic mean
of relatives corresponds is one difficult to grasp. It is that aggregate
in which the price of each commodity included, quoted in terms of
its ordinary commercial unit, has been multiplied by the number of
commercial units which is necessary to make its price in the base
period some predetermined multiple of 100. Now this is a much
more complicated idea to carry in mind and to make clear to readers
than the idea of the price of the commodity multiplied by the num­
ber of units that are ordinarily produced, exchanged, or consumed.
In other words, the arithmetic average of relatives has the same
relation to its corresponding aggregate of actual prices that a com­
plicated mathematical expression has to the same expression reduced
to a simpler form. The difference is one of form, but- simplicity of
form greatly increases the efficiency of thinking.
8. BASE PERIODS, CHAIN INDEX NUM BERS, AND FIXED-BASE SERIES.

When relative prices are used it is necessary to select the quota­
tions of some given period as a base. The actual prices in this base
period are called 100; all antecedent and subsequent prices are
divided by the base prices, and the quotients, multiplied by 100,
make the relatives which are usually summed and divided by the
number of commodities to get the final index number. When aggre1 The explanation given in the text m ay be put in the form of algebraic formulas for readers willing to
study symbols.

Let

Po> Px\

V’o, p'x Irepresent the prices of the commodities from which an index number is to be made in the
n 1 1 base year o and m some other year designated b y the subscript x.
1
**o’ ^x)
Let q, qf and qn respectively represent the physical quantities of these commodities to be used as weights.
Then an unweighted arithmetic mean of relative prices is represented by the following formula, in which
n stands for the number of commodities included:

Vo ' V'o

’

____________ ___*_o
n
A weighted aggregate -of prices reduced to relatives is represented b y the following formula:

Vx q+v'x <?'+••• V l Q
x n
Vo ?+p'o q'+ . . •pn qn
A weighted arithmetic mean of relative prices with m oney weights corresponding to the physical weights
of the expression immediately above is represented by the following formula:

f U ? P o ) +P r ( ? ' P'o) +
fO
Vo
Vo q + v'o q' - f • • • • p

o

qn

B ut in the numerator of this fraction, p 0, p '0, and p h cancel out. Then formula (3) becomes identical with
formula (2). That is, the weighted aggregate of prices gives the same results when turned into relative
as the weighted arithmetic mean of relative prices, and gives them with less work.
1311739




&2

THE M A K IN G AND USING OF INDEX NUMBERS.

gates of actual prices are first made and then turned into relatives the
problem of selecting a proper base period has to be faced at the end
of the computation.
In some cases the prices of a single day have been used as the base,
but as a rule average prices for a year, five years, a decade, or an even
longer period have been preferred. For this preference there is a
simple justification when arithmetic means are used as averages of
the relative prices.2 If the price of any commodity happens to be
unusually high or unusually low in the t>ase period, its relative prices
at other periods will be correspondingly low or high, and the very
high relative prices will exercise much more influence upon arith­
metic means tnan the very low ones. If an appreciable proportion
of the commodities in the list be very high or very low, the final
index number may be distorted. Though numerically correct, the
results have less significance than if they showed changes in terms
of prices that men consider “ normal.” 3 Of course exceptionally
high or exceptionally low quotations are less likely to last for a year
than for a day, and less likely to last for a decade than for a year.
The period chosen as base for the relative prices should be that
period with which accurate comparisons are most significant for the
purpose in hand. Probably most users of general-purpose index
numbers prefer to make their comparisons with recent dates. Hence
the case for “ chain” indexes is very strong— that is, for indexes
like the medians of Table 2, which show the average rise or fall of
prices on the basis of prices in the preceding year.4 Hence, also,
any index number with a fixed base becomes in one respect less signifi­
cant the longer it is maintained. For example, when the Bureau of
Labor Statistics series was established in 1902^ the public was inter­
ested to know how much prices in that year had changed in terms of
average prices in the decade 1890-1899. In 1918, however, when
eople cared less about knowing changes in terms of what prices had
een 19 to 28 years earlier, the Bureau shifted its base to 1913. Sim­
ilarly, Sauerbeck’s index number, which uses prices in 1867-1877 as a
base, suffers in significance for recent comparisons because it forces
one to make all comparisons in terms of prices in a period that ended
before most of the people now living were old enough to know the
meaning of prices.
Index numbers made on a base many years in the past, moreover,
encounter all the difficulties that inhere in the problem of measuring
price variations through long periods of time. As was shown in
Section III of this bulletin (pp. 11 to 23), price variations become
dispersed over a wider range and less concentrated about their mean
as the time covered by the variations increases. That is, the longer
a fixed-base series is maintained, the more scattered as a rule be­
come the relative prices. The difficulty is particularly serious
when arithmetic means are used. The commodities that have a con-

E

a I f geometric means are used the ratios between the index numbers for different dates are not influenced
at all b y the selection o f the base, and if medians are used they are likely to be affected but slightly, provided
the num ber o f commodities included be large.
s The selection of a proper base period, however, does not guarantee im m unity from the exercise o f undue
influence b y certain articles. More important than the base is the choice of proper weights. Or, to speak
with m ore precision, the choice of base is itself part of the problem of weighting in its inclusive sense.
4 This form o f index number was invented b y lJrof. Alfred Marshall. See Contemporary R eview, March




METHODS USED IN M AK IN G INDEX NUMBEBS.

83

sistent long-period trend gradually climb far above or fall far below the
average relative price. Then the high relative prices of the com­
modities that have risen exercise a much stronger pull upon the
position of the arithmetic mean than do the low relative prices of the
commodities that have fallen. For most purposes this constitutes
a defect, since commodities that have increased greatly in price are
likely to have become scarce, and commodities that have become
cheaper are likely to be more abundant. The changes in the influ­
ence exercised on the mean by the relative prices are likely to be in
inverse ratio to the changes in the importance of the commodities.
In other words, the use of the distant base itself introduces a sur­
reptitious set of weights into the figures to be averaged, and a set
which may well counteract in large measure the formal set of weights
which the investigator uses to show the importance of his articles.
It is not uncommon, of course, to shift fixed-based index numbers
from a remote to a recent base. For example, Sauerbeck’s index as
continued by the Statist was 85 in 1913 on the 1867-1877 base. If
one wishes to find how much English prices rose in 1914-1918 as
compared with their prewar level, he may put 85 —100, and recast
the indexes for the years of war on that scale. But this is a purely
formal manipulation of the results. It does not diminish the scat­
tering of the relative prices from which the averages are computed,
and it does not give the same result that recomputing the relative
prices of the 45 commodities on the 1913 base and averaging them
afresh would give. The first point is obvious; the second requires
explanation.
Averages of relative prices on a given base may be regarded as
averages of actual prices made with a peculiar scheme of haphazard
weights. That is, the quotation of every commodity is in effect mul­
tiplied by the factor necessary to make its price in the base period
equal 100.5 To change the base is of course to change this set of im­
plicit haphazard weights for another set, which may be better or
worse— the computer is unlikely to know which—but which will be
different unless the ratio of change in prices between the old and new
base periods has been precisely identical for all the commodities
included. Of course, different sets of weights applied to the same
set of price quotations will probably alter the average variations
somewhat. Hence, if one really wants to know how a given set
of prices have varied with reference to their standing at any given
time, the only way to find out accurately is to weight the varia­
tions of each commodity by the factors which the chosen base de­
termines; that is, in practice, to compute new relative prices article
by article. But if the purpose in hand is such that one set of hap­
hazard weights will serve as well as another, then there is no objection
to shifting the base by the short method of manipulating merely
the averages, provided the results are properly explained.
5 Comoare F. R . Macaulay, “ Index numbers for retail p rice s /1 American Econom ic Review. Decem­
’
ber, 1915, V ol. V , pp. 928,929.




84

TH E M A K IN G AND USING OF INDEX NUM BERS.

It is easy to arrange examples in which, wide discrepancies appear
between the results of the two methods of shifting the base.6 Hut
the difficult and the important thing is to find out how serious the
discrepancies are in actual practice. For to use index numbers
effectively, it is often necessary to shift the base, and sometimes the
short method must be followed, either because recomputation in full
requires a prohibitive amount of labor, or because the original data
necessary for recomputation have not been published. The next
table gives three pertinent examples. In the first case when Sauer­
beck’s index is shifted from 1867-1877 = 100 to 1890-1899 = 100 the
discrepancies are fairly regular and rather small both absolutely and
relatively. In the last case, when the same series is shifted to 1860 =
100, the discrepancies are highly irregular from year to year, and are
rather large both absolutely and relatively— several times exceeding
5 per cent of the recomputed figures. In the remaining case the
discrepancies are small absolutely, though often large relatively to
the recomputed figures, and also highly variable from year to year.7
The conclusion which these experiments suggest is that the two
methods almost always give different results; that the discrepancies
are by no means constant from year to year in a given case, and that
their magnitude both absolutely and relatively differs much from one
case to another. Hence it is well to avoid, the short method of
6 For example, suppose that an index number includes only wheat and corn, and that their prices are as
lollows:

1913
W heat, per b u s h e l.. .
Corn, per bushel........

$1.00
.40

1914
$0.50
.40

I I 1913 be made the base, the relative prices and index numbers will be:
1913

1914

W heat,relative prices.
Corn, relative prices..

100
100

50
100

Sum s...................
Index num bers..........

200
100

150
75

I f now the base be shifted from 1913 to 1914 b y the short m ethod, the index num ber for 1913 will be
(100-i-75) 100=133$. B ut if the figures be recomputed on the basis of prices in 1914, the result is an index
num ber of 150 in 1913:
1913

1914

Wheat,relative prices.
Corn, relative prices..

200
100

100
100

Sum s...................
Index num bers..........

300
150

200
100

7 The discrepancies shown in the table do not result wholly from the mathematical inconsistency of the
short method, but partly from the fact that when an index number is shifted to a new base b y recoinputatio n in full it is com m only impossible or undesirable to utilize all the original data. Some com m odity, for
example, m ay not be quoted for the dates used as the new base, and therefore has either to be dropped or
introduced at a later dal e b y means of some doubtful assumption as to what its price would have betn had
it been quoted for the full period. Of course this observation makes the objection t o using the short m ethod
stronger rather than weaker. It means that this m ethod often leads the statistician into uses of the original
data which he would have avoided had he undertaken the recomputation of the index number.




85

METHODS USED IK M A K IN G INDEX NUMBERS,

shifting bases whenever possible; and when that method must be
used, its results should not be treated as showing what the index
number would have been had it been made originally on the new base.
E X A M P L E S OF D IS C R E PA N C IE S B E T W E E N T H E R E S U L T S OF T W O M E T H O D S
O F SH IFTIN G T H E B A SE S ON W H IC H IN D E X N U M B E R S A R E C O M P U T E D .

T a b l e 1 7 .—

(Arithmetic means.)

Sauerbeck’ s index number,
1890-1913.

