Benford's law

Benford's law describes a logarithmic distribution of leading digits in many datasets spanning several orders of magnitude: digit d appears first with probability log₁₀(1 + 1/d).

Leading digit: The first nonzero digit in a number's usual decimal representation. Benford's law assigns a specific probability to each possible first significant digit.
Forensic accounting: The use of accounting methods to investigate disputes, fraud, and financial misconduct. Investigators sometimes compare ledger figures with Benford's predictions to select records for closer examination.
Simon Newcomb: An American astronomer and mathematician who studied celestial mechanics and mathematical tables. In 1881, Newcomb noticed that early pages of logarithm tables were more worn than later pages.
Uniform distribution: A probability distribution assigning equal probability to every outcome in a specified set. Benford's law favors small leading digits instead of assigning one-ninth probability to each digit.
Logarithm: The inverse operation of exponentiation, measuring how many powers of a base produce a number. The law's digit probabilities come from logarithms of ratios between successive powers of ten.
Fraud detection: The identification of deliberate deception through evidence, analysis, or monitoring. Unexpected leading-digit frequencies can be a screening signal, not proof, of fabricated data.
Frank Benford: An American physicist who documented leading-digit frequencies across diverse datasets. His 1938 study supplied broad empirical evidence that later gave the law its familiar name.
Random number: A value generated by a process intended to prevent prediction from available information. Randomly chosen numbers from a fixed range need not follow Benford's leading-digit distribution.
Scale invariance: A property in which a pattern remains unchanged when quantities are multiplied by a common factor. Multiplying a Benford-distributed dataset by many constants preserves its leading-digit pattern.
Chi-squared test: A statistical test that compares observed categorical counts with counts expected under a specified model. It can quantify whether observed first-digit counts differ from Benford's expected frequencies.

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Equidistribution theoremRelated: Uniformity of logarithms modulo one explains when leading digits follow this law.