Benford's Law still works

Can Benford’s Law practically identify fraud? It’s one of many tests you can use to discover fictitious numbers in supposedly random data sets, such as monetary amounts of purchase transactions. In this case, a comptroller successfully uses Benford’s Law to search for anomalies in warranty claims.

John, the controller of Rafal Inc., is in a quandary. Peter, the company’s financial analyst, says he’s found an abnormal spike in the warranty claims for FY 2019 as compared to FY 2018. John is perplexed as to how a discrepancy could’ve found its way into the payment process when he knows for certain that his team has been following tried-and-true documented “enterprise resource planning” procedures. He discusses the issue with Tim, a forensic consultant, who introduces him to the world of Benford’s Law. The heralded law could settle the issue. (Names and details have been changed in this case history.)

Benford’s Law and its historical evolution

Benford’s Law is a statistical method for detecting any manual intervention in an otherwise automated operational transaction activity.

In 1881, Simon Newcomb, an American astronomer, made an observation in the pattern of usage of logarithm tables. He found that the logarithm pages that began with “1” were more worn out than other pages. He inferred that pages commencing with 1 had far more frequent usage. (See Newcomb’s paper, Note on the Frequency of Use of the Different Digits in Natural Numbers.)

In the 1920s, Frank Benford, a physicist at General Electric, observed — as did Newcomb — the pages of logarithm table books covering numbers with the initial digits “1” and “2” were more worn and dirtier than pages for “7,” “8” and “9.” (See Digital Analysis Using Benford’s Law, by Mark J. Nigrini, Ph.D., Global Audit Publications, 2000.)

Because the first few pages of a logarithm book list multi-digit logs beginning with the digits 1, 2 and 3, Benford theorized that scientists spent more time dealing with logs that began with those numerals. He also found that with each succeeding first digit, the amount of time scientists used it was decreased.

So, he concluded that in a population of naturally occurring multi-digit numbers, those numbers beginning with 1, 2 or 3 must appear more frequently than multi-digit numbers beginning with the digits 4 through 9. Also, the first digit of the numbers will be distributed in a predictable and expected way. Instead of the frequencies of the first digit being equal (a 1 out of 9 chance for each of the digits 1 through 9), the first digit of a multi-digit number typically follows a different pattern. Predictable patterns also occur in the second and third digits of multi-digit numbers. However, in this article, I limit its application to the first digit only.

(For more information, see the online ACFE Fraud Examiners Manual, Section 3: Investigation/Data Analysis and Reporting Tools/Using Data Analysis Software.)