Statistical Science

A Statistical Derivation of the Significant-Digit Law

Theodore P. Hill

Full-text: Open access


The history, empirical evidence and classical explanations of the significant-digit (or Benford's) law are reviewed, followed by a summary of recent invariant-measure characterizations. Then a new statistical derivation of the law in the form of a CLT-like theorem for significant digits is presented. If distributions are selected at random (in any "unbiased" way) and random samples are then taken from each of these distributions, the significant digits of the combined sample will converge to the logarithmic (Benford) distribution. This helps explain and predict the appearance of the significant-digit phenomenon in many different empirical contexts and helps justify its recent application to computer design, mathematical modelling and detection of fraud in accounting data.

Article information

Statist. Sci. Volume 10, Number 4 (1995), 354-363.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


First-digit law Benford's law significant-digit law scale invariance base invariance random distributions random probability measures random $k$-samples mantissa logarithmic law mantissa sigma algebra


Hill, Theodore P. A Statistical Derivation of the Significant-Digit Law. Statist. Sci. 10 (1995), no. 4, 354--363. doi:10.1214/ss/1177009869.

Export citation