# Full text of Economic Letter (Federal Reserve Bank of San Francisco) : Statistical Malpractice

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June 3, 1977

Statistical Mal practice
The Wall Street Journal does it. The
San Francisco Examiner does it. The
Los Angeles Times does it. And so
do the Christian Science Monitor,
the Atlanta Constitution and the
Philadelphia Inquirer, for that matter. But what they do is incorrect
and should not go unnoticed.
These and probably many other
publications incorrectly calculate
the annual rates of change in the
various economic indicators that
government agencies report. Their
mistake is not in converting these
monthly or quarterly figures to
equivalent annual rates, though
some economists question the value of paying close attention to what
happens in short time periods. Indeed, the public tends to think in
terms of annual rates of economic
growth and inflation, so journalists
rightfully render the monthly (or
quarterly) figures more meaningful
by presenting them in an annual
context. The only problem is that
they usually do it incorrectly.

Ruie of thumb
The problem lies in the use of a
simple rule of thumb, which per'forms quite nicely when dealing
with small numbers but not when
dealing with large numbers. Unfortunately, the inflation rate-to take
the obvious example-is not small
today. The rule of thumb involves
multiplying monthly rates of
change by 12 (or quarterly rates by

4) to get the equivalent annual rates
of change.

The consumer-price index, for example, rose by 0.8 percent (seasonally adjusted) in April. Well, 0.8
times 12 is 9.6, so many newspapers
risen at an annual rate of 9.6 percent. This is not too far from the
correct figure of 10.0, but it is off
and the error increases with the
magnitude of the monthly price
change. Thus, when the wholesale
price index increased by 1.1 percent last month, this was equivalent
to an annual rate of 14.0 percent,
not 13.2 percent as reported by
some members of the press. Altogether, the rule of thumb applied
to the consumer price index would
have resulted in at least a smaIi
error in 20 of the past 23 months.

Compounding-the key
Converting a monthly rate to an
annual rate involves finding the
annual rate of change that would
occur if the index rose at that same
monthly rate for 12 months straight.
The rule of thumb goes wrong because it does not capture the compounding effect. It ignores the fact
that the base to which that constant
monthly percentage is applied
grows somewhat each month. Thus,
the annual rate is always larger than
the rule-of-thumb rate, although
this difference may be less than a
tenth of a percentage point when

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Opinions expressed in this newsletter do not
necessarilv reflect the views of the management of the
Federal ReServe Bank of San Francisco, nor of. the Board
of Governors of the Federai Reserve System.

the monthly rate is relatively low.
But compounding is the only way to
go, and without it one has only a
rough approximation.
Frequently, the rule-of-thumb conversion will do the job. There is no
error to a tenth of a percent (which
is all we usually worry about) when
the monthly rate is less than 0.3 and
the quarterly rate is less than 1.0.
When inflation rates were below 3
percent in the 1950's and early
1960's,the rule of thumb was usually a pretty safe bet. But ever since
severe inflation reared its ugly
consistently understated the situation. Come the revolution when
inflation and other evils are banished from the land, the ru Ie of
thumb can be resurrected. In the
meantime, let us go forth and compound.

Deflating statistics
A second common error involves
the calculation of real" or deflated
values of such economic phenomena as wages or GNP. During last
year's California cannery strike, for
that the real wages of cannery
workers had fallen by 13 percent
over the preceding three years. This
occurred, he said, because while
the average cannery worker's wage
increased by only 22.4 percent, the
CPI rose by 35.3 percent. The rule
of thumb, in this case, involves
simply subtracting the percentage
change in prices from the percentage change in wages. While any
decline in real wages is something

calculation is off by several percentage points. The purchasing
power or real wages of cannery
workers actually declined by 9.5
percent, not 13.0 percent.
With this rule of thumb, the roughnessof the approximation increases
with both the difference between
the percentage changes in wages
and prices and the general level of
the percentage changes. Latin
American examples might be instructive, for those are countries
where changes in price and wage
levels historically have been quite
large. For example, in a country
where wages increased at 150 percent and prices at 145 percent, the
rule of thumb would yield a rise in
real wages of 5.0 percent. But actually, there would be only a 2.4percent rise'in real wages-less
than half the rule-of-thumb estimate. So the error rises with the
general level of changes being
examined.

(I

2

If the difference between the wage
and price changes is also large, the
rule of thumb provides estimates
that are essentially worthless. Suppose the' wage gain in our Latin
country remained at 150 percent
but the price rise was, by wise use
of monetary policy, restricted to
only 100 percent. A 50 percent gain
for workers? No. The gain in real
wages is just 25 percent.

