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.