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Observation that in many real life datasets, the leading digit is likely to be small

Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first digit law, is an observation that in many real life sets of numerical data, the leading digit is likely to be small. In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. Uniformly distributed digits would each occur about 11.1% of the time. Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on. The graph to the right shows Benford's law for base 10, one of infinitely many cases of a generalized law regarding numbers expressed in arbitrary (integer) bases, which rules out the possibility that the phenomenon might be an artifact of the base 10 number system. Further generalizations published in 1995 included analogous statements for both the nth leading digit and the joint distribution of the leading n digits, the latter of which leads to a corollary wherein the significant digits are shown to be a statistically dependent quantity. It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, and physical and mathematical constants. Like other general principles about natural data—for example, the fact that many data sets are well approximated by a normal distribution—there are illustrative examples and explanations that cover many of the cases where Benford's law applies, though there are many other cases where Benford's law applies that resist simple explanations. Benford's Law tends to be most accurate when values are distributed across multiple orders of magnitude, especially if the process generating the numbers is described by a power law (which is common in nature). The law is named after physicist Frank Benford, who stated it in 1938 in an article titled "The Law of Anomalous Numbers", although it had been previously stated by Simon Newcomb in 1881. The dotted line shows P(d) were the distribution uniform. (In the SVG image, hover over a graph to show the value for each point.)]] An extension of Benford's law predicts the distribution of first digits in other bases besides decimal; in fact, any base b ≥ 2. The general form is

For 1=b = 2, 1 (the binary and unary) number systems, Benford's law is true but trivial: All binary and unary numbers (except for 0 or the empty set) start with the digit 1. (On the other hand, the generalization of Benford's law to second and later digits is not trivial, even for binary numbers.)

Benford's law tends to apply most accurately to data that span several orders of magnitude. As a rule of thumb, the more orders of magnitude that the data evenly covers, the more accurately Benford's law applies. For instance, one can expect that Benford's law would apply to a list of numbers representing the populations of UK settlements. But if a "settlement" is defined as a village with population between 300 and 999, then Benford's law will not apply. Consider the probability distributions shown below, referenced to a log scale. In each case, the total area in red is the relative probability that the first digit is 1, and the total area in blue is the relative probability that the first digit is 8. For the first distribution, the size of the areas of red and blue are approximately proportional to the widths of each red and blue bar. Therefore, the numbers drawn from this distribution will approximately follow Benford's law. On the other hand, for the second distribution, the ratio of the areas of red and blue is very different from the ratio of the widths of each red and blue bar. Rather, the relative areas of red and blue are determined more by the heights of the bars than the widths. Accordingly, the first digits in this distribution do not satisfy Benford's law at all.)

In 1970 Wolfgang Krieger proved what is now called the Krieger generator theorem. The Krieger generator theorem might be viewed as a justification for the assumption in the Kafri ball and box model that, in a given base with a fixed number of digits 0, 1,  , n,  , , digit n is equivalent to a Kafri box containing n non interacting balls. Other scientists and statisticians have suggested entropy related explanations for Benford's law.

Many real world examples of Benford's law arise from multiplicative fluctuations. For example, if a stock price starts at $100, and then each day it gets multiplied by a randomly chosen factor between 0.99 and 1.01, then over an extended period the probability distribution of its price satisfies Benford's law with higher and higher accuracy. The reason is that the logarithm of the stock price is undergoing a random walk, so over time its probability distribution will get more and more broad and smooth (see above). For numbers drawn from certain distributions (IQ scores, human heights) the Benford's law fails to hold because these variates obey a normal distribution, which is known not to satisfy Benford's law, since normal distributions can't span several orders of magnitude and the mantissae of their logarithms will not be (even approximately) uniformly distributed. However, if one "mixes" numbers from those distributions, for example, by taking numbers from newspaper articles, Benford's law reappears. This can also be proven mathematically: if one repeatedly "randomly" chooses a probability distribution (from an uncorrelated set) and then randomly chooses a number according to that distribution, the resulting list of numbers will obey Benford's law. A similar probabilistic explanation for the appearance of Benford's law in everyday life numbers has been advanced by showing that it arises naturally when one considers mixtures of uniform distributions.

In a list of lengths, the distribution of first digits of numbers in the list may be generally similar regardless of whether all the lengths are expressed in metres, yards, feet, inches, etc. The same applies to monetary units. This is not always the case. For example, the height of adult humans almost always starts with a 1 or 2 when measured in metres and almost always starts with 4, 5, 6, or 7 when measured in feet. But in a list of lengths spread evenly over many orders of magnitude—for example, a list of 1000 lengths mentioned in scientific papers that includes the measurements of molecules, bacteria, plants, and galaxies—it is reasonable to expect the distribution of first digits to be the same no matter whether the lengths are written in metres or in feet. When the distribution of the first digits of a data set is scale invariant (independent of the units that the data are expressed in), it is always given by Benford's law. For example, the first (non zero) digit on the aforementioned list of lengths should have the same distribution whether the unit of measurement is feet or yards. But there are three feet in a yard, so the probability that the first digit of a length in yards is 1 must be the same as the probability that the first digit of a length in feet is 3, 4, or 5; similarly, the probability that the first digit of a length in yards is 2 must be the same as the probability that the first digit of a length in feet is 6, 7, or 8. Applying this to all possible measurement scales gives the logarithmic distribution of Benford's law. Benford's law for first digits is base invariant for number systems. There are conditions and proofs of sum invariance, inverse invariance, and addition and subtraction invariance.

