Benford's law
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
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[3] 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.[4] 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.[5][6] 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",[7] although it had been previously stated by Simon Newcomb in 1881.[8][9]
The law is similar in concept, though not identical in distribution, to Zipf's law.
Definition
A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability[10]
The leading digits in such a set thus have the following distribution:
d | Relative size of | |
---|---|---|
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% |
The quantity is proportional to the space between d and d + 1 on a
For example, a number x, constrained to lie between 1 and 10, starts with the digit 1 if 1 ≤ x < 2, and starts with the digit 9 if 9 ≤ x < 10. Therefore, x starts with the digit 1 if log 1 ≤ log x < log 2, or starts with 9 if log 9 ≤ log x < 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 log x is 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 the fractional part of 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.[5]
In other bases
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[12]
For b = 2, 1 (the
Examples
Examining a list of the heights of the
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 of 2n. 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, ... (sequence A008952 in the OEIS)) 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.[14]
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 astronomer
The phenomenon was again noted in 1938 by the physicist
In 1995, Ted Hill proved the result about mixed distributions mentioned below.[15][16]
Explanations
Benford's law tends to apply most accurately to data that span several
Consider the probability distributions shown below, referenced to a
Thus, real-world distributions that span several
(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.[19])
Krieger–Kafri entropy explanation
In 1970 Wolfgang Krieger proved what is now called the Krieger generator theorem.[20][21] 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[
Multiplicative fluctuations
Many real-world examples of Benford's law arise from multiplicative fluctuations.[25] 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
Unlike multiplicative fluctuations, additive fluctuations do not lead to Benford's law: They lead instead to
Multiple probability 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 is
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.[31][32]
Applications
Accounting fraud detection
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.[33]
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.[34]
Election data
Benford's law
Benford's law has also been misapplied to claim election fraud. When applying the law to Joe Biden's election returns for Chicago, Milwaukee, and other localities in the 2020 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."[46][47]
Macroeconomic data
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.[48][49]
Price digit analysis
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.[50] 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
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 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.[51]
Scientific fraud detection
A test of regression coefficients in published papers showed agreement with Benford's law.[52] 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 the chi-squared test has been used to test for compliance with Benford's law it has low statistical power when used with small samples.
The
- ⍺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 given
Two alternative tests specific to this law have been published: First, the max (m) statistic[55] is given by
The leading factor does not appear in the original formula by Leemis;[55] it was added by Morrow in a later paper.[54]
Secondly, the distance (d) statistic[56] is given by
where FSD is the first significant digit and N is the sample size. Morrow has determined the critical values for both these statistics, which are shown below:[54]
- ⍺Statistic
0.10 0.05 0.01 Leemis's m 0.851 0.967 1.212 Cho & Gaines's d 1.212 1.330 1.569
Morrow has also shown that for any random variable X (with a continuous PDF) divided by its standard deviation (σ), some value A can be found so that the probability of the distribution of the first significant digit of the random variable will differ from Benford's law by less than ε > 0.[54] The value of A depends on the value of ε and the distribution of the random variable.
A method of accounting fraud detection based on bootstrapping and regression has been proposed.[57]
If the goal is to conclude agreement with the Benford's law rather than disagreement, then the
Range of applicability
Distributions known to obey Benford's law
Some well-known infinite
Likewise, some continuous processes satisfy Benford's law exactly (in the asymptotic limit as the process continues through time). One is an exponential growth or decay process: 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
The square roots and reciprocals of successive natural numbers do not obey this law.[63] Prime numbers in a finite range follow a Generalized Benford’s law, that approaches uniformity as the size of the range approaches infinity.[64] Lists of local telephone numbers violate Benford's law.[65] 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.[63] The terminal digits in pathology reports violate Benford's law due to rounding.[66]
Distributions that do not span several orders of magnitude will not follow Benford's law. Examples include height, weight, and IQ scores.[9][67]
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.[68]
- 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".[17] The derivation says that Benford's law is followed if the Fourier transform of 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) for n ≥ 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 the
The uniform distribution, as might be expected, does not obey Benford's law. In contrast, the ratio distribution of two uniform distributions is well-described by Benford's law.
