Math Theory Offers Way to Detect Cooked Books

March 1 -- 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: