Benford's law: why 1 leads

Take the first digit of every number in a big, messy real-world list — river lengths, populations, invoice totals. About 30% begin with 1, and under 5% begin with 9. The digits are not equally likely, and the reason is that the numbers spread across many powers of ten.

Why it matters

Made-up numbers tend to have flat first digits, because people imagine digits as fair. Real quantities that grow or scale do not, and the mismatch is measurable. Forensic accountants test ledgers against the expected digit frequencies, and physicists have checked the same pattern in atomic spectra and half-lives. It also shows something odd about measurement: the pattern holds whether you record rivers in miles or kilometres, which no fair-digit rule could manage.

The mechanism

Write a number as a digit part times a power of ten: 3,400 is 3.4 × 10³. The first digit depends only on the 3.4 — the mantissa — and never on the power. So the question is: how is the mantissa spread between 1 and 10?

Now the key move. Take logarithms base 10. A number's log has a whole-number part (the power of ten) and a fractional part (the log of the mantissa). On a log scale, the numbers from 1 to 2 occupy the stretch from log 1 = 0 to log 2 ≈ 0.301. The numbers from 9 to 10 occupy the stretch from 0.954 to 1. The 1s get a slice of width 0.301; the 9s get 0.046.

So if a set of numbers is spread evenly on the log scale — meaning the fractional parts of their logs are spread uniformly between 0 and 1 — then the chance of a first digit d is log₁₀(d + 1) − log₁₀(d). That gives 30.1% for 1, 17.6% for 2, down to 4.6% for 9. That formula is Benford's law.

When is data spread evenly on the log scale? Whenever it covers several powers of ten and has no particular preferred size. Anything that grows by a percentage each period — populations, prices, compound interest — walks along the log scale at a steady pace, so it lands evenly. Multiplying such a set by any constant, such as changing miles to kilometres, only slides everything along the log scale, and a uniform spread slid along stays uniform. That is why the law survives a change of units.

Data that does not spread — adult heights in centimetres, all between 150 and 200 — has no reason to obey it, and does not.

In one breath

A number's first digit is fixed by where it sits within its power of ten, and on a logarithmic scale the 1s own the widest slice. Data that spans many powers of ten with no favourite size is spread evenly on that scale, so first digits fall out in proportion to slice width: 30% for 1, 5% for 9, and nothing changes when you switch units.