Table of Contents
What Is Benford's Law?
Benford's Law states that in many naturally occurring datasets, the leading digit is more likely to be small. Digit 1 appears as the first digit about 30.1% of the time, while 9 appears only 4.6%. This counterintuitive pattern holds for financial data, population numbers, physical constants, and many other real-world datasets.
Discovered by Simon Newcomb (1881) and later Frank Benford (1938), this law is used in forensic accounting, election fraud detection, and data quality assessment.
Formula
Expected Distribution
| Digit | Probability |
|---|---|
| 1 | 30.1% |
| 2 | 17.6% |
| 3 | 12.5% |
| 4 | 9.7% |
| 5 | 7.9% |
| 6 | 6.7% |
| 7 | 5.8% |
| 8 | 5.1% |
| 9 | 4.6% |
FAQ
When does Benford's Law apply?
It applies to data spanning several orders of magnitude: financial transactions, populations, stock prices, river lengths. It does NOT apply to telephone numbers, dates, or uniformly distributed data.
How is it used to detect fraud?
Fabricated numbers often fail to follow Benford's distribution. Auditors compare actual leading digit frequencies against expected Benford frequencies using chi-square tests.