Benford law distribution

Simon Newcomb published the law first, and nobody noticed. His two-page "Note on the frequency of use of the different digits in natural numbers" appeared in the American Journal of Mathematics in 1881 and gave the logarithmic rule outright: the probability that a number begins with digit d is log10(1 + 1/d). The paper sank without trace.

Fifty-seven years later Frank Benford, a physicist at General Electric, arrived at the same rule and did the empirical work that made it stick. "The Law of Anomalous Numbers" ran in the Proceedings of the American Philosophical Society in 1938 with 20,229 observations drawn from twenty unrelated sources — 335 river surface areas, 3,259 US population figures, 104 physical constants, 1,800 molecular weights, 5,000 entries lifted from a mathematical handbook, 308 numbers plucked from an issue of Reader's Digest, 342 street addresses, death rates and more. Pooled together, the leading digits tracked the logarithmic curve closely.

A story is usually attached to the discovery: that one or other man noticed the early pages of a book of logarithm tables were grubbier than the later ones, because numbers starting with 1 get looked up more often. The anecdote is told about both Newcomb and Benford, and is better treated as a teaching illustration than as documented history.

The theory caught up in stages. Roger Pinkham showed in 1961 that Benford's is the only first-digit law that survives a change of units — if a rule holds for data measured in miles it must also hold in kilometres, and that constraint forces the logarithm. Theodore Hill proved in 1995 that sampling from a random mixture of distributions tends towards Benford even when no single distribution does. And one condition makes the law exact rather than approximate: if the base-10 logarithm of the data is spread evenly across a whole number of decades, the leading digits obey it precisely. That is the construction this page samples from.

Since the law is named after the second person to find it, it is a stock example of Stigler's law of eponymy — which Stephen Stigler, fittingly, credited to Robert Merton.

Key properties

• The probability that a value begins with digit d is log10(1 + 1/d): 30.103% for 1, falling to 4.576% for 9.
• The nine probabilities sum to exactly 1, because the logarithms telescope to log10(10/1) = 1.
• Benford’s law is the only first-digit distribution invariant under a change of scale, so the units the data is recorded in cannot matter (Pinkham, 1961).
• A quantity whose base-10 logarithm is uniformly distributed over a whole number of decades follows the law exactly — that is how this page samples.
• The second-digit distribution is far flatter, from about 12.0% for 0 down to 8.5% for 9, and by the fourth significant digit it is nearly uniform.
• The leading digits of the powers of 2, and of the Fibonacci numbers, follow the law in the limit, because log10(2) and log10(φ) are irrational and so their multiples are equidistributed modulo 1.
• The law is not expected to hold for data confined to one order of magnitude (heights, IQ scores), for normal or uniform distributions, or for assigned numbers such as invoice, cheque or phone numbers.
• Multiplying every value in a conforming dataset by a constant leaves the expected digit distribution unchanged — a direct consequence of scale invariance.

Where they turn up

• Forensic accounting: digit tests on ledgers, expense claims and tax filings are a standard screening step in audit software, flagging sets of numbers worth a closer look rather than proving anything on their own.
• Macroeconomic data screening: a widely cited 2011 study applied digit tests to EU government accounting data and reported that the Greek figures deviated most from Benford’s law.
• Election forensics, where the method is genuinely contested. Joseph Deckert, Mikhail Myagkov and Peter Ordeshook argued in 2011 that the law is misleading as a fraud indicator; Walter Mebane, who defends digit tests, still accepts that many caveats apply.
• Claims that circulated after the 2020 US presidential election ran Benford tests on precinct-level vote counts. Mebane pointed out that precinct totals are not useful for diagnosing fraud, because they span too narrow a range for the law to be expected to hold at all.
• Image forensics research: the DCT coefficients of a JPEG approximately follow a generalised Benford distribution, and departures from it have been proposed as a way to detect re-compressed or manipulated images.