Benford's Law

Discovery and statement of the law

Newcomb (1881) noticed that in books of logarithm tables the early pages, covering numbers beginning with 1, were far more worn than the later pages covering numbers beginning with 9, and he inferred that leading digits are not evenly distributed. The observation was forgotten until Benford (1938) found the same pattern in more than 20,000 observations across 20 categories, including river drainage areas, populations, physical constants and numbers printed in newspapers. Under the law, the probability that the leading digit is d is log10(1+1/d): about 30.1% for 1, 17.6% for 2 and 12.5% for 3, down to about 4.6% for 9. A similar distribution applies to second and later digits, though it grows closer to uniform. It is one of several regularities in natural and social data, alongside Zipf's law and power laws.

Why it holds

Pinkham (1961) showed that if there is a universal law for leading digits, the distribution should not change when every number is multiplied by the same constant, as when units are switched from dollars to yen, and that the only distribution with this scale invariance is Benford's. Raimi (1976) surveyed the research to that point, and Hill (1995) proved that base invariance, a distribution that holds when the number base is changed, implies Benford's law. That same year Hill gave a statistical derivation showing that mixing samples drawn at random from different distributions converges to Benford's law, which explains why the law fits so well for data combined from varied sources. Berger and Hill (2015) set out mathematically the conditions under which the law emerges in growth processes, multiplicative processes and dynamical systems. The more widely data spread across orders of magnitude, the better the fit.

Fraud detection and its limits

Nigrini (1996) proposed testing whether the digit distribution in tax return data departs from Benford's law to flag possible manipulation, and Nigrini (2012) established it as a standard tool of forensic accounting and auditing. The idea is that when people make up numbers, they tend not to use small leading digits often enough or to avoid certain digits. Rauch and colleagues (2011) reported that among EU member states' macroeconomic statistics, Greece's data deviated most from the law, and Diekmann (2007) proposed analyzing the second digits of scientific data such as regression coefficients. In elections, Mebane (2006) proposed election forensics using the second digits of vote counts, but Deckert, Myagkov and Ordeshook (2011) argued that violations of the law in election data are not reliable evidence of fraud. The law does not apply to narrow-range or assigned numbers, so a deviation is only a lead for further investigation and is not in itself evidence of accounting fraud or other wrongdoing.

FAQ

What is Benford's law in simple terms?

If you collect lots of real-world numbers, about 30% start with 1, more than any other digit, and fewer than 5% start with 9. It fits best when the numbers span several orders of magnitude.

Does it apply to all numerical data?

No. It doesn't fit narrow-range data like heights or test scores, or assigned numbers like phone numbers and prices. That's why a departure from the law alone can't prove manipulation.

Sources reviewed: 2026-10-26

References

1. Newcomb, S. (1881). Note on the frequency of use of the different digits in natural numbers. American Journal of Mathematics, 4(1), 39–40.
2. Benford, F. (1938). The law of anomalous numbers. Proceedings of the American Philosophical Society, 78(4), 551–572.
3. Pinkham, R. S. (1961). On the distribution of first significant digits. Annals of Mathematical Statistics, 32(4), 1223–1230.
4. Raimi, R. A. (1976). The first digit problem. American Mathematical Monthly, 83(7), 521–538.
5. Hill, T. P. (1995). Base-invariance implies Benford's law. Proceedings of the American Mathematical Society, 123(3), 887–895.
6. Hill, T. P. (1995). A statistical derivation of the significant-digit law. Statistical Science, 10(4), 354–363.
7. Nigrini, M. J. (1996). A taxpayer compliance application of Benford's law. Journal of the American Taxation Association, 18(1), 72–91.
8. Diekmann, A. (2007). Not the first digit! Using Benford's law to detect fraudulent scientific data. Journal of Applied Statistics, 34(3), 321–329.
9. Deckert, J., Myagkov, M., & Ordeshook, P. C. (2011). Benford's law and the detection of election fraud. Political Analysis, 19(3), 245–268.
10. Rauch, B., Göttsche, M., Brähler, G., & Engel, S. (2011). Fact and fiction in EU-governmental economic data. German Economic Review, 12(3), 243–255.
11. Nigrini, M. J. (2012). Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection. Hoboken, NJ: Wiley.
12. Berger, A., & Hill, T. P. (2015). An Introduction to Benford's Law. Princeton: Princeton University Press.
13. Mebane, W. R. (2006). Election forensics: Vote counts and Benford's law. Paper presented at the Summer Meeting of the Political Methodology Society, University of California, Davis.