Benford's Law

Benford's law, also called the first-digit law, refers to the frequency distribution of digits in many (but not all) real-life sources of data. In this distribution, the number 1 occurs as the first digit about 30% of the time, while larger numbers occur in that position less frequently, with larger numbers occurring less often: 9 as the first digit less than 5% of the time. This distribution of first digits is the same as the widths of gridlines on a logarithmic scale. Benford's law also concerns the expected distribution for digits beyond the first, which approach a uniform distribution.

This 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). It tends to be most accurate when values are distributed across multiple orders of magnitude.

The graph to the right shows Benford's law for base 10. There is a generalization of the law to numbers expressed in other bases (for example, base 16), and also a generalization to second digits and later digits.

It is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881.