What Benford’s law says
In many real-world data sets, such as invoices, populations, river lengths, stock prices and tax returns, the first digit is far more often 1 than 9. Benford’s law gives the expected share of each leading digit d:
P(d) = log₁₀(1 + 1/d)
| Digit | Expected share |
|---|---|
| 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% |
The pattern appears in data that span several orders of magnitude and arise from multiplicative processes.
Why auditors use it
People who invent numbers tend to spread first digits evenly, or to cluster just below approval thresholds. A data set that departs from Benford’s law can point to errors, duplicates or fraud. Forensic accountants, tax authorities and election analysts use it as a screening tool, a reason to look closer, never as proof.
How to read the results
- MAD (mean absolute deviation) between observed and expected shares, with Mark Nigrini’s thresholds for first digits: up to 0.006 close conformity, 0.006 to 0.012 acceptable, 0.012 to 0.015 marginal, and above 0.015 nonconformity.
- Chi-square test with 8 degrees of freedom. With large data sets it flags tiny, harmless deviations, which is why auditors prefer the MAD.
Example
1,000 invoice amounts with first digits counted as 280, 190, 130, 95, 80, 70, 55, 52 and 48:
- 28.0% start with 1, against 30.1% expected; 4.8% start with 9, against 4.6%;
- MAD ≈ 0.0058: close conformity;
- χ² ≈ 3.24, p ≈ 0.92: no significant departure.
When Benford’s law does not apply
- Numbers with a fixed range or a minimum and maximum, such as human heights, ages or percentages.
- Assigned numbers: phone numbers, ZIP codes, invoice or account numbers.
- Prices set by policy, such as $9.99 tags, or amounts capped by rules.
- Small samples: below about 500 numbers, the test has little power.
Going further
Auditors also test the first two digits (90 categories), the second digit, and number duplication, which give finer signals. In practice, a Benford anomaly prompts drilling into the specific digits that are over-represented, for example many invoices starting with 4 just below a $5,000 approval limit.