R ecom ­
Year. Orig­ Shifted puted
to
on basis
inal
1890form, 1899=
189018671899=
100;b y
1877=
100,by
short
100.
method. long
method.
1890..
1891..
1892
1893!!
1894..
1895. .
1896. .
1897..
1898..
1S99..
1900..
1901..
1902..
1903..
1904..
1905..
1906..
1907..
1908..
1909..
1910..
1911..
1912..
1913..

72
72
68
63
63
62
61
62
64
68
75
70
69
69
70
72
77
80
73
74
78
80
85
85

109
109
103
103
95
94
92
94
97
103
114
106
105
105
106
109
117
121
111
112
118
121
129
129

109
109
103
103
95
94
92
93
97
104
115
107
108
106
108
111
119
123
112
114
120
123
130
130

Bureau of Labor Statistics
index number
(old series).

B u­
reau’s
Dis- series on
crep- basis
aii1890eies. 1899=
100.

1
1
1
1
1
1
2
2
2
2
1
2
2
2
1
1

112.9
111.7
105.1
105.6
95.1
93.6
90.4
89.7
93.4
101.7
110.5
108.5
112.9
113.6
113.0
115. 9
122.5
129.5
122. 8
126.5
131.0
129.2
133.6
135.2

Chain
index
made
by
short
method.

-1 .1
- 5 .0
- .5
-9 .0
- 2 .6
- 3 .4
- .8
+ 4 .1
+ 8 .9
+ 8. 7
-1 .8
+ 4 .1
+ .6
- .5
+ 2 .6
+ 5. 7
+ 5.7
-5 .2
+ 3 .0
+ 4 .0
-1 .8
+ 3 .4
+ 1.2

Chain
index
Dis­
m ade crepan­
by
cies.
long
method.

- 0.2
- 4. 4
.2
- 8.7
— 1. 5
- 2.8
+ .2
+ 4.8
+ 10.4
+ 9.4
- 1.1
+ 4.6
+ 1.2
.1
+ 2.9
+ 5.8
+ 6.0
- 5.6
+ 3.2
+ 4.1
- 1.9
+ 3.4
+ 1.2

0.9
.6
.3
.3
1.1
.6
.4
.7
1. 5
.7
.7
.5
.6
.4
.3
.1
.3
.4
.2
.1
.1

Sauerbeck’ s index number,
1860-1891.

Year.

1860.
1861.
1862.
1863.
1864.
1865.
1866.
1867.
1868.
1869.
1870.
1871.
1872.
1873.
1874.
1875.
1876.
1877.
1878.
1879.
1880.
1881.
1882.
1883.
18.84.
1885.
1886.
1887.
1888.
1889.
1890.
1891.

Orig­ Shifted R e­
to
com ­
inal
DisI860
form, = 100, puted crepon
1867by
basis an1877=
short 1860= cies.
100.
method. 100.

99
98
101
103
105
101
102
100
99
98
96
100
109
111
102
96
95
94
87
83
88
85
84
82
76
72
69
68
70
72
72
72

100. 0
99.0
102. 0
104.0
106.1
102.0
103.0
101. 0
100.0
101. 0
99.0
97.0
101. 0
112.1
103. 0
97.0
96.0
95.0
87.9
83.8
88.9
85.9
84.9
82.8
76.8
72. 7
69.7
68.7
70.7
72.7
72.7
72.7

100.0
99.6
105. 5
109.3
112.3
105.8
106.5
103.9
103.1
101.9
100.3
102.6
112.5
116.6
107.0
100.3
97. 5
97.4
91.2
86.7
91.8
88.5
88.0
86.0
79.3
75.4
72.4
70. 7
73.9
76.7
76.0
75.4

0.6
3.5
5.3
6.2
3.8
3.5
2.9
3.1
2.9
3.3
1.6
2.4
4.5
4.0
3.3
1.5
2.4
3.3
2.9
2.9
2.6
3.1
3.2
2.5
2.7
2.7
2.0
3.2
4.0
3.3
2.7

Chain index numbers on the base, prices in the preceding year = 100,
have the advantage pointed out in Section III, that the variations
which they represent are highly concentrated and therefore apt for
averaging. That is, year-to-year variations are relatively easy to
measure with approximate accuracy. It is true that makers of
index numbers find chain relatives more troublesome to compute than
fixed-base series, since most of the prices used as divisors change
every year; but that fact weighs lightly with such laborious folk
in comparison with an improvement in their results. Why, then,
should they not make successive averages of year-to-year variations
covering as long a period as desired and weld the successive links
together by multiplication to form a continuous chain ?
For example, in Table 17 it is shown that the old Bureau of Labor
Statistics index in 1890 on the 1890-1899 base was 112.9 and that
prices fell 0.2 per cent in 1891. On multiplying, we get 112.9 X 0. 998




86

THE

M A K IN G

AN D

U S IN G

OF IN D E X N U M B E R S .

= 112.7. In 1892 the average change of prices was a fall of 4.4
per cent. 112.7x0.956 = 107.7. Once more, in 1893 prices fell
0.2 per cent on the average. Adding this new link to the chain, we
have 107.7x0.998 = 107.5. The next table shows this process
carried through to 1913. The result is a new index number covering
24 years, in which each successive step is taken by averaging rela­
tives which are probably better fitted for averaging, since they are
more highly concentrated, than the corresponding relatives on the
1890-1899 base. Is it not better than the old index on the fixed
base?
One may answer, first, that while each successive step in the chain
index may be taken with confidence, any errors which do inhere in
the steps are likely to accummulate. There is no magic in the
year-by-year computation which makes the final comparison be­
tween prices in 1913 and 1890 more reliable on the one basis than on
the other. Second, the interpretation of the final result is certainly
simpler in the case of the fixed-base than in the case of the chain
index. The figures say in the first case that between 1890 and 1913
there was an average net increase of prices equal to 22.3 per cent of
average prices in 1890-1899. The chain index says that there was an
increase between these two years of 37.1 per cent; but when one asks,
“ P ercent of w hat?” the answer is complicated. Third, the chain
index which was begun arbitrarily on a par with the fixed-base series
drifts away from it upward, and by the end of the period has opened
a gap of nearly 15 points, or more than 11 per cent— a notable dis­
crepancy. Stated in another way, the chain series makes the per­
centage increase in prices from 1890 to 1913 more than half again
as great as the fixed-base series makes it.
T a b l e 1 8 . —A F IX E D -B A S E I N D E X N U M B E R , A C H A IN I N D E X N U M B E R M A D E F R O M

T H E SAM E D A T A , A N D T H E C H AIN IN D E X M A D E IN T O A C O N T IN U O U S S E R IE S .
[Data from Bulletin N o. 149 of Bureau of Labor Statistics.]
(A rith m etic m eans.)

Year.

1890.................
1891.................
1892.................
1893.................
1894.................
1895.................
1896.................
1897.................
1898.................
1899.................
1900.................
1901.................

Bureau’s
index num ­
ber on basis
prices in
1890-1899=
100.
112.9
111.7
106.1
105.6
96.1
93.6
90.4
89.7
93.4
101.7
110.5
108.5

Chain in­
Chain in­
dex num ­
dex num­
ber, on basis ber made
prices in
into a con­
preceding
tinuous
year= 100.
series.

99.8
95.6
99.8
91.3
98.5
97.2
100.2
104.8
110.4
109.4
98.9

112.9
112.7
107.7
107.5
98.2
96.7
94.0
94.2
98.7
109.0
119.3
118.0

Year.

1902.................
1903.................
1804.................
1905.................
1906.................
1907.................
1908.................
1909.................
1910.................
1911.................
1912.
1913.................

Bureau’s
index num ­
ber on basis
prices in
1890-1899=
100.
112.9
113.6
113.0
115.9
122.5
129.5
122.8
126 5
131.6
129.2
133.6
135.2

Chain in­
Chain in­
dex num ­ dex num ­
ber, on basis ber made
prices in
into a con­
preceding
tinuous
year=100.
series.
104.6
101.2
99.9
102 9
105.8
106.0
94.4
103.2
104.1
98.1
103.4
101.2

123.4
124.9
124.8
128.4
135.9
144.1
136.0
140.3
146.1
143.3
148.2
150.0

W hy should the annual shifting of the base on which relatives are
computed make such a difference in the results? On looking at the
figures in Table 17 from which the continuous chain in Tame 18 is
forged, we see that when prices are falling the percentage of change
on the preceding-year base is generally smaller than the corresponding
change on the fixed base. On the contrary, when prices are rising




M E T H O D S U SED I N

M A K IN G

IN D E X N U M B E R S .

87

the preceding year base gives the larger percentage of change. In
two years the percentages are the same (1912 and 1913), and in two
other years the rule is reversed (1908 and 1911); but the rule holds in
19 cases out of 23.8 The problem is to account for the fact that chain
relatives usually rise more than fixed-base relatives when prices are
rising and fall less when prices are falling.
The following numerical examples give the clue to the solution.
We have in the first two columns of each example two relatives on
a fixed base, for two successive years. First the larger of the two
relatives is made to increase 25 per cent in the second year, and then
to fall 25 per cent in the second year, the smaller relative remaining
constant. Afterwards the smaller of the two relatives is made to
rise and then to fall by 25 per cent in the second year, the larger
relative being constant. In the third column the figures for the sec­
ond year are turned into chain relatives. Index numbers are com­
puted for both sets of relatives and the percentages of change on the
two bases are given.
1.
When a relative above the average of the relatives rises, its rise
makes a smaller percentage addition to the chain than to the fixedbase index.
Fixed base.
First year.

Preceding-year base—

Second year.

Second year.

210
160

300
160

125
100

2)400

2)160

2)225

200

230

Per cent of ch an ge........ +15

112.5
Per cent of ch an ge.. +12. 5

2.
When a relative above the average of the relatives falls, its fall
makes a smaller percentage subtraction from the chain than from the
fixed-base index.
Fixed base.
First year.

Preceding-year base—

Second year.

Second year.

240
160

180
160

75
100

2)400

2)340

2)175

200

170

Per cent of ch an ge....... —15

87.5
Per cent of change. . . —12.5

8 The fact was pointed out and the explanation of it suggested b y Professor F. R. Macaulay, in American
Econom ic Review, March, 1916, V oi. V I, pp. 297, 208.




THE M AKING AND USING OF INDEX NUMBERS.

88

3.
When a relative below the average of the relatives rises, its rise
makes a larger percentage addition to the chain than to the fixedbase index.
Fixed base.

Preceding-year base—

First year.

Second year.

240
160

240
200

100
125

2)400

2)440

2)225

200

220

Second year.

112.5
Per cent of ch an ge.. +12. 5

Per cent of change........+ 10

4.
When a relative below the average of the relatives falls, its fall
makes a larger percentage subtraction from the chain than from the
fixed-base index.
Fixed base.

Preceding-year base—

Second year.

Second year.

240
160

240
120

100
75

2)400

2)360

2)175

20-0

180

First year.

