Benford's law

Benford's law, also called the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on. 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 from leading 1 digit to leading n digits. It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, physical and mathematical constants. Like other general principles about natural data—for example the fact that many data sets are well approximated by a normal distribution—there are illustrative examples and explanations that cover many of the cases where Benford's law applies, though there are many other cases where Benford's law applies that resist a simple explanation. It tends to be most accurate when values are distributed across multiple orders of magnitude, especially if the process generating the numbers is described by a power law (which are common in nature). It is named after physicist Frank Benford, who stated it in 1938 in a paper titled 'The Law of Anomalous Numbers', although it had been previously stated by Simon Newcomb in 1881. A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability