Benford or not Benford: new results on digits beyond the first

TL;DR

This paper investigates the distribution of digits beyond the first in data sets, proposing a new model based on an upper bound that better fits observed digit frequencies than traditional Benford's Law extensions.

Contribution

It introduces a novel model for digit distribution beyond the first digit, which depends on an upper bound and offers improved fit over Hill's 1995 law.

Findings

Digit distribution follows a law determined by an upper bound.

The proposed model fluctuates around Hill's theoretical values.

Knowing the upper bound improves the fit of the distribution.

Abstract

In this paper, we will see that the proportion of d as p th digit, where p > 1 and d $\in$ 0, 9, in data (obtained thanks to the hereunder developed model) is more likely to follow a law whose probability distribution is determined by a specific upper bound, rather than the generalization of Benford's Law to digits beyond the first one. These probability distributions fluctuate around theoretical values determined by Hill in 1995. Knowing beforehand the value of the upper bound can be a way to find a better adjusted law than Hill's one.