A Statistical Derivation of the Significant-Digit Law

Abstract

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.