Why so many numbers start with 1

Question

Take a large collection of numbers that span several orders of magnitude: populations of towns, lengths of rivers, company revenues, stock prices. Look at the first digit of each. 1. Naively each digit 1 to 9 should be the first digit about 1/9 of the time. What fraction actually start with 1? 2. Explain why, using the idea that the collection should look the same whether measured in metres or feet, dollars or euros. 3. Why do auditors use this to detect fabricated accounts?

Accepted Answer

About 30 per cent of such numbers start with 1, and only about 4.6 per cent start with 9. This is Benford's law: the fraction starting with digit d is log base 10 of (1 + 1/d). Scale invariance. A collection like "lengths of rivers" has no natural unit: measured in kilometres or miles, it should have the same shape of first-digit distribution. Multiplying every number by a constant c shifts every logarithm by log c. The only distribution of the fractional part of the logarithm that is unchanged by every such shift is the uniform distribution on the interval from 0 to 1. So for a scale-free co…

Why so many numbers start with 1

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