Benford's Mystery

Benford's 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 leading digit about 30% of the time, while larger numbers occur in that position less frequently, with 9 occurring as the first digit less than 5% of the time, ie 6.5 times less often than 1.

This result has been found to apply in diverse contexts, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, depths of earthquakes, rotation rates of pulsars, 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.

This distribution is scale invariant, in that changing from feet to metres, or kilograms to ounces does not change the pattern.

The graph below shows Benford's Law:

d is the leading digit and P(d) its probability

What is the reason behind this?


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