In most naturally occurring datasets, the leading digit is 1 about thirty percent of the time, 9 under five percent. The pattern is mathematical law, and it sends people to prison.
Documented
Newcomb's worn pages to the courtroom
Documented: Simon Newcomb published the observation in 1881; Benford rediscovered and demonstrated it across twenty thousand observations in 1938, rivers, populations, death rates, addresses. The law follows from scale invariance, the pattern must survive multiplication by any constant, and that requirement forces the logarithmic distribution. Mark Nigrini operationalized it for forensic accounting in the 1990s; it has since served as evidence in fraud prosecutions and flagged fabricated expense reports at scale.
Assessment: Fully documented mathematics and documented courtroom use, with a documented caveat: it applies to datasets spanning orders of magnitude, and abusing it on the wrong data convicts the innocent spreadsheet.
Stories
The counterfeit's tell
The folk version is beloved because it is a magic trick that works: humans inventing numbers spread them evenly, nature never does. Ask a room to invent random expenses and Benford will find the humans. It circulates as proof that mathematics has a conscience, or at least a suspicion.
Assessment: Documented psychology: people genuinely default to uniform leading digits when fabricating. The tell is real; the morality is ours. The law detects fabrication style, not intent, and honest anomalies trigger it too.
Interesting claims
The election minefield
The interesting claim is where it overreached: after contested elections, Benford analyses of vote counts circulated as proof of fraud, and statisticians had to publicly explain that precinct-level vote data need not obey the law at all, bounded ranges break the scale invariance. The law became a conspiracy's exhibit and a case study in how a real theorem launders a bad argument.
Assessment: Documented misuse, documented corrections in the statistical literature. Benford stands in this archive as the honest tool most often forged into a weapon: valid law, invalid applications, and a warning that mathematics cited is not mathematics done.
Sources and further reading
Newcomb, Note on the Frequency of Use of the Different Digits in Natural Numbers (1881)
Benford, The Law of Anomalous Numbers (1938)
Nigrini, Digital Analysis Using Benford's Law (2000), the forensic method
Court records of Benford evidence in fraud cases
Post-election statistical statements on Benford misuse, 2020 onward
Worth asking
If fabricated data betrays itself in its first digits, what other fingerprints do lies leave in arithmetic?