Title

Fundamental Flaws in Feller’s Classical Derivation of Benford’s Law

Author Info

Arno Berger, University of Alberta
Theodore P. Hill, Georgia Institute of Technology - Main CampusFollow

Recommended Citation

Available from Mathematics ArXiv, May 14, 2010.

Copyright © 2010 Arno Berger and Theodore P. Hill. The definitive version is available at http://arxiv.org/abs/1005.2598v1.

Abstract

Feller’s classic text An Introduction to Probability Theory and its Applications contains a derivation of the well known significant-digit law called Benford’s law. More specifically, Fellergives a sufficient condition (“large spread”) for a random variable X to be approximately Benford distributed, that is, for log₁₀X to be approximately uniformly distributed moduloone. This note shows that the large-spread derivation, which continues to be widely cited and used, contains serious basic errors. Concrete examples and a new inequality clearly demonstratethat larges pread (or large spread on a logarithmic scale) does not imply that a random variable is approximately Benford distributed, for any reasonable definition of “spread” or measure of dispersion.