Dow 10,000 More Likely Than Dow 9,000???
Following Benford's Law, or
Looking Out for No. 1
By MALCOLM W. BROWNE -- August 4, 1998
Dr. Theodore P. Hill asks his mathematics students
at the Georgia Institute of Technology to go home
and either flip a coin 200 times and record
the results, or merely pretend to flip a coin and fake
200 results. The following day he runs his eye over the
homework data, and to the students' amazement, he
easily fingers nearly all those who faked their tosses.
"The truth is," he said in an interview, "most people
don't know the real odds of such an exercise, so they
can't fake data convincingly."
There is more to this than a
classroom trick.
Dr. Hill is one of a growing
number of statisticians,
accountants and
mathematicians who are
convinced that an astonishing
mathematical theorem known
as Benford's Law is a
powerful and relatively simple
tool for pointing suspicion at
frauds, embezzlers, tax
evaders, sloppy accountants
and even computer bugs.
The income tax agencies of
several nations and several
states, including California, are
using detection software based
on Benford's Law, as are a
score of large companies and
accounting businesses.
Benford's Law is named for
the late Dr. Frank Benford, a
physicist at the General
Electric Company. In 1938 he
noticed that pages of
logarithms corresponding to
numbers starting with the
numeral 1 were much dirtier
and more worn than other
pages.
(A logarithm is an exponent. Any number can be
expressed as the fractional exponent -- the logarithm
-- of some base number, such as 10. Published tables
permit users to look up logarithms corresponding to
numbers, or numbers corresponding to logarithms.)
Logarithm tables (and the slide rules derived from
them) are not much used for routine calculating
anymore; electronic calculators and computers are
simpler and faster. But logarithms remain important in
many scientific and technical applications, and they
were a key element in Dr. Benford's discovery.
Dr. Benford concluded that it was unlikely that
physicists and engineers had some special preference
for logarithms starting with 1. He therefore embarked
on a mathematical analysis of 20,229 sets of numbers,
including such wildly disparate categories as the areas
of rivers, baseball statistics, numbers in magazine
articles and the street addresses of the first 342
people listed in the book "American Men of Science."
All these seemingly unrelated sets of numbers followed
the same first-digit probability pattern as the worn
pages of logarithm tables suggested. In all cases, the
number 1 turned up as the first digit about 30 percent
of the time, more often than any other.
Dr. Benford derived a formula to explain this. If
absolute certainty is defined as 1 and absolute
impossibility as 0, then the probability of any number
"d" from 1 through 9 being the first digit is log to
the base 10 of (1 + 1/d). This formula predicts the
frequencies of numbers found in many categories of
statistics.
Probability predictions are often surprising. In the
case of the coin-tossing experiment, Dr. Hill wrote in
the current issue of the magazine American Scientist, a
"quite involved calculation" revealed a surprising
probability. It showed, he said, that the overwhelming
odds are that at some point in a series of 200 tosses,
either heads or tails will come up six or more times in
a row. Most fakers don't know this and avoid guessing
long runs of heads or tails, which they mistakenly
believe to be improbable. At just a glance, Dr. Hill
can see whether or not a student's 200 coin-toss
results contain a run of six heads or tails; if they
don't, the student is branded a fake.
Even more astonishing are the effects of Benford's Law
on number sequences. Intuitively, most people assume
that in a string of numbers sampled randomly from some
body of data, the first non-zero digit could be any
number from 1 through 9. All nine numbers would be
regarded as equally probable.
But, as Dr. Benford discovered, in a huge assortment of
number sequences -- random samples from a day's stock
quotations, a tournament's tennis scores, the numbers
on the front page of The New York Times, the
populations of towns, electricity bills in the Solomon
Islands, the molecular weights of compounds, the
half-lives of radioactive atoms and much more -- this
is not so.
Given a string of at least four numbers sampled from
one or more of these sets of data, the chance that the
first digit will be 1 is not one in nine, as many
people would imagine; according to Benford's Law, it is
30.1 percent, or nearly one in three. The chance that
the first number in the string will be 2 is only 17.6
percent, and the probabilities that successive numbers
will be the first digit decline smoothly up to 9, which
has only a 4.6 percent chance.
