Benford's Law -- from Wolfram MathWorld

Benford's Law -- from Wolfram MathWorld: Benford's law applies not only to scale-invariant data, but also to numbers chosen from a variety of different sources. Explaining this fact requires a more rigorous investigation of central limit-like theorems for the mantissas of random variables under multiplication. As the number of variables increases, the density function approaches that of the above logarithmic distribution. Hill (1998) rigorously demonstrated that the "distribution of distributions" given by random samples taken from a variety of different distributions is, in fact, Benford's law (Matthews).

One striking example of Benford's law is given by the 54 million real constants in Plouffe's "Inverse Symbolic Calculator" database, 30% of which begin with the digit 1. Taking data from several disparate sources, the table below shows the distribution of first digits as compiled by Benford (1938) in his original paper.

Curious mathematical law is rife in nature - physics-math - 14 October 2010 - New Scientist

Curious mathematical law is rife in nature: A subject of fascination to mathematicians, Benford's law states that for many sets of numbers, the first or "leading" digit of each number is not random. Instead, there is a 30.1 per cent chance that a number's leading digit is a 1. Progressively higher leading digits get increasingly unlikely, and a number has just a 4.6 per cent chance of beginning with a 9 (see diagram).

The law is named after physicist Frank Benford, who in 1938 showed that the trend appears in many number sets, from the surface area of rivers to baseball statistics to figures picked randomly from a newspaper. It later emerged that such distributions are "scale-invariant": if you convert the units of the numbers in the set, from metres to yards, say, the set will still conform to Benford's law.

Benford's Law And A Theory of Everything - Technology Review

Benford's Law And A Theory of Everything - Technology Review: "Today, Lijing Shao and Bo-Qiang Ma at Peking University in China provide a new insight into the nature of Benford's law. They examine how Benford's law applies to three kinds of statistical distributions widely used in physics.

These are: the Boltzmann-Gibbs distribution which is a probability measure used to describe the distribution of the states of a system; the Fermi-Dirac distribution which is a measure of the energies of single particles that obey the Pauli exclusion principle (ie fermions); and finally the Bose-Einstein distribution, a measure of the energies of single particles that do not obey the Pauli exclusion principle (ie bosons)."