Concept
In real-world data, the leading digit is 1 about 30% of the time, 9 only 5%.
Understand it in one breath
For data spanning several orders of magnitude and generated under suitable conditions, the leading digit can be 1 about 30% of the time. Some population and accounting datasets fit, while phone numbers or tightly bounded measurements may not. Auditors can use the law as a screening signal, but a mismatch alone does not prove fabrication.
At a glance
Leading digit d | Probability P(d) | Visual |
|---|---|---|
| 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% | ██ |
Unlike the uniform distribution (1/9 ≈ 11.1%), natural datasets show a pronounced asymmetry.
Key formula
Key moments
Newcomb — the worn pages of logarithm tables
Astronomer Simon Newcomb noticed that pages for numbers beginning with smaller digits were more worn, suggesting a nonuniform distribution of leading digits.
Benford — rediscovery and generalization
GE physicist Frank Benford confirmed the pattern across 20,229 observations from many datasets, and the law came to bear his name.
Benford’s law enters mainstream auditing
High-profile corporate accounting scandals accelerated interest in statistical screening tools, and leading-digit analysis became a standard warning signal in forensic auditing.
Modern applications
Accounting-fraud detection, election-data analysis, scientific-data validation, tax audits, and AI-generated image detection.
Beyond MathVoyage
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