Benford's law · the first significant digit
In a heap of real numbers the leading digit is rigged, and the house always favours the one.
River lengths, city populations, physical constants, the sizes of files on a disk. Read enough of them and tally only the digit each one begins with. The ones arrive near thirty per cent of the time — three times what intuition allows — and the whole count falls away in a fixed logarithmic staircase down to a bare five per cent for the nines. Here the law is spelled in weight: the more often a digit leads, the heavier it is set.
Leading-digit tally
Why the ones win
A quantity that grows by multiplying — interest, populations, the length of a river gathering tributaries — spends far longer climbing from 1 to 2 than it does racing from 9 back to 10. Space the number line by ratio instead of by count and the ones simply own more of it.
The staircase is fixed
Each digit d leads with probability log₁₀(1 + 1/d). That is 30, 18, 12, 10, 8, 7, 6, 5 and 5 per cent — a curve that does not care whether you counted in metres, miles or megabytes.
A second read
Because honest data obeys it, invented data usually doesn't. Auditors run this exact tally over tax returns and expense claims; a ledger whose sevens and eights lead too often is a ledger worth a second look.