What is Benford's Law?
A plain-language guide to Benford's Law, why leading digits follow a logarithmic curve and how auditors use it to spot anomalies.
The first-digit law
Benford's Law observes that in many real-world data sets the leading digit is not uniform. Instead, 1 leads about 30 percent of the time, 2 about 18 percent, and so on down to 9 at roughly 4.6 percent. The expected share of any digit d is log10(1 + 1/d).
Why it happens
Data that spans several orders of magnitude and grows multiplicatively, such as populations, river lengths, stock prices and accounting totals, tends to be uniform on a logarithmic scale. On that scale, low leading digits cover more space, so they appear more often.
Where it is used
Forensic accountants, auditors and election analysts compare a data set's leading-digit pattern to the Benford curve. A large gap can flag invented numbers, since people inventing figures rarely reproduce the logarithmic spread on their own.
Limits to keep in mind
Benford's Law does not apply to every data set. Numbers with a narrow range, sequential IDs, prices clustered near thresholds or heavy rounding can fail the test for innocent reasons. Treat a failed test as a prompt to investigate, never as proof.