What Is Benford's Law and Why Do Fraud Examiners Use It?


John Gill, J.D., CFE
Vice President - Education

Benford’s Law can be a tricky concept to grasp at first. Here are the basic things to remember:

  1. It relies on the idea that the distribution of digits in multi-digit natural numbers is not random; instead, it follows a predictable pattern. More on that later.

  2. It applies only to “natural numbers.” The definition for natural numbers and non-natural numbers in a fraud examination are different than they are in math, so don’t let that throw you off. Here’s how we define them for Benford’s Law.

    • Natural numbers are those numbers that are not ordered in a particular numbering scheme and are not human-generated or generated from a random number system.

    • Non-natural numbers are designed systematically to convey information that restricts the natural nature of the number (e.g., postal codes and telephone numbers).

Figure A

Figure A

Let’s say you have a three-digit payment amount ($XXX). What are the odds that the first digit would be 3? Well, you might think the odds are 1 in 9. In other words, there is an equal chance that the first digit would be 1 or 2 or 3, etc., up to 9.

Benford’s Law says that the odds are not random. In fact, there is a 12.5% chance the first digit will be a 3. There is a 30.1% chance the first digit will be a 1. You can see this in Figure A.

Fraud examiners use Benford’s Law tests on natural numbers, like payment amounts. The theory is that if a fraudster submits fake invoices for payment, he won’t submit invoices for $100 or $200, he will want to go big and submit invoices for $900 or $800. If you do that enough times, it upsets the natural order of the way numbers should occur (according to Benford). For example, if you run a Benford’s Law test on your April payments, and you find the first digit was 9 in 35% of the payments, that’s an anomaly. Bendford’s Law says 9 should be the first digit only 4.6% of the time.

No test is foolproof, but Benford’s Law does provide an extra method for fraud examiners to test data for potentially fraudulent activity. You can read more about Benford’s Law in the online Fraud Examiners Manual.