87.5
Per cent of ch an ge.. —12.5

Per cent of ch an ge........ —10

All that these figures show is that in certain cases the fluctuations
will be greater in the chain relatives and in other cases greater in the
fixed-base relatives. The vital point is, however, that cases 2 and 3
occur in price quotations much more frequently than cases 1 and 4.
Relative prices above the average seem more likely to fall than to rise
further; relative prices below the average seem more likely to rise than
to fall further. That is, the prices of individual commodities tend to
conform to the average movement, and when they have already di­
verged from this average they move back toward it more often than
they move away. These cases that occur more frequently than the
others are those that make the chain relatives rise more (case 3) or
fall less than the fixed-base relatives (case 2) .9
The net difference to be expected on this ground in a large body of
quotations between the movements of the relatives on the two bases
fi Of course this argument can be more generally, as well as more com pactly, stated in algebraic terms.
Prof. W . M. Ogburn contributes the following formulation:
L e tp 'i, p "i, . . . stand for relative prices of commodities during the first year, and
. . .
stand for relative prices of commodities during the second year. Let
be the number o f com m odities
and
the arithmetic mean of the relative prices during the first year.
The fixed-base index is obtained b y getting the average of the relative prices; the fixed-base index for
the first year is:

n

m
i

'Dl' + P l " +

• '

•

1
1
A n d that for the second year is:




& '+ & "+
n

• • •

p’z p"i,
.

METHODS USED IN M AKIN G INDEX NUMBERS,

89

is small in any one year. A glance at the figures in Table 18 will show
that the observed differences are generally less than 1 per cent. But
though small the differences are tolerably constant in direction, and
therefore when cumulated by multiplication they become significant
in 10 or 20 years.
The conclusion is that close agreement is not to be expected
between efforts to measure the change of prices between years far apart
when the measures are made first on a fixed base and then by the chain
method. The chain method is perfectly legitimate, of course, when
its results are carefully interpreted; but, as remarked above, the
interpretation is difficult to put into words. Where means permit it
is well to make from the original quotations two series of index num­
bers, one a chain index, the other a fixed-base series, and then to call
attention to the differences between the two.
The per cent increase, or the rise, is the ratio of the second to the first, or

p/ Pi'+P2n+
' Pi'+Pi"+

• • •
• • •

<0

Let the per cent increase from first year to second in the prices of individual commodities be r ', r " • • •
then the relation between the prices during the first and second year can be expressed b y the following
equations:
p 2'= p i '( f + r ')

P2"=Pi" 0 + r " ) , etc.
B y substitution in (1):

j t f V\V+r’)+Th"<J+r")+
Pir+ P i " +

r>f pi+ p \ "+

f

• • • +Pi'r'+pi"r” +

P i"
i’+P

jy Zpi+Spir
Spi
Putting

• • *

* • •

• • .

‘ 2px

pi— \ X where xi is the size of the relative, we have:
m= i
R f= i+

Z(mi+ii)r
SPi
T iS SZir
O r

Rf=l+

2pi= nmi

m r 2 zir _ 2r Zxr
{L
Rf=
1 + n x+ 2px 1 + n
m

(2)

The chain index is obtained b y averaging the ratios of the individual prices of commodities and is ex­
pressed in the following manner:

V l. P£
Rc^px'^pz'
n
Pi'(l+r')
R c~

&

Rc=*+*L..
n
b y subtracting (3) from

pi"(l+r")
Pi"

n

_ (l+ r Q + (l+ r " ) +

n

2r

• • •

(3 )

($
):
R R> 2xir
f—c 2Pi

In other words the fixed-base index num ber will not equal the chain index number unless 2 ziri= o
(which is true when r is constant). W hen 2ziri is negative the chain-index number will be larger and
when positive the fixed-base index will be larger. 2xiri is positive when z (the size of the relative) is
correlated (positively) with r (the percentage of increase), which is rarely if ever the case. The exact
difference can be measured b y




90

THE M A K IN G AND USING OF INDEX NUMBEBS.

Even this combination, however, is far from meeting all the needs
of users of index numbers. For certain users may require for special
purposes accurate measurements of price fluctuations in terms of the
price level in any given month or year, or any given stretch of time
in the whole period covered by the investigation. If such users are
few as compared with all the people who note or quote the popular
index numbers, they are precise^ the few most interested in price
fluctuations and most likely to increase knowledge by their use of the
figures. But of course compilers can not foresee what base periods
would serve best all these special purposes, and they can not be
expected to work out index numbers on all the bases made possible
by their original data. It is therefore highly desirable to have index
numbers that can be shifted from one base to another readily and
without involving difficulties of interpretation.
It is this desideratum, in large part, that has led to the recent
reaction against index numbers made by striking arithmetic means
of relative prices and in favor of index numbers made by adding
actual prices. For the latter form of index, being a sum of dollars
and cents, with an explicit scheme of weights, can be thrown into
the form of a series of relative prices on any base that is desired, with
slight labor and wdth no ambiguity. Geometric means, of course,
possess the same advantage.
Another problem in base periods has recently been developed by
Prof. Fisher. Should the period to which the weights refer be the
same as the period used as the base for computing relative prices,
or should the weights be taken from a different period? Suppose
that the index number is to be an arithmetic mean of relative prices
weighted by the values of the commodities exchanged in some year.
Then “ if the weights used are the values of th e base year (that is,
the base year for the relative prices) they impart a downward bias
to all the index number^ of any given year calculated thereby, while,
on the other hand, if the weights used are the values of the given
year itself, they impart an upward bias.7
1
To understand tnis effect one must note that the commodities
which have unusually high market prices in the base year will tend to
have both high values (prices multiplied by quantities) in that year
and low relative prices in other years. Vice versa, the commodities
which have unusually low market prices in the base year will tend to
have both low values in that year and high relatives in other years.
Then the multiplication of the low relatives by the high values and
of the high relatives by the low values will tend to reduce the index
numbers for all other years in comparison with the base year. Chang­
ing the weights from values in the base year to values in any other year
will tend to reverse these combinations. For commodities that have
unusually low market prices in the base year and therefore high rela­
tives in other years will tend to have higher values in the latter years,
and the commodities with high market prices in the base year and
low relatives in other years win tend to have lbwer values in the latter
years. The index number with “ given-year” weights will therefore
tend to combine high relatives heavily weighted and low relatives
lightly weighted, arid so give figures that run high for all other years
in comparison with the base year.
How considerable this “ biasing” of the results by the choice of the
period to which the weights refer will prove in practice depends upon




METHODS USED IN MAKING} INDEX NUMBERS.

91

whether the prices and quantities of commodities usually fluctuate in
the same or in opposite directions, for the influence of high and low prices
on the values as weights may be offset, or more than onset, by contrary
changes in the quantities. Little is positively known concerning
the run of these facts. Prof. Fisher believes, however, that the
quantity factor is almost as likely to influence the weights in one
direction as in the other. If so, the price factor has a fair field to in­
fluence the values used as weights and the above argument holds
good.
On this basis Prof. Fisher advises that in making arithmetic means
of relative prices the weights be taken from the base year, in order
that the downward bias of these weights may run counter to the
upward bias of the arithmetic mean (caused by the greater influence
exercised by high than by low relatives upon this form of average).
Harmonic means, on the contrary, have a downward bias (are more
influenced by low than by high relatives) and should therefore be
weighted by values taken from some other year than the base.
Geometric means, medians, and modes, which have no inherent bias,
he holds, should be weighted by values both in the base and in the
given year; for otherwise they will be affected by the bias of the
weights.1
0
9.

TH E “ ID E A L ” FORMULA.

A more complicated formula for making index numbers than those
heretofore discussed has recently been invented independently by
three high authorities and recommended as the best for making
general-purpose series. It may be written thus:
2 p ng n .

Spngo

where 2 indicates “ the sum of such terms as”
pn = the price of any commodity in a given year (or period).
3^ = the quantity of that commodity in the given year.
p0= the price of that commodity in the base year.
g0= the quantity of that commodity in the base year.1
1
To use this formula it is necessary to have data concerning the
prices and the quantities of every commodity in every year covered
by the index number. From these data four sets of aggregates of
actual prices multiplied by quantities are made for each year:
(1)
Prices in the given year times quantities in the given year, (2) the
same prices times quantities in the base year, (3) prices in the base
io Irving Fisher: “ The best form of index num ber.” Quarterly Publication of the American Statistical
Association, March, 1921, pp. 535, 536.
Prof. W . M. Persons has tested Prof. Fisher’s contention that a geometric mean weighted b y prices in the
base year will have a downward bias. He finds that “ Indices of quantity or of prices of agricultural prod­
ucts of the United States for the period 1880-1920 when measured relative to a fixed base (1910 in this case)
show the same general m ovem ent whether the Fisher m ethod or the geometric average is used . . .
no
cumulative divergence of the tw o indices is evident.” —Review of Econom ic Statistics, May, 1921, p. 111.
n Mr. Walsh mentioned this formula in a footnote in his Measurement s of General Exchange Value, 1901,
but did not then exploit its merits. In 1912 Prof. A . C. Pigou published the same formula in W ealtn and
Welfare (p. 46); but failed to note that the square root of the product should be extracted. This oversight
he remedied in his Econom ics of Welfare, 1920 (p . 78). In 1921 Prof. Irving Fisher having invented the for­
mula in his turn, presented it before the American Statistical Association. Meanwhile Mr. Walsh in review­
ing his earlier work had concluded that his footnote formula was perhaps the best of all. (See Quarterly
Publication of the American Statistical Association, March, 1921, pp. 536,539, and “ The Problem of Esti­
m ation,” p . 102.)
I have adopted Prof. Persons’s notation as clearer than that of the inventors.— Review of Econom ic
Statistics, May, 1921, p. 107, note.




