Benford's Law - Mathematical Statement

Mathematical Statement

A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability

Numerically, the leading digits have the following distribution in Benford's law, where d is the leading digit and P(d) the probability:

The quantity P(d) is proportional to the space between d and d + 1 on a logarithmic scale. Therefore, this is the distribution expected if the mantissae of the logarithms of the numbers (but not the numbers themselves) are uniformly and randomly distributed. For example, a number x, constrained to lie between 1 and 10, starts with the digit 1 if 1 ≤ x < 2, and starts with the digit 9 if 9 ≤ x < 10. Therefore, x starts with the digit 1 if log 1 ≤ log x < log 2, or starts with 9 if log 9 ≤ log x < log 10. The interval is much wider than the interval (0.30 and 0.05 respectively); therefore if log x is uniformly and randomly distributed, it is much more likely to fall into the wider interval than the narrower interval, i.e. more likely to start with 1 than with 9. The probabilities are proportional to the interval widths, and this gives the equation above. (The above discussion assumed x is between 1 and 10, but the result is the same no matter how many digits x has before the decimal point.)

An extension of Benford's law predicts the distribution of first digits in other bases besides decimal; in fact, any base b ≥ 2. The general form is:

For b = 2 (the binary number system), Benford's law is true but trivial: All binary numbers (except for 0) start with the digit 1. (On the other hand, the generalization of Benford's law to second and later digits is not trivial, even for binary numbers.) Also, Benford's law does not apply to unary systems such as tally marks.