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The Law of Anomalous Numbers

Frank Benford
Proceedings of the American Philosophical Society
Vol. 78, No. 4 (Mar. 31, 1938), pp. 551-572
Stable URL: http://www.jstor.org/stable/984802
Page Count: 22
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The Law of Anomalous Numbers
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Abstract

It has been observed that the first pages of a table of common logarithms show more wear than do the last pages, indicating that more used numbers begin with the digit 1 than with the digit 9. A compilation of some 20,000 first digits taken from widely divergent sources shows that there is a logarithmic distribution of first digits when the numbers are composed of four or more digits. An analysis of the numbers from different sources shows that the numbers taken from unrelated subjects, such as a group of newspaper items, show a much better agreement with a logarithmic distribution than do numbers from mathematical tabulations or other formal data. There is here the peculiar fact that numbers that individually are without relationship are, when considered in large groups, in good agreement with a distribution law-hence the name "Anomalous Numbers." A further analysis of the data shows a strong tendency for bodies of numerical data to fall into geometric series. If the series is made up of numbers containing three or more digits the first digits form a logarithmic series. If the numbers contain only single digits the geometric relation still holds but the simple logarithmic relation no longer applies. An equation is given showing the frequencies of first digits in the different orders of numbers 1 to 10, 10 to 100, etc. The equation also gives the frequency of digits in the second, third... place of a multi-digit number, and it is shown that the same law applies to reciprocals. There are many instances showing that the geometric series, or the logarithmic law, has long been recognized as a common phenomenon in factual literature and in the ordinary affairs of life. The wire gauge and drill gauge of the mechanic, the magnitude scale of the astronomer and the sensory response curves of the psychologist are all particular examples of a relationship that seems to extend to all human affairs. The Law of Anomalous Numbers is thus a general probability law of widespread application.

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