A General Example for Benford Data

Abstract

Benford’s Law deals, among other things, with the proportion of numbers whose first significant digit is a 1 (e.g., 0.00131 and 19668 both have first significant digit 1) in a variety of datasets. In these datasets, which arise in various compendiums or as mixtures of various sets of numbers, the proportion of numbers with first significant digit one is 0.3010 which is much higher than the commonsense value of 1 / 9. The reasons for this occurrence have been elusive. Mathematical attempts to explain this phenomenon have been relatively fruitless. Methods involving probability have been somewhat more successful. In this article we give some simple reasons for this occurrence and also give an example of a general mixture of distributions which exactly satisfies this Law. Various other examples and counterexamples are also given.

Keywords:  :

• Benford’s Law
• Law of the Stripey Hat
• Significant digits