[D66] Benford's law

[René Oudeweg]

reinold at fedora:~/Projects/fibon$ describe "Benfords law"  en --exact
en:
Benford's law
Benford's law, also known as the Newcomb–Benford law, the law of 
anomalous numbers, or the first-digit law, is an observation that in 
many real-life sets of numerical data, the leading digit is likely to be 
small. In sets that obey the law, the number 1 appears as the leading 
significant digit about 30% of the time, while 9 appears as the leading 
significant digit less than 5% of the time. If the digits were 
distributed uniformly, they would each occur about 11.1% of the time. 
Benford's law also makes predictions about the distribution of second 
digits, third digits, digit combinations, and so on.
The graph to the right shows Benford's law for base 10, one of 
infinitely many cases of a generalized law regarding numbers expressed 
in arbitrary (integer) bases, which rules out the possibility that the 
phenomenon might be an artifact of the base-10 number system. Further 
generalizations published in 1995 included analogous statements for both 
the nth leading digit and the joint distribution of the leading n 
digits, the latter of which leads to a corollary wherein the significant 
digits are shown to be a statistically dependent quantity.
It has been shown that this result applies to a wide variety of data 
sets, including electricity bills, street addresses, stock prices, house 
prices, population numbers, death rates, lengths of rivers, and physical 
and mathematical constants. Like other general principles about natural 
data—for example, the fact that many data sets are well approximated by 
a normal distribution—there are illustrative examples and explanations 
that cover many of the cases where Benford's law applies, though there 
are many other cases where Benford's law applies that resist simple 
explanations. Benford's Law tends to be most accurate when values are 
distributed across multiple orders of magnitude, especially if the 
process generating the numbers is described by a power law (which is 
common in nature).
The law is named after physicist Frank Benford, who stated it in 1938 in 
an article titled "The Law of Anomalous Numbers", although it had been 
previously stated by Simon Newcomb in 1881.The law is similar in 
concept, though not identical in distribution, to Zipf's law.


https://en.wikipedia.org/wiki/Benford%27s_law