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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.
“It applies to quakes, the brightness of gamma rays reaching Earth and the rotations of dead stars”
That Benford’s law pops up in so many natural phenomena won’t surprise mathematicians but may shock some scientists. When Sambridge’s team presented Benford’s law findings at a 2009 geoscience conference, one dubious attendee “thought we were having a laugh”, he recalls.
In 1938, the American physicist Frank Benford revisited the phenomenon, which he called the “Law of Anomalous Numbers” in a survey with more than 20,000 observations of empirical data compiled from various sources, ranging from areas of rivers to molecular weights of chemical compounds, cost data, address numbers, population sizes and physical constants. All of them, to a greater or lesser extent, followed such an exponentially diminishing distribution.
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Image credit: “Numbers follow a surprising law of digits, and scientists can’t explain why” (phys.org...)