92

THE M A K IN G AND USING OF INDEX NUMBEBS.

year times quantities in the given year, and (4) the latter prices
times quantities in the base year. Then the first and second aggre­
gates (prices in the given year weighted in two ways) are reduced to
relatives by dividing them respectively by the third and fourth
aggregates (prices in the base year weighted in the same two ways).
Finally these relatives are multiplied together and the square root
of their product extracted.
What advantages does this formula possess to compensate for the
great amount of labor it entails ?
Prof. Pigou uses it in an index of changes in the volume of “ real
income.” He finds it necessary to use weights for two periods be­
cause of “ The root fact . . . that in the first period our group
expends its purchasing power upon one collection of commodities,
and in the second period it expends it on a second and different col­
lection.” The change in real income can not be accurately measured
unless these alterations in the quantities of goods bought are repre­
sented in the index of prices used in reducing money income to real
income.1
2
Prof. Fisher wants this formula for use in his equation of exchange.
It serves admirably there, because an index number of prices made
by it when-multiplied by a similarly constructed index number of
quantities will show the changes in the total values of goods
exchanged.
Mr. Walsh’s purpose is more general, “ to measure variations in
the exchange value or purchasing power of m oney,” and his argu­
ment concerning its merits is more technical. The first of the two
ratios included in the formula is equivalent to an harmonic mean of
relative prices weighted by values in the given year, while the second
ratio is equivalent to an arithmetic mean of price relatives weighted
by values in the base year. B y using imaginary examples covering
four years, in which the last year has the same prices and quantities as
the first year, Mr. Walsh tests arithmetic and harmonic means
weighted in his way. He finds that they yield different results
which “ lie on opposite sides of the truth, and apparently equally
above and below it proportionately.” This result suggests the pro­
priety of taking the geometric mean between the two averages.
That step yields the “ ideal” formula. Mr. Walsh adds: “ Note that
it involves the arithmetic average, the harmonic average, the weight­
ings of the first and second periods, and the geometric mean. . . .
It
seems to contain everything that could be desired.” 1
3
We may agree with Prof. Pigou that thisiormula is well adapted to
use in a measure of change in real income and with Prof. Fisher that
it is well adapted to use m the equation of exchange. Can we agree
with Mr. Walsh that it is the best formula for making generalpurpose index numbers ?1
4
It the end in view is to compare the change in prices between any
two years, then this formula is more desirable than an aggregate of
actual prices weighted bv quantities in either year alone. That
u ;v u u t ii.
x v v u
proposition holds true of every year-to-year comparison however
h
far extended. Hence the “ ideal” formula is admirably adapted for
making chain index numbers, whenever it is possible to secure the
i* Economics of Welfare, p . 72.
w The Problem of Estimation, p. 102.
h Mr. W alsh is explicit upon this point.




(See The Problem of Estimation, p. 118.)

METHODS USED IN M A K IN G INDEX NUMBERS.

93

necessary annual data for quantities as well as prices and to meet
the necessary expense of computation.
But can the separate links in such a chain index be welded together
to make an equally admirable index covering long periods? Two
objections lie against it on this score. (1) The ideal formula changes
weights in each successive link in the chain. The quantities for 1920
and 1921 used in computing the link for that year are not likely to be
the same as the quantities for 1921 and 1922 used in computing the
latter link. As pointed out in section 6 above a change in the weights
makes it uncertain what part of the net result is due to price fluctua­
tions and what part to fluctuations in quantities. Whenever the
purpose in view requires that the price factor shall be isolated, it is
therefore undesirable to use the “ ideal” formula for any comparisons
except those between two specified years.1 (2) It has been shown
5
in section 8 that an arithmetic mean of relatives on the precedingyear base when forged into a continuous chain drifts away upward
!rom the corresponding fixed-base series made from the same data.
Now the ideal formula does not use relative prices, but is made from
aggregates of actuals which can not drift in this fashion, provided they
are made with constant weights. Does the annual change of weights
required by the “ ideal” formula introduce errors that cumulate and
so cause the chain index to part company from a fixed-base series?
Prof. Persons has answered that question by an actual trial. Taking
the prices and quantities of 12 leaaing crops of each year of the decade
of 1910-1919, he has made first for the quantities and second for the
prices two index numbers, one using the “ ideal” formula computed
directly to the fixed base 1910, another using the “ ideal” formula
chain fashion. Both of the chain indices are found to diverge from
their fixed-base mates by a distance that is rather wide considering
that the errors are cumulated for no more than nine years. The chain
index for quantities drifts upward and the chain index for prices
drifts downward . In both cases the discrepancies reach 4 per cent
in 1919.1 Hence the “ ideal” formula is ill-fitted for making index
6
numbers covering a long period of years, when it is applied in the way
which its logic strictly requires, namely, year-by:year comparisons.
And a fixed-base series made by this formula— that is, one m which
the index for each year is made by compounding the weights of that
year with some base year (instead of the year before) —yields accurate
comparisons only between the base year and any given year and not
comparisons that are accurate as between any two given years. If it
is desired to make possible comparisons between any years of a period
longer than two years aggregates of actual prices or geometric means,
both made with constant weights, seem better than the “ ideal”
formula, as well as far easier to compute.1
7
is This objection is reduced but not rem oved if the indices for each year are computed directly to a fixed
base, say 1913. Then the prices for the year 1920 would be weighted b y quantities in 1913 and 1950, the prices
in 1921 dv the quantities in 1913 and 1921, etc. The weights would still change, but not so much as m the
chain index.
ifl Review of Econom ic Statistics, May, 1921, p p . 113, 114.
1 Concerning the difference in labor of computing Prof. Persons gives an interesting note. The relative
7
times required to com pute the “ ideal ’ index numbers and the geometric means in his test of the tw o were as
follows:
Relative times
required.
Geometric means, constant weights................................................................................................. 27
“ Ideal” index number, fixed b ase................................................................................................... 51
“ Ideal” index number, chain series................................................................................................ 100
Of course the difference would b e m uch larger if the tim e were counted in that is spent in collecting yearly
data concerning quantities called for b y the “ ideal” formula. A sum of actual prices made with fixed
weights takes .still less tim e for com putation than a weighted geometric mean.




94

THE

M A K I N G A N D U S IN G OF IN D E X N U M B E R S .

V.— A COMPARISON OF THE LEADING AMERICAN INDEX
NUMBERS FOR THE YEARS 1890 TO 1918.
Many of the threads running through th e‘preceding sections can
be woven into a comparison of the best-known index numbers cur­
rently published in the United States— a comparison having intrinsic
interest of its own, as well as making a fitting summary of Part I of
this bulletin.
1. ANALYSIS OF THE SIMILARITIES AND DIFFERENCES BY YEARS, 1890
TO 1918.

Three general-purpose index numbers are available for the critical
study proposed, the latest form of the Bureau of Labor Statistics
series, Bradstreet’s index, and Dun’s index. It seems hardly worth
while to include in the comparison index numbers made solely of
the prices of foods, because they do not profess to measure changes
in the commodity markets at large. It has been shown that these
special indexes are not in close agreement with series containing not
only foods but also minerals, forest products, textiles, chemicals, etc.;
and that demonstration need not be repeated.1
8
The first step toward comparing index numbers is to throw them
into similar form and establish mem upon a common base. The
new series of the Bureau of Labor Statistics is a weighted sum of
actual prices, turned into relatives on the base, prices in 1913 = 100.
This series can be shifted to any base desired without appreciable
loss in accuracy. Dun’s and Bradstreet’s series are sums of actual
prices, and have no base of their own. Accordingly they may be
recast into relatives on the base, the average of the original figures
for 1890-1899 = 100. Dun’s figures for this decade average $84.32.
B y dividing the published figures by this sum and multiplying the
results by 100 we can make a new series strictly comparable with the
rest of our material. Shifting Bradstreet’s series is less satisfactory,
because it does not begin until 1892. The best that can be done is
to equate Bradstreet’s average for 1892-1899 with the average made
from the Bureau’s figures for these years— that is, to put $6.7785 =
97.1— and then to apply the rule of three.1
9
These three series in comparable form are assembled in Table 19.2
0
is See subdivision 5, “ The numbers and kinds o f com m odities in clu d e d /' especially pp. 52-56.
1 N o violence is done b y this procedure to Bradstreet’s series; but the eomparision is not quite satis­
9
factory, because our other series were not worked out on the basis, prices in 1892-1899=97.1, and would prob­
ably have shown slightly different results if they had been.
20 The annual aver ages, made from the original figures published b y Dun and Bradstreet’s, ran as follows:
Year.
1890.
1891.
1892.
1893.
1894.
1895.
1898.
1897.
1898.
1899.
1800.
1901.
1902.
1903.
1904.
1905.
1906.




D u n’s.
$90.9
92.2
90.0
92.4
84.7
81.3
76.0
74.0
78.9
82.8
93.4
95.9
100.4
99.0
100.2
100.6
105.3

Brad­
street’s.

$7.78
7.53
6.68
6.43
5.91
6.12
6.57
7.21
7.88
7.57
7.88
7.94
7.92
8.10
8.42

Year.
1907..............................................
1908..............................................
1909..............................................
1910..............................................
1911..............................................
1912..............................................
1913..............................................
1914..............................................
1915..............................................
1916..............................................
1917..............................................
1918..............................................
Averages:
1890-1899..............................
1900-1909.............................
1910-1914..............................
1915-1918..............................

a Average of 1892-1899.

D u n’s.

Brad­
street’s.

$111.8
109.9
117.8
119.2
116.8
124.4
120.9
122.2
128.4
148.8
204.1
229.2

$8.90
8.01
8.52
8.99
8.71
9.19
9.21
8.90
9.85
11.83
15.66
18.73

84.3
103.4
120.7
177.1

« 6.78
8.11
9.00
14.02

95

COMPARISON OF LEADING AMERICAN INDEX NUMBERS.

The second and third divisions of the table bring out certain dif­
ferences among the figures, and the summaries in the latter part
show the average or net movements in various periods.
T able

1 9 .— A

C O M P A R IS O N

O F T H E C H IE F A M E R IC A N
Y E A R S 1890 T O 1918.

The three index num ­
bers shifted to the
1890-1899 base.

Year

IN D E X

Percentage differences
among the threo in­
dex numbers.

Bradstreet’s
greater
( + ) or
Bureau
of
less ( —)
BradLabor
than
street’s. Statis­ D u n’s.
Bureau
tics.
of
Labor
Statis­
tics.

NUM BERS FOR

THE

Percentage b y which
each of the three index
numbers rose ( + ) or
fell ( —) each year.

D u n ’s
greater Brad( + ) or
street’s
Bureau
lass (—> greater
of
than
Brad( + ) or
street’s. Labor D u n’s.
Bureau
less ( —)
Statis­
of
tics.
Labor i than
Statis­ D u n’s.
tics.

P e r i o d o f d eclin e.

1890.......................................
1891.......................................
1892.......................................
1893.......................................
1894.......................................
1895.......................................
1896.......................................

I ll
108
96
92
85

Ill
111
103
106
95
95
90

108
109
107
110
100
96
90

88
94
103
113
108
113
114
113
116
121
127
115
122
129
125
132
132
128
141

91
95
101
109
108
116
117
117
117
121
128
125
133
136
130
138
137
136
139

88
94
98
111
114
119
117
119
119
125
133
130
140
141
139
148
143
145
150

- 3 .3
- 1 .1
+ 2 .0
+ 3. 7
±0
-2 .6
-2 .6
-3 .4
- .9
±0
- .8
-8 .0
-8 .3
- 5 .1
-3 .8
-4 .3
-3 .6

169
224
268

170
241
269

176
242
272

-

+ 7 .8
+ 1.9
+ 1.1
- 3 .2
-5 .6

—2.7
—1.8
+ 3 .9
+ 3 .8
+ 5.3
+ 1.1
±0

±

+
-

3.7
1.8
4.0
4.2
5.6

6

- 2.7
-1 1 .1
- 4.2
- 7.6

— 7.2
+ 2.9
-1 0 .4
± 0
- 5.3

+
+
+
-

.9
1.8
2.8
9.1
4.0
6.3

+
+
+
+
+
+

+
+
+
+
+
+

LI
4.4
6.3
7.9
.9
7.4
.9

+
+
+
+
+
-

±
±
+
+

o

+

.7
.7
2 .2

2.2
6.8
4.3
13.3
2^7
4.4
1.7
+ 1.7
± 0
+ 5.0
+ 6.4
- 2.3
+ 7.7
+
.7
- 1.4
+ 6.5
- 3.4
4 -1 .4
+ 3.4

+22.3
+41.8
+1L6

+ 17.3
+37 5
+ 12! 4

P e r i o d o f gra du a l r ise .

