[Math Lair] Benford's Law

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If we listed the first digits of each of the first 100 powers of 2, we would get:

1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 7, 1, 2, 5, 1, 2, 4, 9, 1, 3, 7, 1, 2, 5, 1, 2, 4, 9, 1, 3, 7, 1, 2, 5, 1, 2, 4, 9, 1, 3, 7, 1, 3, 6, 1, 2, 4, 9, 1, 3, 7, 1, 3, 6, 1, 2, 4, 9, 1, 3, 7, 1, 3, 6

Looking at these digits, not each digit appears with equal frequency. For example, the digit 1 appears 30 times (or 30% of the time), while the digit 9 only appears 5 times.

If you think about it, this makes sense. If you take any number, the first digit of its double will be 1 if the first digit of the original number is anywhere between 5 and 9, while the first digit of its double will be 9 only if the first two digits of the number are between 45 and 49.

If we take a look at the first digits of the first few numbers of the Fibonacci sequence, the results are similar:

1, 1, 2, 3, 5, 8, 1, 2, 3, 5, 8, 1, 2, 3, 6, 9, 1, 2, 4, 6, 1, 1, 2, 4, 7, 1, 1, 3, 5, 8

Looking at the first 30 numbers, 1 appears 9 times, or, again, 30% of the time, while 9 only appears once.

Benford's law, published by Frank Benford in 1938, states that, in various lists of numbers, the digit 1 appears in the leftmost position about 30% of the time, much greater than the 11.1% that would result if each digit occurred with equal probability. In general, in a number list, the probability of a digit appearing in the leftmost position is equal to log(1 + 1/n), where n is the digit. This gives the following probabilities, approximately:
First DigitProbability
130.1%
217.6%
312.4%
49.7%
57.9%
66.7%
75.8%
85.1%
94.6%