Neither the normal distribution nor the ratio distribution of two normal distributions (the Cauchy distribution) obey Benford's law. Although the half-normal distribution does 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 the degrees of freedom (df) with good agreement with df = 1 and decreasing agreement as the df increases. The F-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 the mean and the variance of 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,
Generalization to digits beyond the first
It is possible to extend the law to digits beyond the first.[70] In particular, for any given number of digits, the probability of encountering a number starting with the string of digits n of that length – discarding leading zeros – is given by
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) is log10(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[70]
And the probability that d (d = 0, 1, ..., 9) is encountered as the n-th (n > 1) digit is
The distribution of the n-th digit, as n increases, rapidly approaches a uniform distribution with 10% for each of the ten digits, as shown below.[70] 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
Average and moments of random variables for the digits 1 to 9 following this law have been calculated:[71]
For the two-digit distribution according to Benford's law these values are also known:[72]
A table of the exact probabilities for the joint occurrence of the first two digits according to Benford's law is available,[72] as is the population correlation between the first and second digits:[72] ρ = 0.0561.
In popular culture
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.[30]
- 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.
See also
References
- ^ Arno Berger and Theodore P. Hill, Benford's Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem, 2011.
- ^ Weisstein, Eric W. "Benford's Law". MathWorld, A Wolfram web resource. Retrieved 7 June 2015.
- .
- ^ Paul H. Kvam, Brani Vidakovic, Nonparametric Statistics with Applications to Science and Engineering, p. 158.
- ^ S2CID 202583554.
- S2CID 198147766.
- ^ JSTOR 984802.
- ^ S2CID 124556624.
- ^ PMID 20479878.
- ^ ISBN 978-1-4008-6659-5.
- ^ They should strictly be bars but are shown as lines for clarity.
- ^ Pimbley, J. M. (2014). "Benford's Law as a Logarithmic Transformation" (PDF). Maxwell Consulting, LLC. Archived (PDF) from the original on 9 October 2022. Retrieved 15 November 2020.
- ISBN 978-1-61804-080-0.
- ^ JSTOR 2319349.
- ^ MR 1421567.
- ISSN 0002-9939.
- ^ a b c Steven W. Smith. "Chapter 34: Explaining Benford's Law. The Power of Signal Processing". The Scientist and Engineer's Guide to Digital Signal Processing. Retrieved 15 December 2012.
- ^ (PDF) from the original on 9 October 2022.
- ^ Arno Berger and Theodore P. Hill, Benford's Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem, 2011. The authors describe this argument but say it "still leaves open the question of why it is reasonable to assume that the logarithm of the spread, as opposed to the spread itself—or, say, the log log spread—should be large" and that "assuming large spread on a logarithmic scale is equivalent to assuming an approximate conformance with [Benford's law]" (italics added), something which they say lacks a "simple explanation".
- ISSN 0002-9947.
- ISBN 978-1-139-50087-6.
- .
- S2CID 34151059.
- S2CID 119207367.
- ^ .
- PMID 20479878.
- S2CID 13553246.
- MR 2122815. Archived from the original(PDF) on 4 March 2016. Retrieved 13 August 2015.
- .
- ^ a b Weisstein, Eric W. "Benford's Law". mathworld.wolfram.com.
- ^ Jamain, Adrien (September 2001). "Benford's Law" (PDF). Imperial College of London. Archived (PDF) from the original on 9 October 2022. Retrieved 15 November 2020.
- ^ Berger, Arno (June 2011). "A basic theory of Benford's Law". Probability Surveys. 8 (2011): 1–126.
- .
- Radio Lab. Episode 2009-10-09. 30 September 2009.
- ^ Walter R. Mebane, Jr., "Election Forensics: Vote Counts and Benford’s Law" (July 18, 2006).
- ^ "Election forensics", The Economist (February 22, 2007).
- ISSN 1047-1987.
- .
- ^ Stephen Battersby Statistics hint at fraud in Iranian election New Scientist 24 June 2009
- ^ Walter R. Mebane, Jr., "Note on the presidential election in Iran, June 2009" (University of Michigan, June 29, 2009), pp. 22–23.
- S2CID 88519550.
- ^ Bernd Beber and Alexandra Scacco, "The Devil Is in the Digits: Evidence That Iran's Election Was Rigged", The Washington Post (June 20, 2009).
- ^ Mark J. Nigrini, Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection (Hoboken, NJ: Wiley, 2012), pp. 132–35.
