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.
Available at: http://works.bepress.com/tphill/75/