Benford's law

Benford's law describes the uneven distribution of leading digits in many naturally occurring amounts: 1 appears about 30% of the time, while 9 appears under 5%. It applies to suitable collections of amounts, not every kind of data.

Example

For illustration, in a collection of 100 suitably varied invoice amounts, Benford's law predicts that about 30 would begin with 1 and about 5 would begin with 9. These are expected counts, calculated as 100 × 30.1% and 100 × 4.6%, not a guarantee about any particular collection.

Expected shares for each leading digit

The expected share falls as the leading digit rises. For illustration, the shares are: 1, 30.1%; 2, 17.6%; 3, 12.5%; 4, 9.7%; 5, 7.9%; 6, 6.7%; 7, 5.8%; 8, 5.1%; and 9, 4.6%.

The pattern fits collections that span several orders of magnitude, such as invoice amounts or stated incomes. It is not a good fit for amounts limited by a floor, a ceiling or fixed prices.

Testing amounts against Benford's law

An audit or fraud team can compare the observed leading-digit counts with the expected shares. A chi-square test or a mean absolute deviation test measures how closely the counts fit.

A deviation is a reason to look more closely at the amounts and their context. It is not proof of fraud. The test does not suit data with a floor or ceiling, or fixed-price amounts such as a loan product sold only at set amounts.

Questions we get asked

Decisimo’s Benford law calculator tests whether amounts follow Benford’s law and reports which leading digits stand out, along with a fit score.