Benford's law
Pick a random number from a company's ledger, a country's population table or a river-length list. The chance it starts with 1 is about 30% — not 11%.
The rule
For data that spans several orders of magnitude, the probability that the leading digit is d is log10(1 + 1/d). Astronomer Simon Newcomb noticed it in 1881 from the wear on logarithm tables; physicist Frank Benford tested it on thousands of data sets in 1938.
| Leading digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| Expected share | 30.1% | 17.6% | 12.5% | 9.7% | 7.9% | 6.7% | 5.8% | 5.1% | 4.6% |
Why it happens
Growth is multiplicative. A value growing 10% a year spends a long time between 1,000 and 2,000 (it must double) but a short time between 9,000 and 10,000 (an 11% rise). Mix many such quantities across scales and leading 1s dominate.
Where it applies — and where it doesn't
- Applies: invoice amounts, expense claims, populations, stock volumes, election vote counts at many levels, physical constants.
- Doesn't apply: assigned numbers — phone numbers, PIN and ZIP codes, invoice IDs — and capped or narrow ranges like human heights or exam scores.
How auditors use it
People inventing numbers tend to spread first digits too evenly and avoid "suspicious" repeats. Forensic accountants, popularised by researchers such as Mark Nigrini, compare a ledger's first-digit and first-two-digit distributions with Benford's expectations to decide where to look harder. A deviation is a flag, not proof: legitimate business rules (a ₹4,999 approval limit, say) also distort the curve.
Test your own data
Runs in your browser; nothing is uploaded. Use at least 100 values spanning several orders of magnitude for a meaningful result.