In 1881 the astronomer Simon Newcomb noticed something odd about the books of logarithm tables that scientists used for calculation before calculators existed. The early pages, for numbers beginning with 1, were grubbier and more worn than the later ones. People were looking up numbers that started with 1 far more often than numbers that started with 9.
Newcomb published a two-page note with a formula, and it was forgotten. In 1938 the physicist Frank Benford noticed the same worn pages, apparently independently, and did something Newcomb hadn't. He collected 20,229 numbers from 20 very different sources: river areas, populations, physical constants, molecular weights, street addresses, death rates, numbers printed in Reader's Digest. Nearly all of them followed the same lopsided pattern, and it now carries his name.
| Digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| Share % | 30.1 | 17.6 | 12.5 | 9.7 | 7.9 | 6.7 | 5.8 | 5.1 | 4.6 |
The rule: the share of numbers starting with digit d is log10(1 + 1/d). For d = 1 that is log10(2) = 0.301.
Powers of 2 and Fibonacci numbers fit the law almost perfectly: of the first 1,000 powers of 2, exactly 301 begin with a 1. Products of random numbers drift into it too. Uniform random numbers don't, and heights fail completely, because almost every adult height in centimetres starts with 1. Both failures are clues to what's going on.
Think of something that grows by a steady percentage, such as a town growing 10% a year. To get from 1,000 people to 2,000 it has to double, which takes about 7.3 years. To get from 9,000 to 10,000 it only has to grow by 11%, which takes about 1.1 years. Then it reaches 10,000 and starts with 1 again, for another long stretch. Over any long run, the town's population spends about 30% of its time with a leading 1 and under 5% with a leading 9.
On a ruler, 1 to 2 is the same length as 8 to 9. On a ruler of multiplications, 1 to 2 is a doubling and 8 to 9 is barely a nudge.
That is the whole secret. Benford's law holds for quantities that vary by multiplying rather than by adding, and that spread across several orders of magnitude: tens, thousands, millions. Prices, populations, areas and account balances often behave that way.
Here is a more surprising argument. Suppose there were any universal pattern in the first digits of real-world data. It couldn't depend on the units we happen to use. River areas in square kilometres and in square miles describe the same rivers, and prices in dollars and in yen describe the same economy. So whatever the pattern is, it must survive multiplying every number by a constant.
Try the "Change units" button. Multiply the powers of 2 by 3, again and again, and the digit pattern barely moves. Multiply the uniform numbers and the pattern jumps around. It turns out that Benford's distribution is the only first-digit pattern that is unchanged by every change of units. In 1995 the mathematician Theodore Hill made this precise and went further: if you pick numbers from a random mix of different kinds of data, the combined collection drifts towards Benford's law. That helps explain why Benford's mixed bag of 20 sources fitted so well.
Benford's law is not magic, and the failures matter as much as the fits. It does not hold for:
Narrow ranges. Adult heights, weights and IQ scores cluster within one order of magnitude.
Assigned numbers. Phone numbers and invoice or cheque numbers are labels, not quantities.
Capped or cut-off data. A list of towns "with at least 2,500 people", or prices set just under a round number.
Now the practical part. Genuine financial data, such as payments, sales and expenses, usually spans many orders of magnitude and often fits Benford's law well. Numbers that someone has invented, or nudged to stay under an approval limit, may not. Forensic accountants, following work by Mark Nigrini, run first-digit and first-two-digit tests on large ledgers to decide where to look. It has been applied to tax returns and to company accounts such as payments and sales.
One much-discussed case involved countries. In 2011 a team of economists (Rauch and colleagues, in the German Economic Review) ran a Benford test on the economic data that euro-area governments reported to the EU statistics office. Greece's figures showed the largest deviation of any euro country, which matched what was already known: Eurostat had criticised the reliability of Greek deficit and debt statistics.
But a deviation is a reason to look closer, never proof on its own. Honest data can deviate for innocent reasons, and the method is easily misused. In 2011 the political scientists Deckert, Myagkov and Ordeshook argued that Benford tests are problematic for detecting election fraud. After the 2020 US presidential election, people circulated "Benford analyses" of city vote counts as proof of fraud. They were misapplications: precinct vote totals are similar in size and don't span the orders of magnitude the law needs. The political scientist Walter Mebane, who studies election forensics, said that first digits of precinct counts are not useful for diagnosing fraud.
In many real-world collections of numbers, about 30% start with 1 and under 5% start with 9. This happens because many quantities vary by multiplying, so they spend longer between 1 and 2 than between 9 and 10. It is also the only digit pattern that survives a change of units. It fails for narrow ranges, labels and capped data. Auditors use departures from it to decide where to dig, but a departure is a clue, not a verdict.
Try this: choose "My own numbers" and type 30 invented expense amounts as fast as you can, then compare your bars with the blue marks. Then open a real bank or utility statement and do the same with genuine figures.
Sources. Newcomb (1881, American Journal of Mathematics) and Benford (1938, "The Law of Anomalous Numbers", 20,229 observations from 20 domains): Wikipedia, "Benford's law", and Hill, "The First-Digit Phenomenon", American Scientist (1996). Hill, "Base-Invariance Implies Benford's Law", Proceedings of the AMS 123 (1995), and his 1995 Statistical Science paper on mixtures of distributions. Nigrini, Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection (Wiley, 2012). Rauch, Göttsche, Brähler and Engel, "Fact and Fiction in EU-Governmental Economic Data", German Economic Review 12(3), 2011. Deckert, Myagkov and Ordeshook, "Benford's Law and the Detection of Election Fraud", Political Analysis 19(3), 2011, and Mebane's reply. The 2020 misuse and Mebane's comment: Wikipedia, "Benford's law". Computed in Python and node with exact integer arithmetic: the leading digits of 21…21000 are 301, 176, 125, 97, 79, 69, 56, 52, 45; of the first 1,000 Fibonacci numbers, 301, 177, 125, 96, 80, 67, 56, 53, 45. Growth times at 10% a year: ln 2 ÷ ln 1.1 = 7.27 years; ln(10/9) ÷ ln 1.1 = 1.11 years. "Adult heights" and "products of random numbers" are simulated in your browser, not real records.