Nature of bias
The bias is always in the same direction. The rule of thumb always
overstates the change in real wages.
When wages are rising more rapidly

than prices, as is generally the case,
the rule of thumb makes things
appear rosier for workers than they
really are. However, when wages
happen to be rising less rapidly than
prices, as has been the case during
much of the 1970's,the rule of
thumb overstates the fall in real
wages.
The difference between the rule of
thumb estimate and the actual
change can be seen by analyzing a
simple formula for the actual
change. If the percentage change in
wages is wand the percentage
change in prices is p, then the
percentage change in real wages is
100 times w-p . The rule of thumb
100+p
shortcuts the calculation by eliminating the denominator. As the rate
of inflation, p, increases, the two
calculations diverge.
When is the rule of thumb accurate,
at least to a tenth of a percent? It's

difficult to say, because this depends upon both the level and the
relationship between the two percentage changes. Whenever the
figures involved exceed 7 percent,
or whenever the difference between the two percentage changes
exceeds 6 percentage points, the
rule of thumb will generate an error. But sometimes errors can result
even from smaller numbers.
The conservative approach for the
amateur statistician is always to use
the formula given above, or some
variant. At the least, any use of a
rule-of-thumb calculation should
always be accompanied with a qualification. But when rates of change
are of the double (or greater) digit
variety, even these qualifications
may not be enough. What all this
boils down to, unfortunately, is a
new rule of thumb, to wit: " Whenever economic rates of change are
large, use with caution the trusty
old rules of thumb."
Michael Gorham

NEW PUBLI CATI ON

A supplement to the Spring 1977 issue of the Economic Review entitled
"The Monetarist Controversy" is presently being distributed to subscribers to the Economic Review, and is available free to readers of this
publication by contacting the Public Information Section, Federal
Reserve Bank of San Francisco, P.O. Box 7702,San Francisco 94120,
phone (415)544-2184.
" The Monetarist Controversy)) contains a paper presented by Professor
Franco Modigliani at the Federal Reserve Bank of San Francisco's January
1977 Economic Seminar with discussion by Professor Milton Friedman.

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BANKINGDATA-TWELFTH FEDERAL
RESERVE STRICT
DI
(Dollar amounts in millions)
Seleded Assetsand Liabilities
large Commell'dallBanks

Amount
Outstanding

Change
from

5/18/77

5/11/77

Security loans
Commercial and industrial
Real estate
Consumer instalment
U.S.Treasurysecurities
Other securities
Deposits (lesscashitems)-total*
U.S.Government deposits
Time deposits-total*
Statesand political subdivisions
Savingsdeposits
Other time deposits:j:
Largenegotiable CD's

96,217
74,334
1,962
23,866
22,935
12,938
8,520
13,363
94,247
26,255
425
65,777
5,857
31,929
26,188
9,322

-

Weeldy Averages
of Daily Figures

Weekended

Member Bank ReservePosition
Excess
Reserves(+)/Deficiency (-)
Borrowings
Net free(+)/Net borrowed (-)
Federal Funds-Seven LargeBanks
Interbank Federalfund transactions
Net purchases(+)/Net sales(-)
Transactionswith U.S.security dealers
Net loans (+)/Net borrowings (-)

+
+

-

+
+

-

-

+

-

+

-

+
+

5/18/77

17
498
136
99
79
42
254
261
67
382
52
6
22
124
95
107 .

Change from
year ago
Dollar
Percent
+ 9.77
+ 8,567
+ 12.51
+ 8,266
.+ 610
+ 45.12
+ 1,570
+ 7.04
+ 14.72
+ 2,942
+ 17.10
+ 1,889
755
- 8.14
+ 8.58
+ 1,056
+ 8.42
+ 7,320
+ 11.34
+ 2,674
- 33.49
- 214
+ 7.29
+ 4,467
- 837 - 12.50
+ 21.69
+ 5,691
54
- 0.21
- 1,705
- 15.46

-

Weekended

5/11177

27
4
31

25
8
33

+

117
216

+

248

+

Comparable
year-ago period

+

214

+

47
16
31
244

+

153

*Includes items not shown separately.:j:lndividuals, partnerships and corporations.
lEdito:rial
comments may be addressedto the editor (William Burke) or to the author .•••
Information on this and other publications can be obtained by calling or writing the Public
Information Section, federal Reserve Bank of San Francisco, P.O. Box 7702, San francisco 94120.
Phone (415)544-2184.

.04 BPI
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