Benford's law: the first-digit pattern that catches fraud

Numbers in business. 6-minute read. Updated 2026-10-10.

Take any large set of naturally occurring numbers, such as invoice totals, city populations or river lengths, and count how often each digit appears first. You might expect each of 1 through 9 to show up about 11% of the time. In practice, 1 leads about 30% of the time and 9 leads less than 5% of the time. This is Benford's law.

The formula

The expected share for leading digit d is log₁₀(1 + 1/d).

Why it happens

Quantities that grow multiplicatively, like prices, populations and revenues, spend longer with a leading 1 than with a leading 9. To go from 100 to 200 a value must double; to go from 900 to 1,000 it needs to grow only about 11%. Data that spans several orders of magnitude inherits this pattern.

How auditors use it

People who invent numbers tend to spread first digits evenly or favour mid-range digits. Forensic accountants compare a ledger's first-digit distribution with Benford's expectation; large deviations flag accounts for closer review. Tax authorities and election analysts have used the same idea. It is a screening test, not proof: some legitimate data, such as prices set just under a threshold or numbers with a fixed range, does not follow the law.

When not to use it

• Assigned numbers: phone numbers, ZIP codes, invoice IDs.
• Data with a hard minimum or maximum, such as adult heights.
• Small samples. Aim for at least several hundred values.

Every result in the number decoder notes the leading digit, a quick way to sanity-check individual figures.

Keep reading

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