Finance professor Jialan Wang won the Internet today with a beautiful note on Benford’s law in US accounting data (for completeness of her victory see here, here, here, here, and here).

Here’s the argument. Benford’s Law is a statistical regularity that applies to many collections of numbers of differing orders of magnitude. As Wang writes:

A second earth-shattering fact is that there are more numbers in the universe that begin with the digit 1 than 2, or 3, or 4, or 5, or 6, or 7, or 8, or 9. And more numbers that begin with 2 than 3, or 4, and so on. This relationship holds for the lengths of rivers, the populations of cities, molecular weights of chemicals, and any number of other categories.

The explanation generally seems linked with exponential growth, and the formula is P(d) = log10 (1 + 1/d). So the probability of a number starting with a 1 is log 2, or 30%; the probability of it starting with a 9 is log 1.11, or about 4.6%. Strong men have been driven mad peering into this abyss.

Benford’s law ought to hold for lots of kinds of financial data, particularly if you just take a big unsorted pile of stuff. So Wang took 50 years of various financial data (revenues, assets, and 41 other publicly reported categories) from 20,000 publicly reporting companies and just plotted the number of numbers that started with 1s, 2s, 3s … etc. And it was a pretty good match to the Benford distribution:

So far so good. Now the bad news: the relationship has been moving away from a Benford distribution over time.
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