Benford's Law

Abstract

Some time around 1881, Simon Newcomb (1) was flipping through logarithmic tables and was struck at how much more quickly the first tables wear out than the subsequent ones. After some investigation, he found that the reason lay in the frequency of digits in the natural numbers. To quote “the fist significant figure is oftener 1 than any other digit, and the frequency diminishes up to nine. 1” By noticing that the frequency of lookups in anti-logarithm tables was completely uniform, Newcomb was able to infer the following law: that the first significant digits in a random tabulation of data obey a logarithmic distribution, with 1 being the most common, at 30.1% and frequency decreasing for digits up to 9 with frequency 4.5%. He gave a table of frequencies, but no algebraic formulation of the law and practically no theoretical justification. The paper was duly forgotten, and no more significant developments came to light for 57 more years, until in 1938 a researcher named Frank Benford (2) at the General Electric Company laboratories in Schenectady, New York collected 20 different collected data sets such as the area of rivers, constants, or addresses of the first 342 names mentioned in American Men of Science. Averaging all these numbers out, he observed the same curve as Newcomb.