1897.......................................
1898.......................................
1899.......................................
1900........ : .............................
1901.......................................
1902.......................................
1903.......................................
1904.......................................
1905................................ .......
1906.......................................
1907.......................................
1908.......................................
1909.......................................
1910.......................................
1911.......................................
1912.......................................
1913........ . ...............................
1914.......................................
1915.......................................
P erio d

-3 .3
-1 .1
-3 .0
+ 1.8
+ 5.6
+ 2 .6
±0
+ 1.7
+ 1 .7
+ 3 .3
+ 3 .9
+ 4 .0
+ 5 .3
+ 3 .7
+ 6 .9
+ 7 .2

- 5 .9

+ 4.4
+ 6 .6

+ 1 .4

+ 7 .9

0
0
5.1
1.8
5.3
5.0
2.6
5.0
2.5
- 3.2
- 4.5
- 1 1 .5
- 1 2 .9
- 8.5
-1 0 .1
- 1 0 .8
±
±

+
+
-

-

7 .7

- 1 1 .7
- 6.0

+
+
+

+
+

+
±
+

3.5
6.8
9.6
9.7
4.4
4.6
.9
.9
2.7
4.3
5.0
9.4
6.1
5.7
3.1
5.6
0
3.0
10.2

0
3.4
5.8
- 2.3
+ 6.4
+ 2.3
- 4.4
+ 6.2

o f accelerated r is e
d u e to w a r .

1916.......................................
1917.......................................
1918.......................................




.6

—7.1
- .4

+ 3 .5
+
+

.4

1.1

—
-

4.0
7.4
1.5

+ 19.9
+32. 5
+ 19.6

96
T

TH E M A K IN G AND USING OF INDEX NUMBERS.

a b l e

1 9 .—

A C O M P A R IS O N O F T H E C H IE F A M E R IC A N I N D E X N U M B E R S
Y E A R S 1890 TO 1018—Concluded.

The three index num ­
bers shifted to the
1890-1899 base.

Item .
Bureau
of
B radstreet’ s. Labor
Sta­
tistics.

Averages b y 5 - y e a r
periods:
1890-1894.......................
1895-1899.......................
1900-1904.......................
1905-1909.......................
1910-1314.......................
1915-1918.......................
Averages b y 1 0 - y e a r
periods:
1890-1899.......................
1900-1909.......................
1910-1918.......................
Maxima and m inima:
1890-1914—
M axim a.................
Minima..................
Differences............
1914-1918—
M axim a.................
Minima..................
Differences............
Net rise ( + ) or fall ( —):
1.890-1896.......................
1896-1907.......................
1907-1908.......................
1908-1914.......................
1914-1918.......................
Algebraic averages:
1890-1894.......................
1895-1899.......................
1900-1904......................
1905-1909.......................
1910-1914.......................
1915-1918.......................
1890-1914......................
1890^1918.....................

Percent age variation s
among the three in­
dex numbers.

Bradstreet’ s
com ­
pared
with
Dun’s.
Bureau
of
Labor
Sta­
tistics.

FOR THE

Percentage variations of
the yearly rise and fall
of each of the three
in d ex numbers.

D u n’ s
com ­
Bradpared
Bureau
with street’ s
of
com ­
BradBureau pared
street’ s. Labor Dun’ s.
of
Sta­
with
Labor
tistics.
D un’ s.
Sta­
tistics.

105
92
112
120
129
201

105
94
113
125
135
205

107
93
116
129
143
210

3.6
3.0
2.5
3.6
9.5
2.4

3.5
1.7
2.3
3. 6
5.8
3.2

3.2
3.0
3.9
6.9
10.0
4.7

6.9
6.3
4.1
5.5
3.5
20.6

5.1
3.4
3.4
3.6
2.9
19.5

3.7
4.7
4.8
4.3
2.7
.17.7

(100)
116
161

100
119
166

100
123
173

3.3
3.0
3.6

2.6
3.0
4.6

3.1
5.4
7.5

6. 5
4.8
11.1

4.2
3.5
10.2

4.2
4.5
9.3

132
85
47

138
90
48

148
88
60

8.3
0
8.3

7.2
0
7.2

12.9
0
12.9

11.1
0
11.1

10.4
0
10.4

13.3

268
128
140

269
136
133

272
145
127

7.1
.4
6.7

7.9
. A
7.5

11.7
1.5
10.2

32.5
3.0
29.5

41.8
.7
41.1

37.5
1.4
36.1

+ 3 .6
- 2 .2
—1.0
-3 .6
-4 .5
—1.7

+ 1.7
—1. 3
+2. 3
+ 3 .6
+ 5. 8
+ 3 .2

.7
— .9
— 3.2
— 6.9
-1 0 .0
— 4.7

— 6.9
+ 1.6
+ 2.0
+ 1.7
+ 1.0
+20.6

— 3.7
+ 1.3
+ 3.1
+ 2.7
+ .5
+ 19. 5

— 1.8
— .3
+ 4.1
+ 3.4
+ .8
+17.7

-2 .0
-2 .0

+ 2 .4
+ 2 .5

-

+ .8
+ 3.9

+ 1.0
+ 3.6

+ 1.4
+ 3.7

- 26
+ 42
- 12
+ 13
-1-140

+
+
+

21
38
3
11
133

+
+
+

.i3.3
0

18
43
3
15
127

4.6
4.6

A cursory examination of this table, or a glance at Chart 12, shows
that these index numbers made by three independent organizations
have a marked family resemblance. They all agree that prices fell
heavily in 1890-1896, rose still more sharply in 1896-1900, wavered
uncertainly in 1901-1904, rose rapidly again in 1905-1907, fell in
1908, more than recovered their lost ground in 1909-1910, dropped
back in 1911, rose to a new high record in 1912, receded somewhat
in 1912-1914, and finally shot up at an extraordinary rate during the
war. Further, the three index numbers agree that the general level
about which the yearly oscillations clustered was higher in 1910-1914
than in 1900-1909, and higher in 1900-1909 than in 1890-1899.
About the major facts of pricejiistory, in short, the testimony of the
leading American index numbers is unanimous.
On the other hand, Table 19 shows that the series differ in many
details. For example, not once in the 29 years covered by the pres­
ent record are all three index numbers identical, and in only six years




Ch

t311739 0 —41

a rt

1 2 .— IN D E X

(T o face page 96.)




N U M B E R S O F TH E B U R E A U O F L A B O R STATISTICS, DU N , A N D B R A D S T R E E T , 1890 TO 1918.

COMPARISON OF LEADING AMERICAN INDEX NUM BERS.

97

are any two indexes precisely the same. On the average of the whole
eriod the Bureau of Labor Statistics series varies from Bradstreet’s
y 3.3 per cent, from Dun’s by 3.4 per cent, while Bradstreet’s index
varies from Dun’s by 5.4 per cent. The maximum differences in anv
one year rise to 8.3 per cent between the bureau’s index and Bradstreet’s (1909), 7.9 per cent between the bureau’s and Dun’s (1915),
and 12.9 per cent between Dun’s and Bradstreet’s (1909). Concern­
ing the direction in which prices move from one year to the next, the
bureau’s series contradicts Bradstreet’s in one year (1893) and Dun’s
series in four years, while Dun’s and Bradstreet’s indexes contradict
each other in six years. If we ~'imt cases in which one index re­
mains the same for two successive years while another series rises or
falls, we find four years of partial contradiction when we compare
the bureau’s index with Bradstreet’s, three years when we compare
the bureau’s index with Dun’s, and two years when we set Brad­
street’s against Dun’s. In general, the bureau’s index steers a middle
course between the other two, averaging 2 per cent higher than Brad­
street’s and 2.5 per eent lower than Dun’s, while the margin by which
Dun’s index exceeds Bradstreet’s averages 4.6 per cent.3
1
Most of the detailed differences among the annual figures of the
three index numbers may be regarded as resulting from differences
in respect to (1) secular trend and (2) degree of variability from one
year to the next.
1.
Chart 12 and the averages by decades in Table 19 show that on
the whole Dun’s index number has risen more than the bureau’s,
and the bureau’s more than Bradstreet’s. This long-period shifting
of the level about which the monthly and yearlv oscillations occur
is technically called the secular trend. Graphically it may be repre­
sented bv a straight line. Two turning points occur in the 29 years
covered oy the table. The great fall of prices which began in 1873
ended in 1896 or 1897, and a rise began. In 1915 the rate of this
rise was violently accelerated by the war,* so that the slope of the
straight line representing the direction of the secular trend was sud­
denly made steeper. Of the three periods marked off by these turn­
ing points in the first half of Table 19, the middle one, 1896-1914,
alone is long enough to make the computation of the secular trend
significant.
The secular trends of the three index numbers during this period of
19 years, given in Table 20, are represented collectively on Chart 13
ana are shown separately with their respective curves on Charts 14,
15, and 16. They are summarized in the following table:

E

T a b l e 2 0 . — SE C U L A R T R E N D S O F I N D E X N U M B E R S O F B U R E A U OF L A B O R
TIST IC S, B R A D S T R E E T , A N D D U N , 1896 TO 1914—S U M M A R Y .

Index numbers.

Annual
geometric
increment
of secular
trend in
1896-1914.

Geometric
mean in
1896-1914.

Ratio of
annual
increment
to geome­
tric mean
(per cent).

Terminal
points of the
straight line
representing
the secular
trend.
1896

BradstreeUs...................................................
Bureau of Labor Statistics.........................
Dun’ s...............................................................

1.0230
1.0232
1.0269

113.7
117.1
120.3

0.90
.87
.85

Net per
cent of rise
in lines of
secular
trend,
1896-1914.

1914

92.7
95.3
94.7

139.6
144.0
152.6

« These averages are made, of course, from algebraic sums of the yearly percentage differences.
+311739 0 —41------ 7




STA­

15.1
15.1
16.1

THE MAKING AND USING OF INDEX NUMBERS.

It is primarily these differences in secular trend that make the
bureau’s index number follow a course intermediate between Bradstreet’s and Bun’s indexes.
C hart 1 3.—S E C U LA R T R E N D S O F I N D E X N U M B E R S O F B U R E A U O F L A B O R S T A T IS ­
TICS, D U N , A N D B R A D S T R E E T , 1896-191L




COMPARISON OF LEADING AMERICAN INDEX NUM BERS.

99

C A T 14.—INDEX NUMBERS OF BRADSTREET, COMPARED WITH THEIR SECULAR
HR
TREND, 1890-1914.




