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Last Updated: November 19, 2020

# Benford’s Law and the 2020 Election

Last week, I started to discuss how Benford’s law is being applied incorrectly to allege voter fraud in the 2020 presidential election. That newsletter was running long; so, as promised I am resuming the topic this week.

What got me started was this post at my non-gambling forum Diversity Tomorrow, which states in part that election outcomes in Michigan showed a 99.999% chance of voter fraud per failing a test of Benford’s Law. The quotation evidently comes from this source (https://yournews.com/2020/11/08/1915953/benfords-law-has-been-used-to-prove-election-fraud-in/).

The math in that article is so bad that it would normally not be worth addressing. However, it gives me a chance to teach something, and education is always a good thing. The article links to Detroit election data on a precinct by precinct level. It then goes on to argue that the results are highly inconsistent with expectations according to Benford’s Law.

What is Benford’s Law? Briefly, it states, in part, that the frequency of the first digit in a random set of data should follow a logarithmic distribution, as follows:

Digit Probability
1 30.10%
2 17.61%
3 12.49%
4 9.69%
5 7.92%
6 6.69%
7 5.80%
8 5.12%
9 4.58%
Total 100.00%

This digit frequency can be expected only if the data it is being applied to follow an exponential distribution and each data item is independent. However, not everything that is random follows such an exponential distribution. Especially data that is cherry picked to be a certain size. In the case of the election results in Detroit, there were 629 precincts with at least one presidential vote in 2020. The average number of votes per precinct is 381. If the number of votes followed an exponential distribution with this average, then you would expect 24.6% to fall in the range of 100 to 250 votes.

In reality, 60.6% of precincts fell in that range. With that in mind, what seems to be the case is that the precincts are drawn to have a somewhat consistent population in most of them.

Given these relatively consistent precinct sizes in Detroit, of course the results are going to fail a Benford test! To expect that they would pass a Benford test is an absolutely terrible use of statistics and an example of how the ignorant masses can be easily confused and misled with bad Math. It would be like taking a survey of the average age in Nevada among customers playing bingo at a casino in Laughlin only.

I could easily go on with other analysis I have done of Benford and the election. However, I already present my analysis of other contests besides Detroit at Wizard of Vegas in the thread BENFORD'S LAW AND THE 2020 ELECTION.

This whole topic is putting me in a foul mood, to be honest with you. Using the Benford Test to allege vote fraud just goes to show that anybody can say anything on the Internet with no oversight. Everybody knows that. While I’m all in favor of free speech, I think the audience should be asking questions about the qualifications of whoever is doing the speaking. If that is too time consuming, at least ask whether what is being said passes a smell test that the conclusions are plausible and supported by evidence.

In conclusion, please don’t believe everything you hear. Apply a reasonable amount of skepticism to everything. The more incredible the claim, the more evidence should be needed to prove it. Everybody, including me, should be open-minded that they could be wrong about something and be open to such evidence. Meanwhile, the reality seems to be that the more ridiculous a notion is, the more tenaciously it tends to held.

Getting back to Benford’s Law, here are some sources that argue that it is dangerous to use it to analyze elections. They make their case better than I could.