Benford’s Law was published in 1938, but over 80 years later, events over the past year have resurfaced discussions of Benford’s Law. What is the law? What are its applications?

Benford’s Law is a mathematical formula that calculates the probability of occurrence of the non-zero leading digits in data sets that span several orders of magnitude.

In 1938, physicist Frank Benford published a paper titled “The Law of Anomalous Numbers” in the Proceedings of the American Philosophical Society, Volume 78, pages 551-572, 1938. However, Simon Newcomb, astronomer and mathematician originally stated the law in 1881 based on noticing the wear on pages of logarithmic tables. His paper was titled "Note on the Frequency of the Use of Digits in Natural Numbers." Amer. J. Math. 4, 39-40, 1881.

The formula for the Benford’s law is:

benford's law

Where “d” (the leading digit) is a number from 1 to 9.

Benford’s law is often used is in fraud and error detection. The applications include accounting fraud, price issues, the manipulation of Covid data and other fraud categories.

If a set of numbers spanning several orders of magnitude occur without being modified, they will probably follow Benford’s Law. If they don’t, it’s possible sign that fraud has occurred.

A Numberphile video by video journalist Brady Haran does a nice job of introducing Benford’s Law. 

You can view the Numberphile video here.

NOTE: Whether or not Benford’s Law applies to election fraud is controversial. We won’t address that here. The reader is welcome to do the applicable research pro and con.