Editing Benford's law (section)

==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.<ref>{{Cite journal|first=Hal |last=Varian |author-link=Hal Varian |title=Benford's Law (Letters to the Editor) |journal=[[The American Statistician]]|year=1972 |issue=3 |volume=26 |page=65 |doi=10.1080/00031305.1972.10478934}}</ref>

===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.<ref>{{cite episode| url = https://www.wnycstudios.org/story/91699-from-benford-to-erdos| title =From Benford to Erdös | series = Radio Lab | series-link = Radio Lab | airdate = 2009-09-30 | number = 2009-10-09}}</ref>

===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) in [[election forensics]].<ref>Walter R. Mebane, Jr., "[http://www-personal.umich.edu/~wmebane/pm06.pdf Election Forensics: Vote Counts and Benford’s Law]" (July 18, 2006).</ref> Such analysis is considered a simple, though not foolproof, method of identifying irregularities in election results.<ref>"[https://www.economist.com/science-and-technology/2007/02/22/election-forensics Election forensics]", ''[[The Economist]]'' (February 22, 2007).</ref> 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.<ref>{{cite journal |last1=Deckert |first1=Joseph |last2=Myagkov |first2=Mikhail |last3=Ordeshook |first3=Peter C. |title=Benford's Law and the Detection of Election Fraud |journal=Political Analysis |date=2011 |volume=19 |issue=3 |pages=245–268 |doi=10.1093/pan/mpr014 |url=https://www.cambridge.org/core/journals/political-analysis/article/benfords-law-and-the-detection-of-election-fraud/3B1D64E822371C461AF3C61CE91AAF6D |language=en |issn=1047-1987|doi-access=free }}</ref> 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.<ref>{{cite journal |last1=Mebane |first1=Walter R. |title=Comment on "Benford's Law and the Detection of Election Fraud" |journal=Political Analysis |date=2011 |volume=19 |issue=3 |pages=269–272 |doi=10.1093/pan/mpr024 |url=https://www.cambridge.org/core/journals/political-analysis/article/comment-on-benfords-law-and-the-detection-of-election-fraud/BC29680D8B5469A54C7C9D865029FE7C |language=en |doi-access=free }}</ref>

Benford's law [[Results of the 2009 Iranian presidential election#Initial Digit Distribution/Benford's Law|has been used as evidence of fraud]] in the [[Iranian presidential election, 2009|2009 Iranian elections]].<ref>Stephen Battersby [https://www.newscientist.com/article/mg20227144.000-statistics-hint-at-fraud-in-iranian-election.html Statistics hint at fraud in Iranian election] ''New Scientist'' 24 June 2009</ref> 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 vote|invalid ballots]] had a greater influence on the results, suggesting widespread [[ballot stuffing]].<ref>Walter R. Mebane, Jr., "[http://www-personal.umich.edu/~wmebane/note22jun2009.pdf Note on the presidential election in Iran, June 2009]" (University of Michigan, June 29, 2009), pp. 22–23.</ref> Another study used [[bootstrapping (statistics)|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,<ref>{{cite journal |doi=10.1080/02664763.2013.838664 |arxiv=0906.2789|title=A first-digit anomaly in the 2009 Iranian presidential election|year=2014|last1=Roukema|first1=Boudewijn F.|journal=Journal of Applied Statistics|volume=41|pages=164–199|bibcode=2014JApS...41..164R|s2cid=88519550}}</ref> 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.<ref>Bernd Beber and Alexandra Scacco, "[https://www.washingtonpost.com/wp-dyn/content/article/2009/06/20/AR2009062000004.html The Devil Is in the Digits: Evidence That Iran's Election Was Rigged]", ''[[The Washington Post]]'' (June 20, 2009).</ref> Benford's law has also been applied for forensic auditing and fraud detection on data from the [[2003 California gubernatorial election]],<ref>Mark J. Nigrini, ''Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection'' (Hoboken, NJ: Wiley, 2012), pp. 132–35.</ref> the [[United States presidential election, 2000|2000]] and [[2004 United States presidential election]]s,<ref name="election-forensics">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. [http://www-personal.umich.edu/~wmebane/fraud06.pdf PDF]</ref> and the [[2009 German federal election]].<ref>{{cite journal|first1=Susumu |last1=Shikano |first2=Verena |last2=Mack |title=When Does the Second-Digit Benford's Law-Test Signal an Election Fraud? Facts or Misleading Test Results |journal=Jahrbücher für Nationalökonomie und Statistik |date=2011 |pages=719–732 |volume=231 |issue=5–6 |doi=10.1515/jbnst-2011-5-610 |s2cid=153896048 }}</ref> The Benford's Law Test was found to be "worth taking seriously as a statistical test for fraud," although "the test is not sensitive to distortions we know significantly affected many votes. In particular, the test does not indicate problems for Florida in 2000."<ref name="election-forensics"/>