1 0 0
C

h a r t

TH E M A K IN G AND USING OF INDEX NUMBERS.
15.—INDEX NUMBERS OF BUREAU OF LABOR STATISTICS, COMPARED WITH
THEIR SECULAR TREND, 1896-1914.




(Based on Table 21.)

COMPARISON OF LEADING AMERICAN INDEX NUM BERS.
C

h a b t

101

16.—INDEX NUMBERS OF DUN, COMPARED WITH THEIR SECULAR TREND.
1896-1914.




(Based on Table 21.)

102

THE M A K IN G AND USING OF IN DEX NUM BERS.

Table S}1.—INDEX, NUMBERS OP BRADSTREET, THE BUREAU OF LABOR STATISTICS,
AND DUN* COMPARED WITH THEIR SECULAR TRENDS, BY YEARS, 1896 TO 1914.
Bradstreet’s.

Bureau of Labor Statistics.

Excess of—
Year.

1896.............
1897.............
1898.............
1899.............
1900.............
1901.............
1902.............
1903.............
1904.............
1905.............
1906.............
1907.............
1908.............
1909.............
1910.............
1911.............
1912.............
1913.............
1914.............

Secu­ Index Secular
lar num­ trend
trend. ber. over
index
num­
ber.

92.7
94.9
97.1
99.3
101.6
103.9
106.3
187
Q.
111.2
113.7
116.3
119.0
121.8
124.6
127.0
130.4
133. 4
136.4
139.6

85
88
94
103
113
108
113
114
113
116
121
127
115
122
129
125
132
132
128

5.9
2.1
4.3
1.0
3.3
9.0

Excess of—

Excess of—

Index Secu­ Index Secular Index Secu­ Didex Secular Index
lar num­ trend num­
lar num­ trend num­
num­
ber trend. ber.
ber trend. ber. over
over
ber
index over
index over
over
secular
num­ secular
num­ secular
ber. trend.
trend.
ber.
trend.

P ct. P ct.
er
er
9.1
7.8
3.3

Dun’s.

3.7
11.3
3.9
6.3
4.8
1.6
2.0
4.0
6.7
1.2

95.3
97.5
99.7
102.0
104. 4
106.8
109.3
111. 9
114.5
117.1
119.8
122.6
125.5
128.4
131.3
134.4
137.5
140. 7
144.0

90
91
95
101
109
108
116
117
117
117
121
128
125
133
136
130
138
137
136

P ct. P ct.
er
er
5.8
7.1
5.0
1.0

.1
.4
3.4
2.7
5.9

4.4
1.1
6.1
4.6
2.2
1.0
4.4
3.6
3.5
.4

94.7
97.3
99; 9
102.5
105.3
108.1
111,0
114.0
117.0
120.3
123.5
126.8
130; 2
133.7
137.3
141.0
144.8
148.6
152.6

90
88
04
98
111
114
119
117
119
119
125
133
130
140
141
139
148
143
145

P ct. P ct.
er
er
5,2
10; 6
6.3
4,6

1.1
.2
1. 4
3.9
5.2

5. 4
5,5
7,2
2,6
1.7
1.2
4.9
4; 7
2.7
2.2

2.
While steadier oyer a considerable period of time, Bradstreet’s
index changes more from one year to the next than does either the
bureau’s or Dun’s series. Dun’s index, further, is more variable than
the bureau’s.
Several different ways of measuring year-to-year variations all sup­
port this conclusion: (1) If the “ percentage by which each of the
three index numbers rose or fell each year” as snown in Table 19 be
averaged from 1892 to 1914, the results are Bradstreet’s 5.15 per
cent, Dun’s 4.37 per cent, and the Bureau of Labor Statistics’s 3.71
per cent. (2) The standard deviations of these annual percentages
of rise and fall are, Bradstreet’s 5.79, Dun’s 5.06, and the bureau’s
4.46. (3) If the figures showing the excess of the secular trend over
the index number or the excess of the index number over the secular
trend in Table 21 be averaged for 1896-1914, the results are, Brad­
street’s 4.0 per cent, Dun’s 4.0 per cent, the bureau’s 3.3 per cent.
(4) If the yearly deviations from the secular trend are plotted as in
Chart 17, it appears that Bradstreet’s fluctuates through the widest
and the bureau’s series through the narrowest range, Dun’s index
being intermediate.
To show that these index numbers differ in detail, however, means
little. The significant problem is whether these differences are due to
the inherent difficulty of measuring changes in the price level, to the
crudity of the method of measurement in vogue, or to technical dif­
ferences in the construction of the particular index numbers in
question.
Unfortunately it is not possible to attack this problem effectively
on the lines of analysis suggested in the preceding sections. For the
compilers of Bradstreet’s and Dun’s index numbers do not give suffi-




COMPARISON OF liEABING AMERICAN INDEX NUM BERS.
C

h a r t

103

17.—YEARLY DEVIATIONS FROM SECULAR TREND OF INDEX NUMBERS OF
BUREAU OF LABOR STATISTICS, DUN, AND BRADSTREET, 1896-1911.




(Based on Table 21.)

104

TH E M A K IN G AND USING OF INDEX NUM BERS.

cient data concerning the sources of information drawn upon for
quotations, the commodities included and the weights employed for
each commodity to make possible a close comparison with the bu­
reau’s series. Bradstreet’s publishes quotations for 106 commodities,
but bases its index number on the prices of 96, and does not say
which 10 are omitted. Its prices per pound, which are added up to
give the index number, were published for a short time in 1897, but
are not disclosed in recent years. Dun’s Review does not publish its
list of commodities, to say nothing of their prices, and explains
merely that it weights by per capita consumption, allowing 50 per
cent of the total for foods, 18 per cent for textiles, 16 per cent for
minerals, and 16 per cent for other commodities.2 With such scanty
2
information about these two series, statements concerning the rea­
sons for the relatively slight differences between each of them and the
bureau’s index number would be subject to a relatively wide margin
of error.2
3
After all, the important fact is that the three index numbers agree
with one another very closely. The divergencies which do appear
are smaller than those which result from most attempts to measure
economic quantities. For example, two sets of experts employed
upon a valuation case are likely to arrive at results farther apart than
the maximum differences shown in Table 19. Again it is doubtful
whether the margin of error in the average balance sheets of business
enterprises, or in cost computations is as narrow as the average mar­
gin between Bradstreet’s and Dun’s index numbers, to say nothing
of the narrower margins between the official series and either of these
commercial indexes.
To sum up the comparisons in the most definite form the coefficient
of correlation must be used. This coefficient is the standard statis­
tical device for measuring the degree of agreement or difference be­
tween twro variables. Its extreme limits are —1.0 and -f 1.0, the
latter expressing perfect agreement.2 When such coefficients are
4
computed for the annual index numbers in 1892-1914, inclusive, the
following results are obtained:
Coefficients
of correlation.
Bureau of Labor Statistics index number and Bradstreet’s................................... + 0 . 964
Bureau ctf Labor Statistics index number and D un’s ............................................. -j- •992
Bradstreet’s index number and D un’s........................................................................... 4- . 959

High coefficients of correlation are to be expected, of course, when
the variables compared are different measurements of the same
quantity— in this case the general level of wholesale prices through a
period of 23 years. To get such high coefficients as the preceding
indicates that the measurements made by different hands are in close
agreement and therefore presumably reliable.
A severer test may be applied by computing the coefficients of cor­
relation between the percentage changes in the three index num** Compare I. P. Norton’s article in the Quarterly Journal of Economics, Aug., 1910, Vol. XXIV,p. 754.
8 Most of the analytic comparisons among various American index numbers in Bulletin No. 173
3
dealt with series much more perfectly known than Dun’s or Bradstreet’s. The reader who turns
back to that discussion will probably share the writer’s belief that were all the necessary data available,
the differences among the three series dealt with would be found to result primarily from differences in
the lists of commodities and in the systems of weighting. But that belief will remain a mere probability
so long as the construction of Bradstreet’s and Dun’s indexes is not fully disclosed.
u Most statistical text books explain the method of computing the coefficient of correlation in, detail.
See for example, G. Udney Yule, Introduction to the Theory of Statistics, 2d edition, 1912, chs. IX
and X.




COMPARISON OF LEADING AMERICAN INDEX NUM BERS.

bers from one year to the next.
follows:

105

The results of this operation are as

Coefficients
of correlation.
Bureau of Labor Statistics index number and Bradstreet’s ................................... -j-0 .882
Bureau of Labor Statistics index number and D un’s............................................... -j- . 873
Bradstreet’s index number and D un’s........................................................................... -j- •788

Here the coefficients, though less than in the preceding case, are
still high. Bradstreet's agrees a bit better with the bureau's series
than does Dun's, whereas in the former comparison Dun's had dis­
tinctly the higher correlation. In both comparisons, the bureau's
series makes the best showing. Other things being equal, among
different measures of a given quantity, that measure has the best
claim to acceptance which is nearest the mean of all the measures.
In the present case, however, other things are not equal. The bu­
reaus's series includes more commodities than either of its rivals, its
system of weighting is better, and its method of construction from
start to finish is disclosed with a fullness which justifies confidence.
On these grounds its superiority is clear. The fact that it agrees bet­
ter with both the commercial indexes than they agree with each other
merely confirms the choice which would be made on a priori grounds.
2. C O M PARISO N OF FOUR LEADING AM ERICAN IN D EX NUM BERS, BY
M O N T H S , JULY, 1914, T O D ECEM BER, 1918.

The peculiar interest attaching to the revolution in prices during
the World War makes desirable a more detailed comparison of the
leading American index numbers in 1914-1918. For this period, there
are available besides the three series discussed in the preceding section,
the index number compiled by the Price Section of the War Industries
Board.
Table 22 and chart 18 present the four series on a common base—
namely,,, average prices in the twelve months preceding the outbreak
of war (July, 1913-June, 1914) = 100, giving by months first the index
numbers themselves, and then the percentage by which each of the
four index numbers rose or fell as compared with the month preceding.
Study of the table and of the chart based upon it shows at once a
closeness of agreement for which even the results of the preceding
comparison scarcely prepare one. And this impression of close agree­
ment is abundantly justified when the coefficients of correlation are
worked out. These coefficients, shown on page 108, approach even
more closely to the limit of perfect agreement (4-1.0) than the remark­
ably high coefficients we have found for the yearly index numbers in
times of peace.




106
T ABL E

TH E M A K IN G AND U SIN G OF INDEX N UM BEBS.
33.—A COMPARISON OF FOUR LEADING AMERICAN INDEX NUMBEBS, BY
MONTHS, JULY, 1914, TO DECEMBER, 1918.
The four index numbers shifted
to the base, July, 1913-June,
1914=100.
Year and month.

1914.

Percentage by whioh each of the
four index numbers rose (+) or
fell (—) each month.

War Bureau
War Bureau
of
of
Indus­ Labor Brad- Dun’s. Indus­ Labor Brad- Dun’s.
tries Statis­ street’s.
tries Statis­ street’s.
Board. tics.
Board. tics.

