benford's law

why datasets keep starting with 1 way more than 9

look at the leading digit of a big pile of real-world numbers — populations, river lengths, stock prices, electricity bills, whatever. the first digit isn't uniform. 1 shows up ~30% of the time, 9 only ~4.6%. not a fluke, not a bias in how we count: it falls out of the numbers living on a log scale.

observed benford predicted

the trick: numbers spread evenly across orders of magnitude (1–10, 10–100, 100–1000…) aren't uniform on the number line — they're uniform on a log line. on a log scale, the span from 1 to 2 is as wide as from 2 to 4, so digits "1" gets a bigger slice. the probability the leading digit is d is log10(1 + 1/d).

the uniform-random button is there on purpose: benford isn't universal. draw numbers uniformly from 1 to 1,000,000 and you get a flat first-digit histogram — it only appears when the data spans many magnitudes and grows multiplicatively. that's why it's a forensic tool: fabricated numbers (like a person making up tax figures) tend to be too uniform.