The Mathematics of Benford's Law -- A Primer

TL;DR

This paper offers an accessible overview of the mathematical principles behind Benford's law, helping researchers and students understand when and why it appears in real-world data.

Contribution

It provides a clear, concise explanation of the core mathematical theory of Benford's law, suitable for scientists and students with basic calculus and probability knowledge.

Findings

Clarifies conditions for Benford's law appearance in data

Includes fundamental examples and theorems without proofs

Serves as an educational resource for understanding Benford's law

Abstract

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.