97
101
101
99
98
98

99
103
103
99
98
97

97
103
106
101
100
102

99
102
103
102
102
102

±0.
+4.1
±0
-2 .0
-1 .0
±0

+0.8
+3.2
+ .5
-4 .2
- .9
- .8

* +0.5
+6.3
+2.9
-4 .7
-1 .2
+1.6

-0 .1
+3.0
+1.1
-L I
+ .3
- .1

100
100
100
100
100
100
102
102
102
104
107
111

98
100
99
100
101
99
101
100
99
101
103
106

105
108
109
110
109
110
110
110
111
114
118
120

103
103
103
104
104
103
103
103
103
106
108
111

+2.0
±6
+0
±0
±0
±6
+2.0
±0
±0
+2.0
+2.9

+1.1
+1.7
-L 0

+3 .5

+0.6
±0

January...................... ....................
February.........................................
March..............................................
April...................................... : ........

115
118
121
123

June................................................
July.................................................
August............................................
September.......................................
October...........................................
November.................. ................. .
December........................................

122
123
127
132
141
144

110
111
114
116
118
118
119
123
•127
134
143
146

123
126
129
132
131
130
129
130
133
139
148
153

148
151
156
170
178
183
189
187
186
182
183
182

150
155
160
172
181
185
185
185
182
180
183
182

185
187
188
191

186
187
188
191
191
193
199
203
207
204
207
207

July.................................................
August............................................
September.......................................
October...........................................
November.......................................
December........................................
1915.

January.. . . .....................................
February.........................................
March..............................................
April.............................................. :
May.................................................
June............................. ..................
July.................................................
August.............. ..............................
September..................................... .
October..................................... .
November.......................................
December........................................
1916.

M ay.........................................................

1917.

January...........................................
February.........................................
March..............................................
April...............................................
May.................................................
June................................................
July-...............................................
August............................................
September.......................................
October...........................................
November.......................................
December...... .................................
1918.
January...........................................
February.........................................
March..............................................
April................................................
May.................................................
June................................................
July.................................................
August............................................
September.......................................
October...........................................
November.......................................
December........................................




123

125

190

189
193
196
201
201
201
203

+3.7

+1.0
-1 .6
+1.8
- .7
-1 .3
+2.4
+1.5
+2.9

+2.5
+ .6
+ .9
- .2
+ .4
+ .4
- .3
+ .8
+2.9
+13
+2.6

- .4
- .1
+ .6
+2.3
+2.5
+2.7

115
117
118
120
120
119
119
122
125
130
137
139

+3.6
+2.6
+2.5
+1.7
±0
- .8
+ .8
+1.6
+1.6
+3.9
+6.8
+2.1

+4*2
+L2
+2.3
+2.0
+L 6
+ .4
+ .4
+3.5
+3.3
+5.0
+7.2
+2.0

+2.3
+2.1
+2.8
+1.6
— .3
- .9
-1 .1
+1.1
+2.6
+4.3
+6.5
+3.5

+3.4
+ 1.6
+1.2
+1.4
—
- .4
- .5
+2.4
+2.8
+4.2
+5.0
+1.4

155
157
161
166
171
176
182
185
188
190
194
199

142
149
155
164
173
175
177
178
179
181
181
182

+2.8
+2.0
+3.3
+9.0
+4.7
+2.8
+3.3
-1 .1
- .5
-2 .2
+ .5
- .5

+2.8
+3.4
+3.2
+7.2
+5.4
+2.0
+ .4
- .4
-1 .4
-1 .1
+1.4
- .5 :

+1.0
+1.5
+2.3
+3.4
+3.0
+3.1
+3.0
+1.8
+1.6
+1.3
+2.0
+2.6

+2.4
+4.8
+3.8
+5.9
+5.7
+ .8
+1.5
+ .7
+ .2
+1.3
+ .1
+ .3

202
202
204
209
212
214
215
214
213
212
212
210

190
187
188
188
186
188
191
191
192
191
190
189

+1.6
+1.1
+ .5
+1.6
- .5
- .5
+2.1
+1.5
+2.5
±0
±0
+ 1.0

+2.2 '
+ .7
+ .4
+1.9
- .1
+1.0
+2.9
+2.2
+2.1
-1 .5
+ 1.3
±0

+1.4
+ .3
+1.1
+2.3
+1.5
+ .7
+ .6
- .6
- .3
- .4
+ .1
-1 .0

+4.3
-1 .4
+ .7
- .3
-1 .2
+ 1.3
+1.6
+ .1
+ .3
- .5
- .6
- .1

+ .7

-

.2

+L0
+ .4
- .7

COMPARISON OF LEADING AMERICAN INDEX NUM BERS.

107

Since both Br ads tree t’s and Dun’s index numbers are computed
from prices as of the first of the month while the Bureau of Labor
Statistics and War Industries Board use average prices for the month
or prices at various dates within the month, it is not quite accurate
to compute coefficients of correlation from the figures as they stand

after shifting to a common base. To overcome this difficulty as well
as may be, new monthly figures for Dun’s and Bradstreet’s have
been made by averaging^ the index for July and August to get a new
figure for July, then averaging the figures for August and September
to get a new figure for August, and so on.




108

TH E M AKING AND U SING OF INDEX NUM BERS.

Coefficients o f correlation among four American index numbers in the 54 months July,
1914s to December, 1 9 1 8 .
A . Coefficients of correlation computed from the monthly index numbers.

(1)
(2)
(3)
(4)
(5)
(6)

Coefficients of
correlation.
Bureau of Labor Statistics’ and War Industries Board’s series.................... + 0 . 997
Bureau of Labor Statistics’ and Bradstreet’s series........................................... + •
988
Bureau of Labor Statistics’ and D un’s series....................................................... -j- .994
War Industries Board’s and Bradstreet’s series............................................... + .986
War Industries Board’s and Dun’s series.............................................................. -}- . 995
Bradstreet’s and Dun’s series..................................................................................... -j- •991

B . Coefficients of correlation computed from thepercentage change in prices
one month to the next.
(1)
(2)
(31
(4)
(5)
(6)

Bureau of Labor Statistics’ and War Industries Board’s series...................
Bureau of Labor Statistics’ and Bradstreet’s series...........................................
Bureau of Labor Statistics’ and Dun’s series.......................................................
War Industries Board’s and Bradstreet’s series........................... .......................
War Industries Board’s and Dun’s series..............................................................
Bradstreet’s and Dun’s series.....................................................................................

from

-fO. 866
4- •633
-f •
801
+ •
640
+ •761
-j- •616

Taking both sets of coefficients into account, we find that the
Bureau of Labor Statistics’ index number has the closest agreement
with the other three series. Then, in order, come the War Industries
Board’s series, Dun’s, and Bradstreet’s— which is the most divergent
of the four. But there is a better test of reliability. In view of its
very comprehensive list of commodities (1,366 in number) and its
use of class in addition to commodity weights, the War Industries
Board’s series is probably the nearest approximation to a true measure­
ment of the changes in the wholesale price level during the war.
Accepting it as the standard, we may ask which of the three index
numbers currently published is in closest agreement with it. Once
more the answer is in favor of the bureau’s series, when one considers
the correlation either of the monthly index numbers themselves or of
the monthly percentages of change. Dun’s comes second and Brad­
street’s again ranks lowest.
3. CRITICAL EVALUATION OF T H E BUREAU OF LABOR STATISTICS’,
BRADSTREET’S, AN D D U N ’ S IN D E X NUM BERS.

A few additional remarks are called for on the relative merits of
the three general-purpose index numbers now regularly published in
the United States.
In the publication of actual prices, the Bureau of Labor Statistics’
and Bradstreet’s stand foremost. The contribution they have thus
made to the knowledge of prices possesses great and permanent value
over and above the value attaching to then' index numbers. For, it
is well to repeat, all efforts to improve index numbers, all investiga­
tions into the causes and consequences of price fluctuations, and all
possibility of making our pecuniary institutions better instruments
of public welfare depend for their realization in large measure upon
the possession of systematic and long-sustained records of actual
prices. And much of this invaluable material would be lost if it
were not recorded month by month and year b y year.
Critical users of statistics justly feel greater confidence in figures
which they can test than in figures which they must accept upon




COMPARISON OF LEADING AMERICAN INDEX NUMBERS.

109

faith. Hence the compilers of index numbers who do not publish
their original quotations inevitably compromise somewhat the repu­
tations of their series. They compromise these reputations still
further when they fail to explain in full just what commodities they
include, and just what methods of compilation they adopt. Bradstreet’s index number suffers a bit in comparison because readers are
not told which 96 commodities out of the 106 for which prices arc
published are included in the index number, and because the method
of reducing prices by the yard, the dozen, the bushel, the gallon, etc.,
to prices per pound is not fully explained. Dun’s index number is
more mysterious still, because neither the list of commodities nor the
weights applied to each commodity are disclosed.
The number, of commodities now included in the three series is
ven as follows by the compilers: Bureau of Labor Statistics' 328,
un’s “ about 300, Bradstreet’s 96. Provided the commodities are
equally well chosen, of course the longer the list of commodities
included the better claim has an index number to acceptance as a
measure of changes in the general level of commodity prices.
•The preceding study of the relations among the leading American
index numbers was made in the winter of 1919-20, just before the
great fall in prices began. Early in the course of this fall marked
discrepancies appeared between the Bureau’s series and both the
commercial indexes. These discrepancies presently became wider
than any that had appeared in the preceding 30 years. B y Decem­
ber, 1920, Bradstreevs index was 22.4 per cent lower than the Bureau’s
index and Dun’s was 10.9 per cent lower.2
5

S

» f h e following table continues, b y m onths, from January* 1919, t o M ay, 1921, the ind ex numbers of the
Bureau of Labor Statistics, Bradstreet, and D un in the form given in f a b l e 22:

C p risonofth A erica in exn mers, b m th January, 1 1 ,to May,!9 1
om a
ree m n d u b y on s,
99
2.
Year and m o n th /

1919:
J an u a ry ...
F ebruary..
M arch. . . . .
A p ril.........
M ny...........
J u n e .......
J u ly......... .
August
September
O ctob er...
November.
D ecem ber.
1920:
Jan u a ry..
February..
M arch____
June.
J u ly......... .
A ugust___
September
O ctober. . .
November.
Decem ber,
1921:
J an u a ry ..
February.
M arch___
A p r il.___
M ay.-..'...




Bradstreet.

D un.

202.35
195.02
193.03
193.11
197.64
206.92
217.62
220.84
218.15
220.56
224.22
226.80

185.13
179.74
179.72
181.83
185.12
189.85
195.48
197.38
195,01
195.10
198.71
202.34

230. C8
233.09
232.22
231.87
227.19
219.46
213.60
205.89
195.16
182.30
163,93
147.08

206.08
208.39

140.05
135. 58
130.02
124.17
119.93

158.09
151.23
146.53
140.26
136.80

210.10
214.34
216.09
214; 89
214.95
209.95
199.69
191.03
180,45
168.70

Bureau
o f Labor
Statistics.

