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

Arno Berger; Theodore P. Hill

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arXiv · 1005.2598

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

Arno Berger·Theodore P. Hill

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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, Feller gives a sufficient condition ("large spread") for a random variable $X$ to be approximately Benford distributed, that is, for $\log_{10}X$ to be approximately uniformly distributed modulo one. 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 demonstrate that large spread (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

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Arno Berger, Theodore P. Hill. 2010-05-14. Fundamental Flaws in Feller's Classical Derivation of Benford's Law. https://arxiv.org/abs/1005.2598

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