- ^ a b Walter R. Mebane, Jr., "Election Forensics: The Second-Digit Benford's Law Test and Recent American Presidential Elections" in Election Fraud: Detecting and Deterring Electoral Manipulation, edited by R. Michael Alvarez et al. (Washington, D.C.: Brookings Institution Press, 2008), pp. 162–81. PDF
- S2CID 153896048.
- ^ "Fact check: Deviation from Benford's Law does not prove election fraud". Reuters. 10 November 2020.
- ^ Dacey, James (19 November 2020). "Benford's law and the 2020 US presidential election: nothing out of the ordinary". Physics World.
- ^ William Goodman, The promises and pitfalls of Benford's law, Significance, Royal Statistical Society (June 2016), p. 38.
- ^ Goldacre, Ben (16 September 2011). "The special trick that helps identify dodgy stats". The Guardian. Retrieved 1 February 2019.
- S2CID 154273305.
- PMID 22629319.
- S2CID 117402608.
- Journal of the Royal Statistical Society, Series B. 32 (1): 115–122.
- ^ a b c d Morrow, John (August 2014). Benford's Law, families of distributions and a test basis. London, UK. Retrieved 11 March 2022.
{{cite book}}
: CS1 maint: location missing publisher (link) - ^ S2CID 122607770.
- S2CID 7938920.
- ^ Suh, I. S.; Headrick, T. C.; Minaburo, S. (2011). "An effective and efficient analytic technique: A bootstrap regression procedure and Benford's Law". J. Forensic & Investigative Accounting. 3 (3).
- S2CID 126293429.
- The Fibonacci Quarterly. 19 (2): 175–177.
- The Fibonacci Quarterly. 5: 137–140.
- ^ Sarkar, P. B. (1973). "An Observation on the Significant Digits of Binomial Coefficients and Factorials". Sankhya B. 35: 363–364.
- ^ a b In general, the sequence k1, k2, k3, etc., satisfies Benford's law exactly, under the condition that log10 k is an irrational number. This is a straightforward consequence of the equidistribution theorem.
- ^ JSTOR 2319349.
- ^ Zyga, Lisa; Phys.org. "New Pattern Found in Prime Numbers". phys.org. Retrieved 23 January 2022.
- S2CID 7938920. Retrieved 8 March 2022.
- S2CID 206987736.
- ^ Singleton, Tommie W. (May 1, 2011). "Understanding and Applying Benford’s Law", ISACA Journal, Information Systems Audit and Control Association. Retrieved Nov. 9, 2020.
- ^ Durtschi, C; Hillison, W; Pacini, C (2004). "The effective use of Benford's law to assist in detecting fraud in accounting data". J Forensic Accounting. 5: 17–34.
- ^ S2CID 2596996.
- ^ JSTOR 2974952.
- ^ Scott, P.D.; Fasli, M. (2001) "Benford's Law: An empirical investigation and a novel explanation" Archived 13 December 2014 at the Wayback Machine. CSM Technical Report 349, Department of Computer Science, Univ. Essex
- ^ a b c Suh, I. S.; Headrick, T. C. (2010). "A comparative analysis of the bootstrap versus traditional statistical procedures applied to digital analysis based on Benford's law" (PDF). Journal of Forensic and Investigative Accounting. 2 (2): 144–175.
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. ISBN 978-0-691-16306-2.
- Alex Ely Kossovsky. Benford's Law: Theory, the General Law of Relative Quantities, and Forensic Fraud Detection Applications, 2014, World Scientific Publishing. ISBN 978-981-4583-68-8.
- "Benford's Law – Wolfram MathWorld". Mathworld.wolfram.com. 14 June 2012. Retrieved 26 June 2012.
- 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. .
- Sehity; Hoelzl, Erik; S2CID 154273305.
- S2CID 88518074.
- Bernhard Rauch; Max Göttsche; Gernot Brähler; Stefan Engel (August 2011). "Fact and Fiction in EU-Governmental Economic Data". S2CID 155072460.
- Wendy Cho & Brian Gaines (August 2007). "Breaking the (Benford) Law: statistical fraud detection in campaign finance". S2CID 7938920.
- JSTOR 2280379.
External links
- Benford Online Bibliography, an online bibliographic database on Benford's law.
- Testing Benford's Law An 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. ISSN 0003-049X. - Benford's original publication