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."<ref>{{cite news|url=https://www.reuters.com/article/uk-factcheck-benford/fact-check-deviation-from-benfords-law-does-not-prove-election-fraud-idUSKBN27Q3AI|title=Fact check: Deviation from Benford's Law does not prove election fraud|date=November 10, 2020|work=[[Reuters]]}}</ref><ref>{{cite web|url=https://physicsworld.com/a/benfords-law-and-the-2020-us-presidential-election-nothing-out-of-the-ordinary/|title=Benford's law and the 2020 US presidential election: nothing out of the ordinary|first= James|last= Dacey|date=November 19, 2020|publisher=[[Physics World]]}}</ref>

===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.<ref>William Goodman, [https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1740-9713.2016.00919.x The promises and pitfalls of Benford's law], ''[[Significance (magazine)|Significance]]'', Royal Statistical Society (June 2016), p. 38.</ref><ref name="Goldacre">{{cite news | title= The special trick that helps identify dodgy stats |url= https://www.theguardian.com/commentisfree/2011/sep/16/bad-science-dodgy-stats| last= Goldacre| first= Ben| author-link= Ben Goldacre | date= 16 September 2011| work= [[The Guardian]] | access-date= 1 February 2019}}</ref>

=== 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.<ref>{{Cite journal|last1=Sehity|first1=Tarek el|last2=Hoelzl|first2=Erik|last3=Kirchler|first3=Erich|date=2005-12-01|title=Price developments after a nominal shock: Benford's Law and psychological pricing after the euro introduction|journal=International Journal of Research in Marketing|volume=22|issue=4|pages=471–480|doi=10.1016/j.ijresmar.2005.09.002|s2cid=154273305 }}</ref> 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 [[Introduction of the euro|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.

===Genome data===
The number of [[open reading frame]]s and their relationship to genome size differs between [[eukaryote]]s and [[prokaryote]]s 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.<ref name=Friar2012>{{cite journal | last1 = Friar | first1 = JL | last2 = Goldman | first2 = T | last3 = Pérez-Mercader | first3 = J | year = 2012 | title = Genome sizes and the benford distribution | journal = PLOS ONE | volume = 7 | issue = 5| page = e36624 | doi = 10.1371/journal.pone.0036624 |arxiv = 1205.6512 |bibcode = 2012PLoSO...736624F | pmid=22629319 | pmc=3356352| doi-access = free }}</ref>

===Scientific fraud detection===
A test of regression coefficients in published papers showed agreement with Benford's law.<ref name=Diekmann2007>{{cite journal | last1 = Diekmann | first1 = A | s2cid = 117402608 | year = 2007 | title = Not the First Digit! Using Benford's Law to detect fraudulent scientific data | journal = J Appl Stat | volume = 34 | issue = 3| pages = 321–329 | doi = 10.1080/02664760601004940 | bibcode = 2007JApSt..34..321D | hdl = 20.500.11850/310246 | hdl-access = free }}</ref> 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.

=== Academic publishing networks ===
Testing the number of published scientific papers of all registered researchers in Slovenia's national database was shown to strongly conform to Benford's law.<ref>{{Cite journal |last1=Tošić |first1=Aleksandar |last2=Vičič |first2=Jernej |date=2021-08-01 |title=Use of Benford's law on academic publishing networks |url=https://www.sciencedirect.com/science/article/pii/S1751157721000341 |journal=Journal of Informetrics |volume=15 |issue=3 |pages=101163 |doi=10.1016/j.joi.2021.101163 |issn=1751-1577}}</ref> Moreover, the authors were grouped by scientific field, and tests indicate natural sciences exhibit greater conformity than social sciences.

=== Ecological application ===
A 2025 PLOS ONE journal article argues Benford probability distribution of species in certain ecological systems can detect impending transitions of the system. <ref>{{Cite journal|last=Davic|first=RD|journal=PLOS ONE|date=March 28, 2025|title=Newcomb-Benford number law and ecological processes|volume=20 |issue=3 |pages=e0310205 |doi = 10.1371/journal.pone.0310205 |doi-access=free |pmid=40153682 |pmc=11952760 |bibcode=2025PLoSO..2010205D }}</ref>