Benford's law

Not to be confused with the unrelated adageBenford's law of controversy.

Benford's law,also known as theNewcomb–Benford law,thelaw of anomalous numbers,or thefirst-digit law,is an observation that in many real-life sets of numericaldata,theleading digitis 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 distribution of first digits, according to Benford's law. Each bar represents a digit, and the height of the bar is the percentage of numbers that start with that digit.

Frequency of first significant digit of physical constants plotted against Benford's law

The graph to the right shows Benford's law forbase 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 1995included analogous statements for both thenth leading digit and the joint distribution of the leadingndigits, the latter of which leads to a corollary wherein the significant digits are shown to be astatistically dependentquantity.

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, andphysicalandmathematical constants.Like other general principles about natural data—for example, the fact that many data sets are well approximated by anormal 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 multipleorders of magnitude,especially if the process generating the numbers is described by apower law(which is common in nature).

The law is named after physicistFrank Benford,who stated it in 1938 in an article titled "The Law of Anomalous Numbers",although it had been previously stated bySimon Newcombin 1881.

The law is similar in concept, though not identical in distribution, toZipf's law.

Definition

Alogarithmic scalebar. Picking a randomxpositionuniformlyon this number line, roughly 30% of the time the first digit of the number will be 1.

A set of numbers is said to satisfy Benford's law if the leading digitd(d∈ {1,..., 9}) occurs withprobability

$P(d)=\log _{10}(d+1)-\log _{10}(d)=\log _{10}\left({\frac {d+1}{d}}\right)=\log _{10}\left(1+{\frac {1}{d}}\right).$

The leading digits in such a set thus have the following distribution:

d	⁠$P(d)$⁠	Relative size of⁠$P(d)$⁠
1	30.1%	30.1
2	17.6%	17.6
3	12.5%	12.5
4	9.7%	9.7
5	7.9%	7.9
6	6.7%	6.7
7	5.8%	5.8
8	5.1%	5.1
9	4.6%	4.6

The quantity⁠$P(d)$⁠is proportional to the space betweendandd+ 1on alogarithmic scale.Therefore, this is the distribution expected if thelogarithmsof the numbers (but not the numbers themselves) areuniformly and randomly distributed.

For example, a numberx,constrained to lie between 1 and 10, starts with the digit 1 if1 ≤x< 2,and starts with the digit 9 if9 ≤x< 10.Therefore,xstarts with the digit 1 iflog 1 ≤ logx< log 2,or starts with 9 iflog 9 ≤ logx< log 10.The interval[log 1, log 2]is much wider than the interval[log 9, log 10](0.30 and 0.05 respectively); therefore if logxis uniformly and randomly distributed, it is much more likely to fall into the wider interval than the narrower interval, i.e. more likely to start with 1 than with 9; the probabilities are proportional to the interval widths, giving the equation above (as well as the generalization to other bases besides decimal).

Benford's law is sometimes stated in a stronger form, asserting that thefractional partof the logarithm of data is typically close to uniformly distributed between 0 and 1; from this, the main claim about the distribution of first digits can be derived.

In other bases

Graphs ofP(d) for initial digitdin various bases.The dotted line showsP(d) were the distribution uniform. (Inthe 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 otherbasesbesidesdecimal;in fact, any baseb≥ 2.The general form is

$P(d)=\log _{b}(d+1)-\log _{b}(d)=\log _{b}\left(1+{\frac {1}{d}}\right).$

Forb= 2, 1(thebinaryandunary) 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, thegeneralization of Benford's law to second and later digitsis not trivial, even for binary numbers.)

Examples

Distribution of first digits (in %, red bars) in thepopulation of the 237 countriesof the world as of July 2010. Black dots indicate the distribution predicted by Benford's law.

Examining a list of the heights of the58 tallest structures in the world by categoryshows that 1 is by far the most common leading digit,irrespective of the unit of measurement(see "scale invariance" below):

Leading digit	m		ft		Per Benford's law
	Count	Share	Count	Share
1	23	39.7 %	15	25.9 %	30.1 %
2	12	20.7 %	8	13.8 %	17.6 %
3	6	10.3 %	5	8.6 %	12.5 %
4	5	8.6 %	7	12.1 %	9.7 %
5	2	3.4 %	9	15.5 %	7.9 %
6	5	8.6 %	4	6.9 %	6.7 %
7	1	1.7 %	3	5.2 %	5.8 %
8	4	6.9 %	6	10.3 %	5.1 %
9	0	0 %	1	1.7 %	4.6 %

Another example is the leading digit of2ⁿ.The sequence of the first 96 leading digits (1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1,... (sequenceA008952in theOEIS)) exhibits closer adherence to Benford’s law than is expected for random sequences of the same length, because it is derived from a geometric sequence.

Leading digit	Occurrence		Per Benford's law
	Count	Share
1	29	30.2 %	30.1 %
2	17	17.7 %	17.6 %
3	12	12.5 %	12.5 %
4	10	10.4 %	9.7 %
5	7	7.3 %	7.9 %
6	6	6.3 %	6.7 %
7	5	5.2 %	5.8 %
8	5	5.2 %	5.1 %
9	5	5.2 %	4.6 %

History

The discovery of Benford's law goes back to 1881, when the Canadian-American astronomerSimon Newcombnoticed that inlogarithm tablesthe earlier pages (that started with 1) were much more worn than the other pages.Newcomb's published result is the first known instance of this observation and includes a distribution on the second digit as well. Newcomb proposed a law that the probability of a single numberNbeing the first digit of a number was equal to log(N+ 1) − log(N).

The phenomenon was again noted in 1938 by the physicistFrank Benford,who tested it on data from 20 different domains and was credited for it. His data set included the surface areas of 335 rivers, the sizes of 3259 US populations, 104physical constants,1800molecular weights,5000 entries from a mathematical handbook, 308 numbers contained in an issue ofReader's Digest,the street addresses of the first 342 persons listed inAmerican Men of Scienceand 418 death rates. The total number of observations used in the paper was 20,229. This discovery was later named after Benford (making it an example ofStigler's law).

In 1995,Ted Hillproved the result about mixed distributions mentionedbelow.

Explanations

Benford's law tends to apply most accurately to data that span severalorders 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 alog 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.

A broad probability distribution of the log of a variable, shown on a log scale. Benford's law can be seen in the larger area covered by red (first digit one) compared to blue (first digit 8) shading.

A narrow probability distribution of the log of a variable, shown on a log scale. Benford's law is not followed, because the narrow distribution does not meet the criteria for Benford's law.

Thus, real-world distributions that span severalorders of magnituderather uniformly (e.g., stock-market prices and populations of villages, towns, and cities) are likely to satisfy Benford's law very accurately. On the other hand, a distribution mostly or entirely within one order of magnitude (e.g.,IQ scoresor heights of human adults) is unlikely to satisfy Benford's law very accurately, if at all.However, the difference between applicable and inapplicable regimes is not a sharp cut-off: as the distribution gets narrower, the deviations from Benford's law increase gradually.

(This discussion is not a full explanation of Benford's law, because it has not explained why data sets are so often encountered that, when plotted as a probability distribution of the logarithm of the variable, are relatively uniform over several orders of magnitude.)

Krieger–Kafri entropy explanation

In 1970Wolfgang Kriegerproved 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$B$with a fixed number of digits 0, 1,...,n,...,$B-1$,digitnis equivalent to a Kafri box containingnnon-interacting balls. Other scientists and statisticians have suggested entropy-related explanationsfor Benford's law.

Multiplicative fluctuations

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 thelogarithmof the stock price is undergoing arandom walk,so over time its probability distribution will get more and more broad and smooth (seeabove).(More technically, thecentral limit theoremsays that multiplying more and more random variables will create alog-normal distributionwith larger and larger variance, so eventually it covers many orders of magnitude almost uniformly.) To be sure of approximate agreement with Benford's law, the distribution has to be approximately invariant when scaled up by any factor up to 10; alog-normallydistributed data set with wide dispersion would have this approximate property.

Unlike multiplicative fluctuations,additivefluctuations do not lead to Benford's law: They lead instead tonormal probability distributions(again by thecentral limit theorem), which do not satisfy Benford's law. By contrast, that hypothetical stock price described above can be written as theproductof many random variables (i.e. the price change factor for each day), so islikelyto follow Benford's law quite well.

Multiple probability distributions

Anton Formannprovided an alternative explanation by directing attention to the interrelation between thedistributionof the significant digits and the distribution of theobserved variable.He showed in a simulation study that long-right-tailed distributions of arandom variableare compatible with the Newcomb–Benford law, and that for distributions of the ratio of two random variables the fit generally improves.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 theSignificandof 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 aprobability 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.

Invariance

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 isscale-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 isbaseinvariant for number systems. There are conditions and proofs of sum invariance, inverse invariance, and addition and subtraction invariance.

Applications

Accounting fraud detection

In 1972,Hal Variansuggested that the law could be used to detect possiblefraudin 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.

Use in criminal trials

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

Election data

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) inelection 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, andPeter C. Ordeshookargued 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 lawhas been used as evidence of fraudin the2009 Iranian elections.An analysis by Mebane found that the second digits in vote counts for PresidentMahmoud Ahmadinejad,the winner of the election, tended to differ significantly from the expectations of Benford's law, and that the ballot boxes with very fewinvalid ballotshad a greater influence on the results, suggesting widespreadballot stuffing.Another study usedbootstrapsimulations to find that the candidateMehdi Karroubireceived almost twice as many vote counts beginning with the digit 7 as would be expected according to Benford's law,while an analysis fromColumbia Universityconcluded 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 the2003 California gubernatorial election,the2000and2004 United States presidential elections,and the2009 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."

Benford's law has also been misapplied to claim election fraud. When applying the law toJoe Biden's election returns forChicago,Milwaukee,and other localities in the2020 United States presidential election,the distribution of the first digit did not follow Benford's law. The misapplication was a result of looking at data that was tightly bound in range, which violates the assumption inherent in Benford's law that the range of the data be large. The first digit test was applied to precinct-level data, but because precincts rarely receive more than a few thousand votes or fewer than several dozen, Benford's law cannot be expected to apply. According to Mebane, "It is widely understood that the first digits of precinct vote counts are not useful for trying to diagnose election frauds."

Macroeconomic data

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

Price digit analysis

Researchers have used Benford's law to detectpsychological pricingpatterns, 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.

Whenthe 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.

Genome data

The number ofopen reading framesand their relationship to genome size differs betweeneukaryotesandprokaryoteswith 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.

Scientific fraud detection

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.

Statistical tests

Although thechi-squared testhas been used to test for compliance with Benford's law it has low statistical power when used with small samples.

TheKolmogorov–Smirnov testand theKuiper testare more powerful when the sample size is small, particularly when Stephens's corrective factor is used.These tests may be unduly conservative when applied to discrete distributions. Values for the Benford test have been generated by Morrow.The critical values of the test statistics are shown below:

⍺ Test	0.10	0.05	0.01
Kuiper	1.191	1.321	1.579
Kolmogorov–Smirnov	1.012	1.148	1.420

These critical values provide the minimum test statistic values required to reject the hypothesis of compliance with Benford's law at the givensignificance levels.

Two alternative tests specific to this law have been published: First, the max (m) statisticis given by

$m={\sqrt {N}}\cdot \max _{k=1}^{9}\left\{\left|\Pr \left(X{\text{ has FSD}}=k\right)-\log _{10}\left(1+{\frac {1}{k}}\right)\right|\right\}.$

The leading factor${\sqrt {N}}$does not appear in the original formula by Leemis;it was added by Morrow in a later paper.

Secondly, the distance (d) statisticis given by

$d={\sqrt {N\cdot \sum _{l=1}^{9}\left[\Pr \left(X{\text{ has FSD}}=l\right)-\log _{10}\left(1+{\frac {1}{l}}\right)\right]^{2}}},$

where FSD is the first significant digit andNis the sample size. Morrow has determined the critical values for both these statistics, which are shown below:

⍺ Statistic	0.10	0.05	0.01
Leemis'sm	0.851	0.967	1.212
Cho & Gaines'sd	1.212	1.330	1.569

Morrow has also shown that for any random variableX(with a continuousPDF) divided by its standard deviation (σ), some valueAcan be found so that the probability of the distribution of the first significant digit of the random variable$|X/\sigma |^{A}$will differ from Benford's law by less thanε> 0.The value ofAdepends on the value ofεand the distribution of the random variable.

A method of accounting fraud detection based on bootstrapping and regression has been proposed.

If the goal is to conclude agreement with the Benford's law rather than disagreement, then thegoodness-of-fit testsmentioned above are inappropriate. In this case the specifictests for equivalenceshould be applied. An empirical distribution is called equivalent to the Benford's law if a distance (for example total variation distance or the usual Euclidean distance) between the probability mass functions is sufficiently small. This method of testing with application to Benford's law is described in Ostrovski.

Range of applicability

Distributions known to obey Benford's law

Some well-known infiniteinteger sequencesprovably satisfy Benford's law exactly (in theasymptotic limitas more and more terms of the sequence are included). Among these are theFibonacci numbers,thefactorials,the powers of 2,and the powers ofalmostany other number.

Likewise, some continuous processes satisfy Benford's law exactly (in the asymptotic limit as the process continues through time). One is anexponential growthordecayprocess: If a quantity is exponentially increasing or decreasing in time, then the percentage of time that it has each first digit satisfies Benford's law asymptotically (i.e. increasing accuracy as the process continues through time).

Distributions known to disobey Benford's law

Thesquare rootsandreciprocalsof successive natural numbers do not obey this law.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.The terminal digits in pathology reports violate Benford's law due to rounding.

Distributions that do not span several orders of magnitude will not follow Benford's law. Examples include height, weight, and IQ scores.

Criteria for distributions expected and not expected to obey Benford's law

A number of criteria, applicable particularly to accounting data, have been suggested where Benford's law can be expected to apply.

Distributions that can be expected to obey Benford's law

• When the mean is greater than the median and the skew is positive
• Numbers that result from mathematical combination of numbers: e.g. quantity × price
• Transaction level data: e.g. disbursements, sales

Distributions that would not be expected to obey Benford's law

• Where numbers are assigned sequentially: e.g. check numbers, invoice numbers
• Where numbers are influenced by human thought: e.g. prices set by psychological thresholds ($9.99)
• Accounts with a large number of firm-specific numbers: e.g. accounts set up to record $100 refunds
• Accounts with a built-in minimum or maximum
• Distributions that do not span an order of magnitude of numbers.

Benford’s law compliance theorem

Mathematically, Benford’s law applies if the distribution being tested fits the "Benford’s law compliance theorem".The derivation says that Benford's law is followed if theFourier transformof the logarithm of the probability density function is zero for all integer values. Most notably, this is satisfied if the Fourier transform is zero (or negligible) forn≥ 1. This is satisfied if the distribution is wide (since wide distribution implies a narrow Fourier transform). Smith summarizes thus (p. 716):

Benford's law is followed by distributions that are wide compared with unit distance along the logarithmic scale. Likewise, the law is not followed by distributions that are narrow compared with unit distance… If the distribution is wide compared with unit distance on the log axis, it means that the spread in the set of numbers being examined is much greater than ten.

In short, Benford’s law requires that the numbers in the distribution being measured have a spread across at least an order of magnitude.

Tests with common distributions

Benford's law was empirically tested against the numbers (up to the 10th digit) generated by a number of important distributions, including theuniform distribution,theexponential distribution,thenormal distribution,and others.

The uniform distribution, as might be expected, does not obey Benford's law. In contrast, theratio distributionoftwo uniform distributionsis well-described by Benford's law.

Neither the normal distribution nor the ratio distribution of two normal distributions (theCauchy distribution) obey Benford's law. Although thehalf-normal distributiondoes not obey Benford's law, the ratio distribution of two half-normal distributions does. Neither the right-truncated normal distribution nor the ratio distribution of two right-truncated normal distributions are well described by Benford's law. This is not surprising as this distribution is weighted towards larger numbers.

Benford's law also describes the exponential distribution and the ratio distribution of two exponential distributions well. The fit of chi-squared distribution depends on thedegrees of freedom(df) with good agreement with df = 1 and decreasing agreement as the df increases. TheF-distribution is fitted well for low degrees of freedom. With increasing dfs the fit decreases but much more slowly than the chi-squared distribution. The fit of the log-normal distribution depends on themeanand thevarianceof the distribution. The variance has a much greater effect on the fit than does the mean. Larger values of both parameters result in better agreement with the law. The ratio of two log normal distributions is a log normal so this distribution was not examined.

Other distributions that have been examined include the Muth distribution,Gompertz distribution,Weibull distribution,gamma distribution,log-logistic distributionand theexponential power distributionall of which show reasonable agreement with the law.TheGumbel distribution– a density increases with increasing value of the random variable – does not show agreement with this law.

Generalization to digits beyond the first

Log–log graph of the probability that a number starts with the digit(s)n,for a distribution satisfying Benford's law. The points show the exact formula,P(n) = log₁₀(1 + 1/n). The graph tends towards the dashed asymptote passing through(1, log₁₀e)with slope −1 in log–log scale. The example in yellow shows that the probability of a number starts with 314 is around 0.00138. The dotted lines show the probabilities for a uniform distribution for comparison. (Inthe SVG image,hover over a point to show its values.)

It is possible to extend the law to digits beyond the first.In particular, for any given number of digits, the probability of encountering a number starting with the string of digitsnof that length – discarding leading zeros – is given by

$\log _{10}(n+1)-\log _{10}(n)=\log _{10}\left(1+{\frac {1}{n}}\right).$

Thus, the probability that a number starts with the digits 3, 1, 4 (some examples are 3.14, 3.142,π,314280.7, and 0.00314005) islog₁₀(1 + 1/314) ≈ 0.00138,as in the box with the log-log graph on the right.

This result can be used to find the probability that a particular digit occurs at a given position within a number. For instance, the probability that a "2" is encountered as the second digit is

$\log _{10}\left(1+{\frac {1}{12}}\right)+\log _{10}\left(1+{\frac {1}{22}}\right)+\cdots +\log _{10}\left(1+{\frac {1}{92}}\right)\approx 0.109.$

And the probability thatd(d= 0, 1,..., 9) is encountered as then-th (n> 1) digit is

$\sum _{k=10^{n-2}}^{10^{n-1}-1}\log _{10}\left(1+{\frac {1}{10k+d}}\right).$

The distribution of then-th digit, asnincreases, rapidly approaches a uniform distribution with 10% for each of the ten digits, as shown below.Four digits is often enough to assume a uniform distribution of 10% as "0" appears 10.0176% of the time in the fourth digit, while "9" appears 9.9824% of the time.

