Get M.A.D. with the Numbers!

Moving Benford's Law from Art to Science

By By David G. Banks, CFE, CIA

Until recently, using Benford's Law was as much of an art as a science. Fraud examiners and auditors performed digital frequency analyses (DFA) and subjectively viewed the resulting data.

In my article, "Benford's Law Made Easy," in the Sept./Oct. 1999 issue of The White Paper, I described how to use Microsoft Excel's macro functions to extract the initial digits from a data table for analysis. In this article I go a step further and show how to use commonly available spreadsheet software to quantify data from a digital frequency analysis and distill it down to a single meaningful number. A fraud examiner or auditor can use that number to quickly perform time period or unit comparisons of DFA results and compile evidence against suspects.

Digital Frequency Analysis and Benford's Law

When people are asked the chances that the first digit of any number in a table will be the digit 9, most people readily assume that the odds are one in nine (or 11.1 percent). However, Dr. Frank Benford, a physicist, demonstrated in the 1930s that the odds actually were less than one in 20.

Without the aid of a computer, Benford examined first-digit frequencies of 20 lists covering 20,299 observations of natural numbers in a diverse group of tables.

He worked only with tables of numbers which weren't manipulated by a particular numbering scheme and weren't generated by a random number generator. The data in the tables included, among others, street numbers of scientists listed in an edition of American Men of Science, the numbers contained in the articles of one issue of Reader's Digest, and such natural phenomena as the surface areas of lakes and molecular weights.

Benford noted that the frequency of the first digits in any table of unmanipulated data followed a predictable pattern, which now bears his name. He calculated the expected rate of occurrence for the first digit with this logarithmic distribution formula:

Probability (X is the first digit) = Log 10(x+1) – Log 10(x) 

When data is manipulated, as it is in a fraud, the frequency of appearance of the initial digits usually differs from Benford's predicted frequency, which makes his law a potentially powerful tool for fraud detection. Using Benford's formula, these are the probabilities of a number appearing as the initial digit:

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