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UNITED STATES DEPARTMENT OF LABOR F rances P erk in s, Secretary 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 , Commissioner 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 1891J9 But this also was a single investigation, and the United States 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, 1900.10 Another short 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.11 In January, 1901, the second continuous Ameri 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.12 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. 10 11 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;13 but after 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 13 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.14 The demonstration that a formerly 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 § 14 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.15 Since this sec 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.16 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.17 It shows for each year how many 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. 15 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 +46.1 -1 0 .2 + 3.5 +25.8 + 7.7 94.0 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 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 1 2.0 to 4.9 per cent. 21 22 35 21 12 21 16 9 109 40 13 23 24 13 32 14 35 11 105 9 9 32 47 25 70 35 55 22 16 7 4 13 21 22 26 19 27 34 40 30 27 12 22 28 24 13 18 30 28 10 50 15 21 20 18 31 43 18 16 ‘A 29 30 25 23 28 27 1 2 5 3 3 16 14 2 11 2,0 30 39 44 44 41 26 3 9 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 20 23 10 14 18 32 26 25 22 Number of com m odities 5.0 to to 50.0 per to to that rose cent or 4.9 per 9.9 per 19.9 per 49.9 per in price. cent. cent. cent. cent. more. 17 9 17 15 20 22 12 20 25 38 20 19 20.0 10.0 2.0 17 19 15 4 15 15 17 7 6 18 16 28 25 22 15 13 21 3 17 18 30 34 45 59 16 35 44 28 37 16 12 10 3 12 18 11 40 39 57 21 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 88 30 73 14 3 13 1 1 2 6 17 22 2 2 16 2 26 33 14 3 9 27 2 1 10 16 15 19 20 2 22 10 20 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 CO 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 wrhich the 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.18 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 18 Owing to the constant shifting of the base line, no fixed scale of relative prices can be shown on the 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.19 When it can be shown that phe 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.20 19 See, for example, Prof. F. Y . Edgeworth’s article “ Probability,” Part II, Encyclopaedia Britannica, 11th ed., and the references there given. 89Table 4 and Chart 2 m ight be im proved b y a change in form. If the ‘ ‘price variations” in each 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.21 These departures of the actual distribu 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 21 That the arithmetic mean is slightly above zero arises partly from the fact that there are 33 percentages 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 1913.22 These 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.23 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 28The bureau quoted 252 commodities in 1913; but 11 could not be included in the present comparison 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.” 24 But this position is untenable. (1) As a statistical 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 24 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.” 25 These three aims, which at first 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.” 26 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 by the end of the year. The collection of contract prices on a {are>rices arger scale and the analysis of their relation with open-market prices matters to which the makers of index numbers may profitably direct greater attention.27 27 The best presentation of contract and open-market prices yet m ade is in T he Prices of Coal and Coke, 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.28 Index numbers made from these two 23 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.29 These figures are the total prices of 55 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.30 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.31 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.32 Four years later, Dun’s Review 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.33 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 31 Gibson’s index number is such a sum. See pp. 172 to 175. The difference between sums of relative 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. 33 The confidence merited b y this index num ber is discussed in Section V . 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.34 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. 1908.35 Many of the commodities 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 145.36 The third column, which includes 50 commodities, is made up from the list adopted for 34 These statements refer to the number of series of relative prices averaged to get the final results as m m 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. 35 T o facilitate comparison, decimals have been dropped and the index for each year rounded off to the 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. 36 This*experimental list of 145 com m odities is given below. W hen the relative prices of closely related 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.37 The fourth series is 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.38 The last two columns contain 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. 37 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. 33 The remaining 17 pairs are corn ancl corn meal, cotton and cotton textiles, flaxseed and linseed oil, 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.39 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............. 39 114 114 105 105 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. . . ......................................................... .................................................... ± ± + ± + + + + 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............................................................................................................. ± o 1 0 1 1 1 4- 4 4- 4 + 8 ± db o o 97 f 73 + + + 2.5 1.8 1.0 .4 129 +105 + + 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 0 2 1 5 3 1 2 1 2 6 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 modities, modities, com m od modities. modities. second modities. 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 wrere 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, howrever, 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: 40 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.41 The second pair of columns contains index numbers made from the prices of 20 raw materials and of 20 products manufactured from these same materials.42 Then *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. 42 The articles included here are those from which the index number of 40 commodities in Table 6 was 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.43 The later sections of the table give 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.44 It follows that index numbers made 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 aw Man teri tured R m a ufac als. prod teri tured ucts. als. goods. Five triplets. Average W heat group. Inter R aw m edi F in ma ished Wheat. W heat Bread. teri ate goods. flour. 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. ool Steel W orst Wen Raw Cot Cot Steel ton Raw 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.45 Here the 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.) 4r>The lists of com m odities are as follows: 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.46 An index number 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.’*7 Though consisting more largely of the erratically 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.) 47 The consumers* goods are bread, crackers, butter, cheese, salt fish, evaporated apples, prunes, raisins, 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,48 but the analysis has been carried far enough 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.49 Hence the inclusion 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.50 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 49 A com pilation of the differences among the relative prices in question taken seriatim for each of the 24 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,77 or to measure changes in “ the cost of living.77 If these aims were always 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,51 and the preceding analysis shows 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 Sauerbeck7s has strong claims to preference 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 Average change from j^ear to year................ 4.0 25 com modities, first list. 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 + + -4- , . ; 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 96 100 106 118 107 115 114 111 121 128 127 137 134 147 139 100 122 136 100 114 138 100 115 139 27 44 8 16 2 - 29 + 40 + 6 + 17 4 - 28 + 40 + 7 + 19 8 54 63 66 5.0 7.1 7.3 48 com modities, Bureau of Labor Sta tistics list. ; , 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.52 The bureau 52 Of the 56 articles included altogether, only 11 are com m on to all three lists. The Gibson list has 8 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,53 and the Gibson figures agree with the other two series in 19 years out of the 24. On the other hand, the three food indexes &3 Even in 1903-4 the bureau’s figures record a slight advance of prices in harm ony with the Annalist 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.54 A food index num 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.55 Forest 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 04 The exceptions are salt and soda, and of these articles the Annalist and Gibson quote neither. 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.56 To give another illustration, 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.51 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.58 Again, 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.59 Nor 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 58 For details, see p p. 161-168. 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.60 Per haps “ unconscious weighting71 would be an even better expression. 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.61 Second, there are high authorities who hold that the results turn out much the same whether or not formal weights are used.62 Certainly u tEe weights are of * * * less impor tance in determining an index number of prices than the prices themselves.” 63 But whether their importance is negligible is a ques tion best answered by a study of actual cases such as are shown in the next table.61 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 60 C- M. Walsh, The Measurement of General Exchange-Value, pp. 81 and 82. Haphazard weighting is 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. C1 See p . 26. W hen the (then) Departm ent of Labor started its former index number it canvassed the 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. 62 Compare A . L . B ow ley, Elem ents of Statistics, 2d ed., pp. 113 and 220-224. 03 Irving Fisher, The Purchasing Power of Money, revised edition, p . 406. For further details see the 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, tain ex 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 between A ll articles tance, cer tance, com 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.65 Here an attempt is made to ascertain the aggre 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.” 66 Somewhat different weights would 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.” 67 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.” 68 On 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. 66 Report on Wholesale and Retail Prices in the United Kingdom in 1902. London, 1903, p. 411. The 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.69 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 69 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.71 Yet it is clear that a system of fixed 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 70 For details see Bulletin N o. J8i of the Bureau of Labor Statistics. 71 See the criticism of index imbibers made from import-export values, p p . 29-31. f311739 0 —41------5 66 THE M A K IT O A3STD 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.72 A further practical reason in 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.” 73 But, since the importance of price fluc 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.74 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 72 73 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.75 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.’ ” 76 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 75 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.77 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,78 geometric 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. 78 W . 8. Jevons, “ A serious fall in the value o f gold ascertained,” 1863. Reprinted in his Investigation 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.79 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 100.80 The cases of extraordinary advance, 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.81 Accordingly, the arithmetic mean (217) stands high above the geo metric mean (194) and median (191).82 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: 1913.................... 1914.................. ; Arithmetic means. 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: 4913...................... 1914...................... Arithmetic means. 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 =141.42.83 By 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.84 Finally, geometric means 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.85 Statisticians 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. 84 This point is more fully explained on pp. 76 and 77. 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(Po) 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.88 It is generally claimed that of all averages 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.87 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.88 (3) When the number of items to be 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 86Compare, for example, F. Y . E dgew orth ,“ Index num bers,” Dictionary of Political E conom y, V ol. II, 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. 87 See his paper “ On the use of analytical geometry to represent certain kinds of statistics, ” Journal of the R oyal Statistical Society, June, 1914, V ol. L X X V I I , p. 733. 