197.24
200.90
203.43
206.85
206.91
218.74
226.42
229.55
229.89
238.28
243.59
249.19
253.97
272.14
263.20
249.89
241.93
2 2 5 .3 6
2 0 7 .3 3

189.44
177.92
167.41
83
153.72
151.09

m.

110

TH E M A K IN G AND USING OF INDEX NUMBERS.

These wide discrepancies mean, not that the index numbers had
become suddenly worse, but that the diversity among price fluctua­
tions had become greater, so that differences among index numbers
in respect to the number of commodities included and methods of
weighting produced wider differences in the results. In other words,
we have here the demonstration of a significant fact about price
fluctuations: The great drop of prices in 1920-21 was characterized
by much more irregularity in the promptness and degree of readjust­
ment of different markets to the new situation than was the great rise
of prices in 1915-1919. Presumably these great irregularities will
prove to be a feature of the transition period only, and the three
index numbers will approach one another again as tne readjustments
are gradually worked out in all markets.
With reference to weighting, Bradstreet’s index number takes low
rank, for the plan of reducing all quotations to prices per pound
grossly misrepresents the relative importance of many articles.
That figures made thus should give results in close agreement with the
Bureau of Labor Statistics’ series is really remarkable and proves
that if prices, the raw materials from which index numbers are made,
are accurate the particular method used in computing the index
nun her is of secondary importance. Dun’s system of weighting is
distinctly better than Bradstreet’s in theory. Whether the practice
is as good as the theory is doubtful, for anyone familiar with the
deficiencies of American statistics of consumption must wonder whence
the compilers derive their estimates of the quantities of “ about 300”
commodities “ annually consumed by each inhabitant.” Moreover,
what little is known concerning the actual weights is not unobjec­
tionable. Fifty per cent of the total is too large a weight to allow
to foods in a wholesale-price series. Even in the great collection of
budgets of workingmen’s families made by the Commissioner of
Labor in 1901 the average expenditure for food was less than 45 per
cent of total family expenditure, and in 1918 it was found to be
only 38.2 per cent.2 The bureau’s practice of weighting wholesale
6
prices by the quantities of commodities that enter into trade is
preferable to weighting by consumption. Moreover, the bureau
publishes its weights, and shows each year the percentage which
each weighted price makes of the total for the group in which the
commodity is put, as well as of the total for all commodities.
2« Eighteenth A nnual Report of the Commissioner of Labor, 1903, p . 66. The data represented 25,400
families and 124,108 persons, both natives and immigrants. Also the M onthly Labor Review of the Bureau
of Labor Statistics, August, 1919, p. 118. The data represented 12,096 white families in 92 industrial centers.




COMPARISON OF LEADING AMERICAN INDEX NUM BERS.

I ll

In the form of presenting results, Bradstreet’s set an admirable
example, which was wisely followed by Dun’s. Their sums of
actual prices can readily be turned into relatives on any base desired,
and hence can be made to yield direct comparisons between any two
dates. The bureau’s series shares this advantage, since it too is made
by adding actual prices multiplied by weights; but it is presented in
a form more convenient for comparison than the other two series.
The relatives on the scale of 100, into which the bureau throws its
figures in the last step of compilation, are easier to use than the
awkward sums of dollars and cents which Dun’s Review and Bradstreet’s publish.
It is interesting, finally, to test the reliability of the several index
numbers as 1 business barometers.” Monthly figures would be better
1
for this purpose than our yearly averages, but since they are not
available for all three series in the 1890’s, we must do the best we can
with the rougher gauge. In 17 of the 26 years since 1892 (when
Bradstreet’s index m its present form begins), the three series agree
concerning the direction m which prices were moving; they differ in
nine years. In the following schedule these nine years are repre­
sented by columns in which each index number is credited with + 1
when its change accords with the character of the alteration in
business conditions, debited with —1 in case of disagreement, and
marked ± 0 when it recognizes no appreciable change in the price
level.2 The net scores made by easting up the plus and minus entries
7
indicate roughly the relative faithfulness with which these series have
reflected changes in business conditions in the past quarter of a
century.
Index number.

1S93

1895

1897

1901

1903

1904

1905

1913

1914

Net
score.

Bradstreet’ s...................................................
Bureau of Labor Statistics’ .......................
D un’ s..............................................................

+1
-1
-1

-1
±0
-1

+1
+1
-1

+1
+1
-1

-1
-1
+1

+1
±0
-1

+1
±0
±0

±0
+1
+1

-1-1
+ 1
-1

+4
-+-2
-4

Of the three indexes, Bradstreet’s makes the best showing. Pre­
sumably the poor quality of Dun’s index as a business barometer is
due chiefly to the heavy w eight (50 per cent) which it ascribes to
T
foods. For foods are largely farm crops whose prices in a given year
depend at least as much upon the w
reather as upon the condition of
business. The bureau’s series in this respect stands intermediate
between the two commercial series, giving a lighter weight to foods
than Dun’s and a heavier weight than Bradstreet’s. Probably that
is why it is a better business barometer than the one and not so good
as the other.
Of course this conclusion that Bradstreet’s index number is a
better business barometer than the bureau’s series does not invalidate
the preceding conclusion that the bureau’s series is the best measure
of changes in the general level of prices. For when farm crops are
given their due weight in an index number, it is not to be expected
that the index will always rise with business prosperity and decline
2 For a sum m ary of the changes in business conditions during these years, see Business Cycles, by
7
W esley C. Mitchell, p. 88.




112

TH E M A K IN G AND USING OF INDEX NUMBERS.

with business depression. In making a wholesale price index number
for use as a business barometer, indeed, one shoula exclude altogether
commodities whose price fluctuations are determined largely by the
weather. We have no such series at present, and it is high time that
this lack should be supplied. But if some one does make a wholesale
price index that is a nearly infallible business barometer, it will not
be as reliable a measure of changes in the general level of prices as
the present Bureau of Labor Statistics series.
VI.— CONCLUSIONS.
1. Variations in the level of wholesale prices from one year to the
next are capable of being measured by a close approximation to
accuracy, for these variations are highly concentrated about a central
tendency. There are two American chain index numbers which for
a quarter of a century never differ by more than 5 per cent, and differ
on the average by only 2 per cent, although they were compiled from
start to finish quite mdependently of each other, based upon dis­
similar sets of price variations, constructed by unlike methods, and
covered a period of violent fluctuations.2
8
2. Variations in prices that have been cumulating through several
or many years show much less concentration about a central tend­
ency than variations from one year to the next. Hence, index num­
bers become less accurate the greater the time over which they are
extended. Nevertheless, the discrepancies observed between the two
series just referred to (Dun’s and the Bureau of Labor Statistics’
new series of index numbers) do not reach 8 per cent in a period of
26 years, and average 3.4 per cent. The coefficient of correlation
between these two series in 1892-1914 is +0.992, a close approach to
+ 1.0, the expression of perfect agreement.
3. The choice of methods to be employed in making an index
number should be guided by the purpose for which the results are to
be used. These purposes are so numerous and so diverse that it is
impossible to make a single series Avell adapted to them all. Prob­
ably the time is near when certain uses will be so standardized that
several divergent types of index numbers will be regularly compiled
to serve the needs of various groups of users. Even now we have
special index numbers of the prices of foods, of farm products, of
metals, etc. To this list there might well be added a series especially
designed to throw changes in business conditions into high relief, and
assist in the bettering of business forecasts. Most of the currently
2; These figures are com puted from Table 19, b y turning the percentages b y which each index number
rose or fell each year into relatives on the preceding-year bass and com puting the percentage differences
between the resulting indexes. The results for three series are as follows:
Average difference.
Index numbers.
1893-1914. 1893-1918.

Bureau of Labor Statistics and Bradstreet’ s ......................................................................
Bureau of Labor Statistics and D u n 's .....................................................................................
Bradstreet's and D u n 's ............................................................................




P er cent. P er
2 .2o
1.95
2.92

cent.
2.82
2.00
3.15

CONCLUSIONS.

113

published index numbers, however, are what may be called generalpurpose series, which undertake to measure changes in the wholesale
price level at large.
4. The best form for these general-purpose series is a weighted
aggregate of actual prices or a weighted geometric mean. The latter
is preferable for measuring average ratios of change in prices; the
former is preferable for measuring average change in the amount of
money required to buy goods.
5. The more commodities that can be included in such an index
number the better, provided that the system of weighting is sound.
Certainly, each of the following classes of commodities should be
represented, and represented as fully as is feasible: Raw mineral,
forest, animal, and farm products, and manufactured products in
various stages of elaboration, bought for family consumption and for
business use.
6. Probably the best weights to apply are the average physical
quantities of the commodities bought and sold over a period of years
without reference to the number of times their ownership is changed.
In making an aggregate of actual prices these weights should be ap­
plied directly to the quotations of each commodity in making up the
totals for the several groups that have been mentioned, and then, if
the necessary data can be secured the totals for the several groups
should be weighted again in making up the grand totals for “ all
commodities.”
7. In presenting such an index number, it is well to publish the
aggregate actual prices, both for the several groups and for the
grand totals. But it is highly desirable to publish also relatives
made from these actual prices on a percentage scale, since comparisons
can be made more easily from such figures than from the aggregates
of actual prices, which are likely to run in awkward quantities.
Indeed, several sets of these relatives, computed on the basis of actual
prices at different times, can readily be provided for readers inter­
ested in knowing how prices have changed with reference to recent
or to past years. Among the relatives of greatest significance is the
set which shows the annual percentage of rise or fall as compared
with prices in the preceding year. In such chain index numbers it
is usually possible to include some commodities for which quotations
are lacking in certain of the years covered by the whole investigation.
8. Chain index numbers are best made by the “ ideal” formula,
when the chief aim is to attain the greatest possible accuracy in
measuring fluctuations from one year to the next. But when the
annual percentages of rise or fall in prices made in this way are
forged into a continuous series, their errors cumulate and vitiate
comparisons between the earlier and the later years. Such series
are also faulty for some purposes in that one can not tell what part
of the net results is due to changes in prices and what part to changes
in the quantities used as weights. When the chief aim is to forge a
chain which will yield reliable comparisons between prices in any
two years it is best to use constant weights and make aggregates of
actual prices or geometric means of price fluctuations, the choice
turning once again upon the specific purpose in mind.
1311739 0 —41------3




114

TH E M A K IN G AND USING OF INDEX NUMBERS.

9.
While index numbers are a most convenient concentrated
extract of price variations, they are far from being a competent
representation of all the facts which they summarize. Most “ con­
sumers of statistics” lack the time to go back of the finished products
to the data from which they are made. But the increase of knowl­
edge concerning the causes and consequences of price variations
depends much more upon intensive study of the ultimate data than
upon the manipulation of averages or aggregates. Upon the exten­
sion of knowledge in this field depend in turn large issues of public
welfare. Hence it is highly important to collect and to publish in
full the actual prices of as many commodities as possible, even
though some of the quotations may not now be available for use in
making an index number.