trump biden
Benford's law (or the first-digit law) is an observation about the frequency of the most significant digit of naturally occurring numbers. Somewhat counter-intuitively (but with a well-understood explanation), small digits are more common than large ones, with around 30% of all values in a large collection beginning with 1, while only 4.6% start with 9. The probability for digit d is given by:


Benford's law is widely used in forensic accounting to detect phoney numbers, and evidence based upon these analyses has been admitted in a number of criminal trials. It is also used to detect scientific fraud in published data.

User "cjph8914" on GitHub has created a repository, 2020_benfords, which analyses the 2020 presidential election results for various districts and compares them with the expectation from Benford's law. Here are the results for precincts and wards in Fulton County, Georgia (containing Atlanta).
benfords law

The red curve plots the expectation from Benford's law, while the blue bars show the observed frequency of first digits in vote counts reported by the precincts and wards within the county. This is a well-behaved distribution which is consistent with Benford's law for a sample of this size and doesn't show evidence of hanky-panky.

Now let's head north and take a peek at the results for the charming metropolis of Milwaukee, Wisconsin.
benfords law
The results for Donald Trump and the minor party candidates are consistent with Benford's law, but whoa! โ€” take a look at that chart for Biden. This is wildly anomalous, and the kind of thing you get when you ask people to "write down random numbers" as opposed to measurements of actual phenomena.

What about the Windy City, that paragon of electoral probity? Off to Chicago we go....
benfords law
Once again, we see Benford-like numbers for Donald Trump and the minor party candidates, but wildly discordant results in the votes for Biden.

This analysis is not dispositive, but they're the kind of thing that if you saw it in sales figures reported by a public company, it would make the eyebrows of the more cynical kind of Wall Street security analyst head for the hairline.

If you want to check the results or do your own analysis with publicly-available election data for other jurisdictions, all of the raw data and Jupyter notebook tools are available from the GitHub repository.

Particularly interesting would be a Benford's law analysis of early results versus those incorporating late-arriving votes "discovered" well after the polls closed, and comparing votes from regions expected to lean toward one candidate or the other.