Editing Benford's law (section)

===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>