Quick Answer: Benford's Law states that in many naturally occurring datasets, the leading digit is 1 about 30.1% of the time, 2 about 17.6% of the time, and decreasing through to 9 at about 4.6%. This applies to population figures, river lengths, financial data, and street addresses — but NOT to uniform random numbers.
What Is Benford's Law?
Benford's Law (also called the First-Digit Law) describes the frequency distribution of leading digits in many real-world numerical datasets. It states that lower digits appear as the leading digit far more often than higher digits. The probability of leading digit d is: P(d) = log₁₀(1 + 1/d).
| Leading Digit | Benford's Law Prediction | Example: 1,000 numbers |
|---|---|---|
| 1 | 30.1% | 301 numbers |
| 2 | 17.6% | 176 numbers |
| 3 | 12.5% | 125 numbers |
| 4 | 9.7% | 97 numbers |
| 5 | 7.9% | 79 numbers |
| 6 | 6.7% | 67 numbers |
| 7 | 5.8% | 58 numbers |
| 8 | 5.1% | 51 numbers |
| 9 | 4.6% | 46 numbers |
Why Benford's Law Exists
Benford's Law applies to datasets that span multiple orders of magnitude and arise from multiplicative processes. Populations, financial transactions, physical constants, and river lengths all involve multiplicative growth — and multiplicative processes naturally produce logarithmic (Benford) distributions of leading digits. Data generated uniformly at random does NOT follow Benford's Law.
Fraud Detection
Forensic accountants use Benford's Law to detect fraudulent financial data. When people fabricate numbers, they tend to distribute leading digits more uniformly (intuiting uniformity as "random"). If a dataset of expenses shows too many 7s and 8s as leading digits and too few 1s and 2s, it may indicate data manipulation.