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Benford's law, also called the First-Digit Law, is a phenomenological law about the frequency distribution of leading digits in many (but not all) real-life sets of numerical data. That law states that in many naturally occurring collections of numbers the small digits occur disproportionately often as leading significant digits.〔 For example, in sets which obey the law the number would appear as the most significant digit about 30% of the time, while larger digits would occur in that position less frequently: would appear less than 5% of the time. If all digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also concerns the expected distribution for digits beyond the first, which approach a uniform distribution. It has been shown that this result applies 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,〔Paul H. Kvam, Brani Vidakovic, ''Nonparametric Statistics with Applications to Science and Engineering'', p. 158〕 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 here 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 is named after physicist Frank Benford, who stated it in 1938,〔 (subscription required)〕 although it had been previously stated by Simon Newcomb in 1881.〔 (subscription required)〕 ==Mathematical statement== A set of numbers is said to satisfy Benford's law if the leading digit ''d'' (''d'' ∈ ) 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Benford's law」の詳細全文を読む スポンサード リンク
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