The Newcomb–Benford law: Scale invariance and a simple Markov process based on it

Andrea Burgos,Andrés Santos

doi:10.1119/10.0004957

Abstract

The Newcomb–Benford law, also known as the first-digit law, gives the probability distribution associated with the first digit of a dataset so that, for example, the first significant digit has a probability of 30.1% of being 1 and 4.58% of being 9. This law can be extended to the second and next significant digits. This article presents an introduction to the discovery of the law and its derivation from the scale invariance property as well as some applications and examples. Additionally, a simple model of a Markov process inspired by scale invariance is proposed. Within this model, it is proved that the probability distribution irreversibly converges to the Newcomb–Benford law, in analogy to the irreversible evolution toward equilibrium of physical systems in thermodynamics and statistical mechanics.

Highlights

In the late 19th century, an astronomer and mathematician visits his institution’s library and consults a table of logarithms to perform certain astronomical calculations
In a long list of records frng obtained from measurements or observations, the fraction pd of records beginning with the significant digit d 1⁄4 1; 2; ...; 9 is not pd 1⁄4 1=9, as one might naively expect, but rather follows a logarithmic law
Why is the first digit not evenly distributed among the nine possible values? A simple argument shows that, if a robust distribution law exists, it cannot be the uniform distribution whatsoever

Summary

INTRODUCTION

In the late 19th century, an astronomer and mathematician visits his institution’s library and consults a table of logarithms to perform certain astronomical calculations. A little over half a century later, a physicist and electrical engineer, who is unaware of his predecessor’s discovery, observes the same curious phenomenon on the pages of logarithm tables and arrives at the same conclusion. Both have understood that, in a long list of records frng obtained from measurements or observations, the fraction pd of records beginning with the significant digit d 1⁄4 1; 2; ...; 9 is not pd 1⁄4 1=9, as one might naively expect, but rather follows a logarithmic law.

ORIGIN OF THE LAW

APPLICATIONS AND EXAMPLES

CONCLUDING REMARKS

Findings

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