Inappropriate Applications of Benford’s Law Regularities to Some Data from the 2020 Presidential Election in the United States

According to Wikipedia: “Walter Richard Mebane, Jr. (born November 30, 1958) is a University of Michigan professor of political science and statistics and an expert on detecting electoral fraud. He has authored numerous articles on potentially fraudulent election results, including a series of notes on the results of the Iranian presidential election, 2009.[1] He authored a paper[2] disputing the Organization of American States’s claim of fraud in the 2019 Bolivian general election as well.[3] He also drafted a paper regarding the possibility of frauds on both of 2 major parties in the 2020 South Korean general election from a statistical point of view.”

Walter Mebane is the best known political scientist who has used Benford’s Law to try to detect voter fraud.

Professor Mebane writes November 10, 2020:

As vote counting is drawing to a close in the 2020 presidential election in the United States, some1 are claiming that application of Benford’s Law to the precinct vote counts from a few counties and cities give evidence of election fraud…

It is widely understood that the first digits of precinct vote counts are not useful for trying to diagnose election frauds. See for example the discussion in Carter Center (2005) and Pericchi and Torres (2011). The first digit is largely determined by the number of voters in each precinct, as usually—and especially in small jurisdictions such as individual cities and counties—the share of the votes received by parties or candidates does not vary all that greatly across precincts. Consider for example the densities in Figure 1 for votes from Chicago. The Biden/Harris ticket on average received a proportion of about .82 of the votes, and the Figure shows that the shape of the Biden/Harris vote count distribution pretty closely mirrors the shape of the distribution of votes cast. Trump/Pence received on average about .17 of the votes, but the Trump/Pence vote count distribution has a couple of hitches: for low counts the distribution reflects that in many precincts Trump/Pence vote counts are single digits.

Clearly the first digits of the Biden/Harris counts will most frequently be 3, 4 or 5. That non-Benford’s Law pattern simply relects the distribution of precinct sizes (presuming turnout did not vary that much across the city),2 given the strong support for Biden/Harris across the whole city. The first-digit distribution has nothing whatsoever to do with any kind of election fraud…

To date I’ve not heard of any substantial irregularities having occurred anywhere, and the particular datasets examined in this paper give essentially no evidence that election frauds occurred.

About Luke Ford

I've written five books (see My work has been noted in the New York Times, the Los Angeles Times, and 60 Minutes. I teach Alexander Technique in Beverly Hills (
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