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Topic: [math-learn] Looking for 1's helps detect 'cooked' books [ENRON]
Replies: 2   Last Post: Apr 3, 2002 4:58 PM

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Jerry P. Becker

Posts: 16,576
Registered: 12/3/04
[math-learn] Looking for 1's helps detect 'cooked' books [ENRON]
Posted: Apr 3, 2002 3:18 PM
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From, Commentary By John Allen Paulos, Special to , March 1, 2002. See
.Thanks to Caroly Fry Bohlin for bringing this article to our
SUMMARY -- Columnist John Allen Paulos explains why figures beginning
with the number 1 are more common and how this can help detect cooked
Looking Out for No. 1

Math Theory Suggests Looking for 1s Can Help Detect Cooked Books

Commentary by John Allen Paulos

Was there any way of looking at Enron's books and - not knowing
anything about the company's specific accounting practices -
determining whether the books had been cooked?

There may have been, and the mathematical principle involved is
easily stated, but counterintuitive.

Benford's Law states that in a wide variety of circumstances numbers
as diverse as the drainage areas of rivers, physical properties of
chemicals, populations of small towns, figures in a newspaper or
magazine, and the half-lives of radioactive atoms begin
disproportionately with the digit "1."

Specifically, they begin with "1" about 30 percent of the time, with
"2" about 18 percent of the time, with "3" about 12.5 percent of the
time, and with larger digits progressively less often. Less than 5
percent of the numbers in these circumstances begin with the digit

(This is in stark contrast to many other situations - say where a
computer picks a number between 0 and 100 at random - where each of
the digits from "1" to "9" has an equal chance of appearing as the
first digit.)

Tipped Off by Dirty Pages

Benford's law goes back more than a century to astronomer Simon
Newcomb, who noticed that books of logarithm tables were much dirtier
near the front, indicating that people more frequently looked up
numbers with a low first digit.

Without any proof of why this odd phenomenon should occur, it
remained a little-known curiosity until it was rediscovered in 1938
by physicist Frank Benford. It wasn't until 1996, however, that Ted
Hill, a mathematician at Georgia Tech, established what sorts of
situations generate numbers in accord with Benford's Law.

Then, a mathematically inclined accountant, Mark Nigrini, generated
considerable buzz when he noted that Benford's Law could be used to
catch fraud in income tax returns and other accounting documents.

The following example suggests why collections of numbers governed by
Benford's Law arise so frequently:

Imagine that you deposit $1,000 in a bank at 10 percent compound
interest per year. Next year you'll have $1,100, the year after that
$1,210, then $1,331, and so on. The first digit of your bank account
remains a "1" for a long time.

When your account grows to more than $2,000, the first digit will
remain a "2" for a shorter period as your interest increases. And
when your deposit finally grows to more than $9,000, the 10 percent
growth will result in more than $10,000 in your account the following
year and a long return to "1" as the first digit.

If you record the amount in your account each year for a large number
of years, these numbers will thus obey Benford's Law.

The law is also "scale-invariant" in that the dimensions of the
numbers don't matter. If you expressed your $1,000 in euros or francs
or drachmas and watched it grow at 10 percent per year, about 30
percent of the yearly values would begin with a "1," about 18 percent
with a "2," and so on.

More generally, Hill showed that such collections of numbers arise
whenever we have what he calls a "distribution of distributions," a
random collection of random samples of data. Big, motley collections
of numbers follow Benford's Law.

Suspiciously High Digits

And this brings us back to Enron, accounting, and Nigrini, who
reasoned that the numbers on accounting forms, which come from a
variety of company operations, each from a variety of sources, fit
the bill and should be governed by Benford's Law.

That is, these numbers should begin disproportionately with the digit
"1," and progressively less often with bigger digits, and if they
don't, that is a sign that the books have been cooked. When people
fake plausible-seeming numbers, they generally use more "5s" and "6s"
as initial digits, for example, than would be predicted by Benford's

Nigrini's work has been well-publicized and has no doubt been noted
by accountants and by prosecutors. Whether the Enron and Arthur
Andersen people have heard of it is unclear, but investigators might
want to check if the percentage of leading digits in the Enron
documents is what Benford's Law predicts. Such checks are not
fool-proof and sometimes lead to false positive results, but they
provide an extra tool that might be useful in certain situations.

It would be amusing if, in looking out for No. 1, the culprits forgot
to look out for their "1s." Imagine the Andersen shredders muttering
that there weren't enough leading "1s" on the Enron documents they
were feeding into the machines. A 1-derful fantasy!
Professor of mathematics at Temple University and adjunct professor
of journalism at Columbia University, John Allen Paulos is the author
of several best-selling books, including Innumeracy and A
Mathematician Reads the Newspaper. His Who's Counting? column on appears every month.
Jerry P. Becker
Curriculum & Instruction
Southern Illinois University
Carbondale, IL 62901-4610
Phone: (618) 453-4241 [O]
(618) 457-8903 [H]
Fax: (618) 453-4244

[Non-text portions of this message have been removed]

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