Published in Statistical Science, Volume 10, Issue 4, November 1, 1995, pages 354-363. Copyright © 1995 Institute of Mathematical Statistics. The definitive version is available at http://www.jstor.org/stable/2246134.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
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 combines sample will converge to the logarithmic (Benford) distribution. This helps explain and predict the appearance of the significant0digit phenomenon in many different empirical contexts and helps justify its recent application to computer design, mathematical modeling and detection of fraud in accounting data.