August 25, 2015

Benford’s Law and Forensic Data Analysis

This is somewhat of an indulgent post, but I have always been fascinated about the consistency of a forensic accounting investigative concept known as Benford’s Law.

Named after Frank Benford, an American electrical engineer and physicist, it is a phenomenological law about the frequency distribution of leading digits in many (but not all) real-life sets of numerical data. Also known as the First-Digit Law, Benford’s Law states that, in a naturally occurring set of numbers, the leading digit will appear in a very predictable distribution. More specifically, the leading digit is most likely to be a 1 and then the likelihood decreases as you move to 2 and then to 3 and so on, with 9 being the least likely leading digit.

Watch this brief video to learn a little more about Benford’s Law with some excellent graphics for you visual learners out there.

Key Definitions

There are two key definitions that we need be clear on before we can start discussing the power of Benford’s Law in forensic data analysis. The first is “what is a leading digit?” The leading digit is the left-most digit in a number. In the number 1,234 the leading digit is the 1.

The second is “what is a naturally occurring numerical set?” This question is most easily answered by describing what is not a naturally occurring numerical set. Any set of numbers that is based on pre-determined or formulaic basis is not naturally occurring. Zip codes, social security numbers, area codes, account numbers – these are not naturally occurring numerical sets. As the video states, most financial data is presented in naturally occurring data sets, and for that reason Benford’s Law is a very useful tool for financial data analysis.

Near the end of the video, the narrator states “you can imagine how a forensic accountant might be able to use Benford’s Law to identify some red flags.” That is absolutely true. I will often take the amounts noted in a check register and compare those against the Benford distribution.

Introducing the “Unnatural”

Several data mining and analysis software programs will graphically display any numerical data set against the Benford distribution. If something in that data set is unusual, it will stick out and be visibly clear. For instance, let’s say a company has three facilities and pays rent of $6,700 for each facility on a monthly basis. The annual check register for this company would have a lot of payments in the 6 column and most likely create a spike. Those spikes are the things that need to be investigated, because they aren’t always as easily explained as a recurring rent check. Sometimes they are the result of fraud.

When a fraudster decides to perpetrate a fraud, they are essentially introducing an unnatural number into a natural data set. They may also choose the same amount each time they do it. Both of these elements will cause a spike against the Benford distribution. Even if the fraudster always takes $1,000, it will cause the 1 column to spike against the normal distribution. Making things even more difficult for our fraudsters is that Benford’s Law works with second digits as well (e.g., a 1 is most likely to be the second left digit in a number, then a 2, and so on). The second-digit predictability is not quite as accurate as with first digits but for this purpose it works just fine.

A Smarter Mousetrap

Benford’s Law is a not a fool-proof way to catch a fraud, and just because you see a spike where you don’t expect one doesn’t mean that a fraud is occurring either. However, with today’s availability of tremendous amounts of information, this is another way for the caretakers of corporate assets to take a cross-section of data and find unusual items. Benford’s Law is a tool at our disposal that is well beyond the reasonable set of precautions a fraudster can be expected to take. For that reason, it represents a smarter mousetrap in a world of very savvy mice. Build this into your periodic data analysis routines. Even if you don’t see signs of fraud, you will undoubtedly learn something about your company as a result of the unique cross-section brought to us by Frank Benford and his law of leading digits.

© Clark Nuber PS and Focus on Fraud, 2015. Unauthorized use and/or duplication of this material without express and written permission from this blog’s author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Clark Nuber PS and Focus on Fraud with appropriate and specific direction to the original content.

This article contains general information only and should not be construed as accounting, business, financial, investment, legal, tax, or other professional advice or services. Before making any decision or taking any action, you should engage a qualified professional advisor.