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I love the Radiolab podcast. Today I listened to an excellent episode that introduced me to a principle of mathematics I was unfamiliar with: Benford’s law.
Benford’s law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty. This distribution of first digits arises logically whenever a set of values is distributed logarithmically. Measurements of real world values are often distributed logarithmically (or equivalently, the logarithm of the measurements is distributed uniformly).
This counter-intuitive result has been found to apply 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, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, although the exact proportions change.
That’s right: in many, many random areas 1’s are consistently and predictably more likely to be the first digit of numbers, 2’s more than 3’s… etc.
In fact, it’s reliable to such a degree that it’s used in forensic accounting. That’s right: when the IRS checks your returns, if there are too many numbers leading with 9’s and not enough starting with 1’s, it’s a very good sign you’re cheating on your taxes. So next time you make up a number keep in mind that “random numbers” are not always that random.
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