Benford's Law

Benford’s Law states that, in a naturally occurring set of numbers, the smaller digits appear disproportionately more often as the leading digits. The leading digits have the distribution shown in the following table, where the number 1 appears slightly more than 30% of the time as the leading digit, and the number 9 appears as the leading digit less than 5% of the time (which is a difference of 6x).

 1 = 30.1% frequency of occurrence

 2 = 17.6% frequency of occurrence

 3 = 12.5% frequency of occurrence

 4 = 9.7% frequency of occurrence

 5 = 7.9% frequency of occurrence

 6 = 6.7% frequency of occurrence

 7 = 5.8% frequency of occurrence

 8 = 5.1% frequency of occurrence

 9 = 4.6% frequency of occurrence

If all digits were to appear as the leading digit in a uniform manner, then each one would appear about 11.1% of the time. Since there is quite a disparity between the distributions stated in Benford’s Law and what a uniform distribution would indicate, this disparity can be used to locate instances of fraud.

The analysis involves calculating the distribution on the first digit in a series of numbers. If the distribution varies from the proportions indicated by Benford’s Law, then it is possible that someone is engaged in fraud. The reason for the difference is that someone committing fraud will create randomly generated numbers, rather than following Benford’s distribution.

It is important to understand the situations to which Benford’s Law can be applied. The frequency distribution only occurs for naturally occurring numbers. In a business, examples of these numbers are the grand total billed on an invoice, the compiled cost of a product, or the number of units in stock. It does not apply in situations where numbers are assigned, such as a sequentially assigned check number or invoice number.