Grubby logarithm tables revealed that real-world numbers favour the digit 1
In 1881 the astronomer Simon Newcomb noticed something odd about the books of logarithm tables everyone used for calculation: the early pages, for numbers beginning with 1, were far more worn than the later ones. He had stumbled on a strange regularity. In much real data, about 30 percent of numbers start with 1.
If leading digits were spread evenly, each of the nine would turn up about 11.1 percent of the time. Instead, in data sets that follow the pattern, 1 leads roughly three times in ten while 9 leads less than one time in twenty. Newcomb proposed a formula for it, and even described the second digit. His note was largely forgotten until the physicist Frank Benford rediscovered the effect in 1938.
Benford gathered 20,229 observations from 20 sources: surface areas of 335 rivers, 104 physical constants, 1800 molecular weights, 418 death rates, even 308 numbers from one issue of Reader's Digest. The pattern held across them, and the law took his name, a neat example of Stigler's law, the joke that discoveries are rarely named after whoever found them first. Later tests found it in electricity bills, street addresses, stock and house prices and the heights of the world's tallest structures.
The explanation lies in logarithms. On a log scale the stretch from 1 to 2 is much wider than the stretch from 9 to 10, about 0.30 against 0.05. If numbers are spread evenly on that scale, far more of them land in the wide slot. So the law works best for data spanning several orders of magnitude, and it holds whether heights are measured in metres or feet. Restrict the range, say to villages with between 300 and 999 residents, and it collapses.
Some sequences obey it almost perfectly. The leading digits of successive powers of two run 1, 2, 4, 8, 1, 3, 6 and follow the distribution more closely than random numbers would. In 1995 Ted Hill proved a result about mixtures of distributions that helps explain the law's reach, and in binary the rule is true but trivial, since every number starts with 1.
Source: Benford's law
