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NYT Science August 4, 1998
Following Benford’s Law, or Looking Out for No. 1
By MALCOLM W. BROWNE

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 h
is
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 numb
er
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 evader
s,
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 Gener
al
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.
Logarithm tables (and the slide rules derived from them) are not much used for
routine calculating anymore; electronic calculators and computers are simpler a
nd
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 so
me
special preference for logarithms starting with 1. He therefore embarked on a
mathematical analysis of 20,229 sets of numbers, including such wildly disparat
e
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 follow
ed
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 per
cent
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 formul
a
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 Scient
ist,
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 toss
es,
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 do
n’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 fr
om
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 — rand
om
samples from a day’s stock quotations, a tournament’s tennis scores, the number
s on
the front page of The New York Times, the populations of towns, electricity bil
ls
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 peo
ple
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 number
s in
the sample, the first digit of the sequence is more likely to be 1 than any oth
er.
The larger and more varied the sampling of numbers from different data sets,
mathematicians have found, the more closely the distribution of numbers approac
hes
what Benford’s Law predicted.
One of the experts putting this discovery to practical use is Dr. Mark J. Nigri
ni,
an accounting consultant affiliated with the University of Kansas who this mont
h
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 m
atch
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 predi
cted
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 one
s
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 Attor
ney,
recalled in an interview that he had read an article by Dr. Nigrini that fascin
ated
him.
“He had done his Ph.D. dissertation on the potential use of Benford’s Law to de
tect
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. I
t
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 th
at
it probably contained some rounded-off estimates rather than precise numbers, b
ut
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 the
y
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 app
ear
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 subm
it
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 compan
y’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 itse
lf,”
Dr. Nigrini said. “For me, Benford is a great hero. His law is not magic, but
sometimes it seems like it.”
---
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