Exponential Growth Series and Benford's Law

TL;DR

This paper explores the relationship between exponential growth series and Benford's Law, providing empirical evidence, theoretical insights, and mathematical proofs to explain why such data often follows Benford's distribution.

Contribution

The paper offers a comprehensive analysis linking exponential growth to Benford's Law, including empirical validation and rigorous proofs for continuous and discrete cases.

Findings

Nearly all high-growth exponential series are Benford.

Empirical data from half a million series confirms theoretical predictions.

Mathematical proof shows continuous growth exactly obeys Benford's Law.

Abstract

Exponential growth occurs when the growth rate of a given quantity is proportional to the quantity's current value. Surprisingly, when exponential growth data is plotted as a simple histogram disregarding the time dimension, a remarkable fit to the positively skewed k/x distribution is found, where the small is numerous and the big is rare. Such quantitative preference for the small has a corresponding digital preference known as Benford's Law which predicts that the first significant digit on the left-most side of numbers in typical 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. Exponential growth series with high growth rate are nearly perfectly Benford given that plenty of elements are considered. An additional constraint is that the logarithm of the growth factor must be an irrational number. Since the irrationals vastly outnumber the rationals, on the face of it, this constraint seems to constitute the explanation of why almost all growth series are Benford, yet, in reality this is all too simplistic, and the real and more complex explanation is provided in this article. Empirical examinations of close to a half a million growth series via computerized programs almost perfectly match the prediction of the theoretical study on rational versus irrational occurrences, thus in a sense confirming both, the empirical work as well as the theoretical study. In addition, a rigorous mathematical proof is provided in the continuous growth case showing that it exactly obeys Benford's Law. A non-rigorous proof is given in the discrete case via uniformity of mantissa argument. Finally cases of discrete series embedded within continuous series are studied, detailing the degree of deviation from the ideal Benford configuration.