The first digit of almost any naturally occurring dataset follows a predictable, logarithmic distribution. Accountants use it to catch fraud. Nature uses it because it can't help itself.
Open a newspaper. Pick any number — a population, a stock price, a river length, an election result. The first digit is six times more likely to be 1 than 9. This isn't a coincidence. It's a mathematical law so reliable that the IRS uses it to flag suspicious tax returns.
Discovered by astronomer Simon Newcomb in 1881 (he noticed the early pages of logarithm books were more worn), then rediscovered by physicist Frank Benford in 1938, who tested it across 20,229 observations from river areas to baseball statistics.
Benford's Law predicts the probability of any first digit d as: P(d) = log₁₀(1 + 1/d). This gives us:
The intuitive explanation: numbers don't grow linearly, they grow multiplicatively. To go from a first digit of 1 to a first digit of 2, you need to grow by 100% (from 1000 to 2000). But to go from 8 to 9, you only need to grow by 12.5% (from 8000 to 9000). Numbers spend more "time" in the lower digits because there's more logarithmic distance to cover.
Think of an odometer. It stays in the 100,000s for as long as it was in all of the 10,000s through 99,999 combined. Every power of ten begins with 1.
"God may not play dice with the universe, but He appears to have a strong preference for small first digits." — Adapted from Mark Nigrini, forensic accountant
When humans fabricate numbers, they tend toward uniformity — they think each digit should appear about 11% of the time. This is psychologically intuitive but mathematically wrong. A forensic accountant applying Benford's Law to a company's expense reports can spot fabricated entries because they'll have too many 5s, 6s, 7s, 8s, and 9s and not enough 1s and 2s.
The technique is now standard in auditing, accepted as evidence in courts, and used by tax authorities worldwide. If you're going to commit fraud, at least make your fake numbers logarithmic.