83 It is a convenient feature of arithmetic means com puted from relatives based on average prices over a 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.89 (4) If the num 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.90 On the other 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,91 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.92 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 99 “ This objection is m et,” says Prof. Edgeworth, “ b y denying that the interval between tw o adjacent 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. 91 See section 8 below. 92 Take, for example, the following data: Wheat, per bushel......................................... Corn, per bushel............................................. 1913 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.93 In the chain index 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. 93 Irnr mimorionl uvorrmloc nfof rYnf\T rmfiu’n AM rl arithmetic means ~_.__4 __3 r___ Ai__the same I11G dOftE S9'6 F F. Y For numerical examples geometric and com puted from data', see A defense of index numbers,” Econom ic Journal, V ol. V I (1896), p. 137,, and A . W ~ lux, w. F 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). 93 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.95 The figures for prices dur 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.............................................. ±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 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 Averages, 1890-1899................... 1900-1909................... 1910-1913................... Average change from one year to the n ex t.............................. a Compare Tables 2 and 17. 2. 22 3.64 Relative prices of 145 com m odities (aver age prices in 18901899=100) b Medians. 112 Arithmetic means. Relative prices of 25 com m odities (aver age prices in 18901899= 100). c Medians. Arithmetic means. 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 96 107 113 111 116 118 122 123 130 133 124 133 133 129 140 142 100 115 126 100 118 132 101 123 130 100 122 136 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 AJsTD U SI¥G OF ISTDEX J5TUMBEBS. large upward variation should count for more than a small down ward variation, for it requires more currency;” 96 provided always 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.97 When such 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.98 It is true that sums in dollars and cents are likely to 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.) 98 See the exam ple from H un t’s Merchant’s Magazine given on p p . 31 and 32. However, a very rough 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. 97 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 99 The legitimacy of shifting these relatives b y the “ short” m ethod is best shown b y the use of symbols. 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 111 base year o and m some other year designated b y the subscript x. **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 lx Qn 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: fVo U ? P o ) +Pf rO( ? ' P'o) + 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= 1877= 100;b y 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 Disputed crepform, =I860 100, on 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 mi 'Dl' + P l " + • ' • 11 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” + Pi’+Pi" jy Zpi+Spir Spi Putting • • * * • • • • . ‘ 2px pi—m\=Xi where xi is the size of the relative, we have: R f= i+ Z(mi+ii)r SPi TOiSr SZir Rf=l+ 2pi= nmi m{Lr 2 zir _ 2r Zxr Rf= 1 + nm x+ 2px 1 + n (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 ) ($): Rf—Rc> 2xir 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.71 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.10 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.11 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.12 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.” 13 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 ?14 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 true of every year-to-year comparison however proposition holds 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.15 (2) It has been shown 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 1919.16 Hence the “ ideal” formula is ill-fitted for making index 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.17 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. 17 Concerning the difference in labor of computing Prof. Persons gives an interesting note. The relative 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.18 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.19 These three series in comparable form are assembled in Table 19.20 is See subdivision 5, “ The numbers and kinds o f com m odities in clu d e d /' especially pp. 52-56. 19 N o violence is done b y this procedure to Bradstreet’s series; but the eomparision is not quite satis 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 D u n’s. than street’s. Statis 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.31 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. BradstreeUs................................................... Bureau of Labor Statistics......................... Dun’ s............................................................... Annual geometric increment of secular trend in 1896-1914. 1.0230 1.0232 1.0269 Geometric mean in 1896-1914. 113.7 117.1 120.3 Ratio of annual increment to geome tric mean (per cent). 0.90 .87 .85 Terminal points of the straight line representing the secular trend. 1896 1914 92.7 95.3 94.7 139.6 144.0 152.6 Net per cent of rise in lines of secular trend, 1896-1914. « 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 CHART 14.—INDEX NUMBERS OF BRADSTREET, COMPARED WITH THEIR SECULAR 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 1Q8. 7 111.2 113.7 116.3 119.0 121.8 124.6 127.0 130.4 133. 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. Perct. Per ct. 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 Per ct. Perct. 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 Perct. Per ct. 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.22 With such scanty 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.23 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.24 When such coefficients are 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. 83 Most of the analytic comparisons among various American index numbers in Bulletin No. 173 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 BradBradIndus Labor Indus tries Statis street’s. Dun’s. tries Labor street’s. Dun’s. Statis 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.25 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: ComparisonofthreeAmericanindexnumbers, bymonths,January, 1919,to May,!921. 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.26 The bureau’s practice of weighting wholesale 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 11business barometers.” Monthly figures would be better 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.27 The net scores made by easting up the plus and minus entries 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 wTeight (50 per cent) which it ascribes to foods. For foods are largely farm crops whose prices in a given year depend at least as much upon the wreather 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 27 For a sum m ary of the changes in business conditions during these years, see Business Cycles, by 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.28 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.