Digit	0	1	2	3	4	5	6	7	8	9
1st	—	30.1%	17.6%	12.5%	9.7%	7.9%	6.7%	5.8%	5.1%	4.6%
2nd	12.0%	11.4%	10.9%	10.4%	10.0%	9.7%	9.3%	9.0%	8.8%	8.5%
3rd	10.2%	10.1%	10.1%	10.1%	10.0%	10.0%	9.9%	9.9%	9.9%	9.8%

Moments

Averageandmomentsof random variables for the digits 1 to 9 following this law have been calculated:

• mean3.440
• variance6.057
• skewness0.796
• kurtosis−0.548

For the two-digit distribution according to Benford's law these values are also known:

• mean38.590
• variance621.832
• skewness0.772
• kurtosis−0.547

A table of the exact probabilities for the joint occurrence of the first two digits according to Benford's law is available,as is the population correlation between the first and second digits:ρ= 0.0561.

In popular culture

Benford's law has appeared as a plot device in some twenty-first century popular entertainment.

• Television crime dramaNUMB3RSused Benford's law in the 2006 episode "The Running Man" to help solve a series of burglaries.
• The 2016 filmThe Accountantrelied on Benford's law to expose theft of funds from a robotics company.
• The 2017NetflixseriesOzarkused Benford's law to analyze a cartel member's financial statements and uncover fraud.
• The 2021Jeremy RobinsonnovelInfinite 2applied Benford's law to test whether the characters are in a simulation or reality.
• In the novelTom Clancy Point of ContactbyMike MaidenPaul Brown (Forensic Accountant at Hendley Associates) explains Benford's law to Jack Ryan Jr when discussing methods to unveil fraud in accounting books.

See also

• Zipf's law

Further reading

• Arno Berger; Theodore P. Hill (2017)."What is...Benford's law?"(PDF).Notices of the AMS.64(2): 132–134.doi:10.1090/noti1477.
• Arno Berger & Theodore P. Hill (2015).An Introduction to Benford's Law.Princeton University Press.ISBN978-0-691-16306-2.
• Alex Ely Kossovsky.Benford's Law: Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications,2014, World Scientific Publishing.ISBN978-981-4583-68-8.
• "Benford's Law – Wolfram MathWorld".Mathworld.wolfram. 14 June 2012.Retrieved26 June2012.
• Alessandro Gambini; et al. (2012)."Probability of digits by dividing random numbers: A ψ and ζ functions approach"(PDF).Expositiones Mathematicae.30(4): 223–238.doi:10.1016/j.exmath.2012.03.001.
• Sehity; Hoelzl, Erik;Kirchler, Erich(2005). "Price developments after a nominal shock: Benford's law and psychological pricing after the euro introduction".International Journal of Research in Marketing.22(4): 471–480.doi:10.1016/j.ijresmar.2005.09.002.S2CID154273305.
• Nicolas Gauvrit;Jean-Paul Delahaye(2011).Scatter and regularity implies Benford's law...and more.pp. 58–69.arXiv:0910.1359.Bibcode:2009arXiv0910.1359G.doi:10.1142/9789814327756_0004.ISBN978-9814327756.S2CID88518074.
• Bernhard Rauch; Max Göttsche; Gernot Brähler; Stefan Engel (August 2011). "Fact and Fiction in EU-Governmental Economic Data".German Economic Review.12(3): 243–255.doi:10.1111/j.1468-0475.2011.00542.x.S2CID155072460.
• Wendy Cho & Brian Gaines (August 2007). "Breaking the (Benford) Law: statistical fraud detection in campaign finance".The American Statistician.61(3): 218–223.doi:10.1198/000313007X223496.S2CID7938920.
• Geiringer, Hilda;Furlan, L. V. (1948). "The Law of Harmony in Statistics: An Investigation of the Metrical Interdependence of Social Phenomena. by L. V. Furlan".Journal of the American Statistical Association.43(242): 325–328.doi:10.2307/2280379.JSTOR2280379.

External links

• Benford Online Bibliography,an online bibliographic database on Benford's law.
• Testing Benford's LawAn open source project showing Benford's law in action against publicly available datasets.
• Benford, Frank (1938)."The Law of Anomalous Numbers".Proceedings of the American Philosophical Society.78(4): 551–572.ISSN0003-049X.- Benford's original publication