Benford's law

Benford's law, also called the first-digit law, states that in lists of numbers from many 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 almost one third 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 one time in twenty. The basis for this "law" is that the values of real-world measurements are often distributed logarithmically, thus the logarithm of this set of measurements is generally distributed uniformly.

This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, although the exact proportions change.

It is named after physicist Frank Benford, who stated it in 1938,^[1] although it had been previously stated by Simon Newcomb in 1881.^[2] Although many "proofs" of this law have been put forth (starting with Newcomb himself), none were mathematically rigorous^[3] until Theodore P. Hill's in 1995.

Benford's Law is a powerful and relatively simple tool for pointing suspicion at frauds, embezzlers, tax evaders, sloppy accountants and even computer bugs.

Dr. Benford derived a formula to explain this. If absolute certainty is defined as 1 and absolute impossibility as 0, then the probability of any number "d" from 1 through 9 being the first digit is log to the base 10 of (1 + 1/d). This formula predicts the frequencies of numbers found in many categories of statistics.