# Is there a proof of Benford's Law? [duplicate]

As stated by Wikipedia (here):

Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time.

I find this fascinating and confusing. Benford's original contribution came after the first statement of the phenomenon, and in it he used a data sample with over 20 thousand entries. It has since been tested on data samples that number in the hundreds of thousands.

Statistically, it seems sound. But why should it be true? What is the intuition behind this phenomenon?

Most importantly, is there a proof?

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## marked as duplicate by Raskolnikov, vadim123, O.L., AWertheim, Julian KuelshammerJun 13 '13 at 21:00

The intuition behind ist is scale invariance. The distribution of leading digits of distances between cities, say, should be the same if we measure in centimeters or in inch or in alpha-centaurian gargle-feet. This is only possible if the fractional parts of the logarithms of the values are equidistributed. Under this assumtion, the law can be proved. But how will you prove it about "real-life sources of data" if there is no mathematicla definition of "real-life source of data". –  Hagen von Eitzen Oct 7 '12 at 16:39
@HagenvonEitzen I have heard this argument a number of times, but I think it begs the question. Why should the distribution of the leading digit be independent of the unit of measure? It is certainly not true of the height of human beings, for example. I think an important requirement is that the distribution spans several orders of magnitude, and that the probability density is not “too wild” within this range. –  Harald Hanche-Olsen Oct 7 '12 at 16:49
Related question: Why does Benford's Law (or Zipf's Law) hold? –  Nate Eldredge Oct 7 '12 at 17:13
See the open-access paper by T. Hill, "A Statistical Derivation of the Significant-Digit Law," Statist. Sci. Volume 10, Number 4 (1995), pp. 354-363. –  Sasha Oct 7 '12 at 17:18
Terence Tao on the subject: terrytao.wordpress.com/2009/07/03/… –  Shitikanth Oct 7 '12 at 17:47

To 'demystify' Benford's Law ... http://www.benfords-law.com/

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Hi, @doug, welcome to the site. We welcome these links for answers, but I personally prefer a summary of relevant points; if the summary is good and the link is to a strong reference, then you get my upvote. –  peoplepower Oct 24 '12 at 0:22