Benford's Law and Digit Analysis

You'd think, if you picked a random number from a big pile of financial data, that its leading digit, the very first digit, ignoring sign and decimal point, would be equally likely to be any of 1 through 9. It isn't. For many real-world datasets that span several orders of magnitude, stock prices, populations, invoice amounts, physical constants, the digit 1 appears first about 30% of the time, while 9 appears first only about 4.6% of the time. This is Benford's Law, and it holds because such data tends to grow multiplicatively: a value has to pass through the "1-something" range and stay there proportionally longer (in log terms) before doubling into the "2-something" range, and so on up to 9, where it barely lingers before rolling back over to 1.

The formula for the expected frequency of leading digit $d$ is $P(d) = \log_{10}(1 + 1/d)$, which gives roughly 30.1%, 17.6%, 12.5%, ..., down to 4.6% for digits 1 through 9. Auditors exploit this: real, unmanipulated financial figures, genuine expense reports, tax filings, election vote counts, tend to follow this distribution closely, while numbers that have been invented or altered by a human (who unconsciously favors digits like 5 and 7 and avoids repetition) deviate from it noticeably. A chi-squared test comparing observed digit frequencies to Benford's predicted ones is a standard first-pass fraud screen.

Benford's Law isn't proof of fraud on its own, it only applies to datasets with enough spread across magnitudes, and legitimate data can still deviate for structural reasons (prices capped near round numbers, for instance). It's a flag for closer scrutiny, not a verdict.