In 1972, Hal Varian suggested that the law could be used to detect possible fraud in lists of socio economic data submitted in support of public planning decisions. Based on the plausible assumption that people who fabricate figures tend to distribute their digits fairly uniformly, a simple comparison of first digit frequency distribution from the data with the expected distribution according to Benford's law ought to show up any anomalous results.

In the United States, evidence based on Benford's law has been admitted in criminal cases at the federal, state, and local levels.

Walter Mebane, a political scientist and statistician at the University of Michigan, was the first to apply the second digit Benford's law test (2BL test) in election forensics. Such analysis is considered a simple, though not foolproof, method of identifying irregularities in election results. Scientific consensus to support the applicability of Benford's law to elections has not been reached in the literature. A 2011 study by the political scientists Joseph Deckert, Mikhail Myagkov, and Peter C. Ordeshook argued that Benford's law is problematic and misleading as a statistical indicator of election fraud. Their method was criticized by Mebane in a response, though he agreed that there are many caveats to the application of Benford's law to election data. Benford's law has been used as evidence of fraud in the 2009 Iranian elections. An analysis by Mebane found that the second digits in vote counts for President Mahmoud Ahmadinejad, the winner of the election, tended to differ significantly from the expectations of Benford's law, and that the ballot boxes with very few invalid ballots had a greater influence on the results, suggesting widespread ballot stuffing. Another study used bootstrap simulations to find that the candidate Mehdi Karroubi received almost twice as many vote counts beginning with the digit 7 as would be expected according to Benford's law, while an analysis from Columbia University concluded that the probability that a fair election would produce both too few non adjacent digits and the suspicious deviations in last digit frequencies as found in the 2009 Iranian presidential election is less than 0.5 percent. Benford's law has also been applied for forensic auditing and fraud detection on data from the 2003 California gubernatorial election, the 2000 and 2004 United States presidential elections, and the 2009 German federal election; the Benford's Law Test was found to be "worth taking seriously as a statistical test for fraud," although "is not sensitive to distortions we know significantly affected many votes."

Similarly, the macroeconomic data the Greek government reported to the European Union before entering the eurozone was shown to be probably fraudulent using Benford's law, albeit years after the country joined.

Researchers have used Benford's law to detect psychological pricing patterns, in a Europe wide study in consumer product prices before and after euro was introduced in 2002. The idea was that, without psychological pricing, the first two or three digits of price of items should follow Benford's law. Consequently, if the distribution of digits deviates from Benford's law (such as having a lot of 9's), it means merchants may have used psychological pricing. When the euro replaced local currencies in 2002, for a brief period of time, the price of goods in euro was simply converted from the price of goods in local currencies before the replacement. As it is essentially impossible to use psychological pricing simultaneously on both price in euro and price in local currency, during the transition period, psychological pricing would be disrupted even if it used to be present. It can only be re established once consumers have gotten used to prices in a single currency again, this time in euro. As the researchers expected, the distribution of first price digit followed Benford's law, but the distribution of the second and third digits deviated significantly from Benford's law before the introduction, then deviated less during the introduction, then deviated more again after the introduction.

The number of open reading frames and their relationship to genome size differs between eukaryotes and prokaryotes with the former showing a log linear relationship and the latter a linear relationship. Benford's law has been used to test this observation with an excellent fit to the data in both cases.

A test of regression coefficients in published papers showed agreement with Benford's law. As a comparison group subjects were asked to fabricate statistical estimates. The fabricated results conformed to Benford's law on first digits, but failed to obey Benford's law on second digits.

Some well known infinite integer sequences satisfy Benford's law exactly (in the asymptotic limit as more and more terms of the sequence are included). Among these are the Fibonacci numbers, the factorials, the powers of 2, and the powers of almost any other number. Prime numbers in a finite range follow a Generalized Benford’s law, that approaches uniformity as the size of the range approaches infinity. Lists of local telephone numbers violate Benford's law. Benford's law is violated by the populations of all places with a population of at least 2500 individuals from five US states according to the 1960 and 1970 censuses, where only 19 % began with digit 1 but 20 % began with digit 2, because truncation at 2500 introduces statistical bias. Distributions that do not span several orders of magnitude will not follow Benford's law. Examples include height, weight, and IQ scores.

Benford's law has appeared as a plot device in some twenty first century popular entertainment. Television crime drama NUMB3RS used Benford's law in the 2006 episode "The Running Man" to help solve a series of burglaries. The 2016 film The Accountant relied on Benford's law to expose theft of funds from a robotics company. The 2017 Netflix series Ozark used Benford's law to analyze a cartel member's financial statements and uncover fraud. The 2021 Jeremy Robinson novel Infinite 2 applied Benford's law to test whether the characters are in a simulation or reality. In the novel Tom Clancy Point of Contact by Mike Maiden Paul Brown (Forensic Accountant at Hendley Associates) explains Benford's law to Jack Ryan Jr when discussing methods to unveil fraud in accounting books.