Benford’s Law: How to Use It to Spot Fraud 

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Kate M. Pospisil, CFE 

Communications Specialist 

This blog serves as part three in a three-part series on Benford’s Law. To read part one, click here. To read part two, click here. 

Conventional wisdom suggests that each digit has an equal chance of being the first digit, but Benford's law shows that lower numerals always have a greater chance at being the leading digit compared to higher numerals. Counting jobs start with low numbers and progress to include increasingly higher numbers; for example, when counting to 25, 11 numbers have a leading digit of 1, seven numbers have the leading digit of 2 and only one number leads with the digit 3. 

In general, a series of numerical records follow Benford’s Law when they: 

  • Signify records such as populations, river water flow or sizes of celestial bodies. 

  • Don’t have pre-determined upper or lower limits. 

  • Are not made up of numbers used as identifiers, such as identity or social security numbers, bank accounts or telephone numbers.  

  • Have a lower mean than median, and data that are not concentrated around the mean. 

Scientifically, Benford's Law is based on base-10 logarithms that show the probability that the leading digit of a number will be n can be calculated as log10(1+1/n). By substituting the numbers 1 through 9 for n, you can calculate that each subsequent number 1 through 9 has a diminishing probability that it will be the leading digit.  

Computer-generated numbers give each numeral from 1 to 9 equal probability of being the leading digit. Using the above algorithm, equally weighted numbers would create a graph with straight-line results, thus potentiall indicating fraud. Analyzing data entered using the horizontal number keys on a keyboard could show a bell curve, using the reasonable assumption that dominant fingers — the index and middle fingers — would be hitting more numbers in the middle of the bar, 4, 5, 6 and 7. 

Using Benford’s Law is one way to investigate fraudulent behavior and is a valuable tool in an anti-fraud toolkit, but it is not foolproof and should not be relied upon solely in an investigation. The process of counting leading digits will never determine without a doubt that fraud has occurred. However, if an expected Benford’s Law curve doesn’t show up in a data analysis, it does indicate that further investigation is warranted.