Quantitative Partition Models and Benford's Law

TL;DR

This paper explores how various quantitative partition models relate to Benford's Law, revealing that the law applies under specific conditions linked to skewness and partitioning behavior, and proposing a broader perspective on its universality.

Contribution

It introduces a new viewpoint that Benford's Law is a subset of the positive skewness phenomenon in partition models, expanding understanding of its widespread occurrence.

Findings

Benford's Law applies only in certain partition cases and constraints.

Universal feature: many small parts and few large parts across models.

Benford's Law linked to positive skewness in partitioning phenomena.

Abstract

Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. The two essential prerequisites for data configuration with regards to compliance with Benford's Law are high order of magnitude and positive skewness with a tail falling to the right of the histogram, so that quantitative configuration is such that the small is numerous and the big is rare. In this article various quantitative partition models are examined in terms of the quantitative and digital behavior of the resultant set of parts. The universal feature found across all partition models is having many small parts but only very few big parts, while Benford's Law is valid only in some particular partition cases and under certain constraints. Hence another suggested vista of Benford's Law is viewing it as a particular subset of the broader positive skewness phenomenon in quantitative partitioning. Significantly, such a vista is true in all other causes and explanations of Benford's Law where the small consistently outnumbers the big also in partial structures of the model or well before full convergence to Benford is achieved - endowing the principle universality in a sense. In conclusion, either the active act of partitioning or the passive consideration of a large quantity as the composition of smaller parts can be considered as another independent explanation for the widespread empirical observation of Benford's Law in the physical sciences.