The mathematics of Benford’s law: a primer


This article provides a concise overview of the main mathematical theory of Benford’s law in a form accessible to scientists and students who have had first courses in calculus and probability. In particular, one of the main objectives here is to aid researchers who are interested in applying Benford’s law, and need to understand general principles clarifying when to expect the appearance of Benford’s law in real-life data and when not to expect it. A second main target audience is students of statistics or mathematics, at all levels, who are curious about the mathematics underlying this surprising and robust phenomenon, and may wish to delve more deeply into the subject. This survey of the fundamental principles behind Benford’s law includes many basic examples and theorems, but does not include the proofs or the most general statements of the theorems; rather it provides precise references where both may be found.

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The first author was partially supported by an NSERC Discovery Grant. Both authors are grateful to the Joint Research Centre of the European Commission for the invitation to speak at their Cross-domain conference on Benford’s Law Applications in Stresa, Italy in July 2019, and especially to the organizers of that conference, Professors Domenico Perrota, Andrea Cerioli, and Lucio Barabesi for their warm hospitality. They also wish to thank two anonymous referees for several thoughtful comments and suggestions.

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Correspondence to Arno Berger.

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Berger, A., Hill, T.P. The mathematics of Benford’s law: a primer. Stat Methods Appl (2020).

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  • Benford’s law
  • Significant
  • Scale invariance
  • Base invariance
  • Random probability measure