A strange feature of these probabilities is that they
are "scale invariant" and "base invariant." For
example, it doesn't matter whether the numbers are
based on the dollar prices of stocks or their prices in
yen or marks, nor does it matter if the numbers are in
terms of stocks per dollar; provided there are enough
numbers in the sample, the first digit of the sequence
is more likely to be 1 than any other.
The larger and more varied the sampling of numbers from
different data sets, mathematicians have found, the
more closely the distribution of numbers approaches
what Benford's Law predicted.
One of the experts putting this discovery to practical
use is Dr. Mark J. Nigrini, an accounting consultant
affiliated with the University of Kansas who this month
joins the faculty of Southern Methodist University in
Dallas.
Dr. Nigrini gained recognition a few years ago by
applying a system he devised based on Benford's Law to
some fraud cases in Brooklyn. The idea underlying his
system is that if the numbers in a set of data like a
tax return more or less match the frequencies and
ratios predicted by Benford's Law, the data are
probably honest. But if a graph of such numbers is
markedly different from the one predicted by Benford's
Law, he said, "I think I'd call someone in for a
detailed audit."
Some of the tests based on Benford's Law are so complex
that they require a computer to carry out. Others are
surprisingly simple; just finding too few ones and too
many sixes in a sequence of data to be consistent with
Benford's Law is sometimes enough to arouse suspicion
of fraud.
Robert Burton, the chief financial investigator for the
Brooklyn District Attorney, recalled in an interview
that he had read an article by Dr. Nigrini that
fascinated him.
"He had done his Ph.D. dissertation on the potential
use of Benford's Law to detect tax evasion, and I got
in touch with him in what turned out to be a mutually
beneficial relationship," Mr. Burton said. "Our office
had handled seven cases of admitted fraud, and we used
them as a test of Dr. Nigrini's computer program. It
correctly spotted all seven cases as involving probable
fraud."
One of the earliest experiments Dr. Nigrini conducted
with his Benford's Law program was an analysis of
President Clinton's tax return. Dr. Nigrini found that
it probably contained some rounded-off estimates rather
than precise numbers, but he concluded that his test
did not reveal any fraud.
The fit of number sets with Benford's Law is not
infallible.
"You can't use it to improve your chances in a
lottery," Dr. Nigrini said. "In a lottery someone
simply pulls a series of balls out of a jar, or
something like that. The balls are not really numbers;
they are labeled with numbers, but they could just as
easily be labeled with the names of animals. The
numbers they represent are uniformly distributed, every
number has an equal chance, and Benford's Law does not
apply to uniform distributions."
Another problem Dr. Nigrini acknowledges is that some
of his tests may turn up too many false positives.
Various anomalies having nothing to do with fraud can
appear for innocent reasons.
For example, the double digit 24 often turns up in
analyses of corporate accounting, biasing the data,
causing it to diverge from Benford's Law patterns and
sometimes arousing suspicion wrongly, Dr. Nigrini said.
"But the cause is not real fraud, just a little
shaving. People who travel on business often have to
submit receipts for any meal costing $25 or more, so
they put in lots of claims for $24.90, just under the
limit. That's why we see so many 24's."
Dr. Nigrini said he believes that conformity with
Benford's Law will make it possible to validate
procedures developed to fix the Year 2000 problem --
the expectation that many computer systems will go awry
because of their inability to distinguish the year 2000
from the year 1900. A variant of his Benford's Law
software already in use, he said, could spot any
significant change in a company's accounting figures
between 1999 and 2000, thereby detecting a computer
problem that might otherwise go unnoticed.
"I foresee lots of uses for this stuff, but for me its
just fascinating in itself," Dr. Nigrini said. "For me,
Benford is a great hero. His law is not magic, but
sometimes it seems like it."
Copyright 1998 The New York Times Company |
