Math Theory Offers Way to Detect Cooked Books


March 1, 2002 -- 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 "9."

(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 Law.

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.