Intuitive Considerations Clarifying the Origin and Applicability of the Benford Law

TL;DR

This paper explains that the Benford law originates from the positional notation system and introduces alternative formulas for digit distribution, demonstrating their application to infrared spectra analysis and discussing cases where Benford's law does not hold.

Contribution

It reveals the fundamental origin of Benford's law from positional notation and derives alternative predictive formulas for digit distribution in data sets.

Findings

Benford law stems from positional notation principles.

Alternative formulas effectively predict digit distribution.

Infrared spectra analysis illustrates application and violations.

Abstract

The diverse applications of the Benford law attract investigators working in various fields of physics, biology and sociology. At the same time, the groundings of the Benford law remain obscure. Our paper demonstrates that the Benford law arises from the positional (place-value) notation accepted for representing various sets of data. An alternative to Benford formulae to predict the distribution of digits in statistical data are derived. Application of these formulae to the statistical analysis of infrared spectra of polymers is presented. Violations of the Benford Law are discussed.