The Digit 1 Shows Up About 30 Percent of the Time

In 1881, the astronomer Simon Newcomb noticed that the early pages of logarithm tables were more worn than the later ones. People were looking up logs of numbers starting with 1 far more often than logs of numbers starting with 9. He published a short note proposing that for many natural data sets, the probability of the leading digit being d is log₁₀(1 + 1/d).

The note was forgotten. In 1938 the physicist Frank Benford rediscovered the rule, tested it on 20,229 numbers — river drainage areas, atomic weights, baseball statistics, addresses on a Reader's Digest page — and gave it the name it now carries.

The formula predicts that 1 leads about 30.1 percent of the time, 2 about 17.6 percent, and 9 only about 4.6 percent. The pattern holds whenever the underlying data spans several orders of magnitude — populations, stock prices, file sizes — so the logarithm of the values is roughly uniform.

Forensic accountants now use Benford's law as a fraud screen. People making up plausible numbers tend to overuse middle digits. In 1993, Mark Nigrini's PhD work showed the rule could flag suspicious tax returns, and the Internal Revenue Service has used the technique since. It has been cited in audits of corporate filings (notably an Enron analysis) and in detecting election fraud, though its use as legal evidence remains contested.