Benford’s Law and Vote Totals

Have vote totals been tampered with?

Benford’s law is described in Wikipedia as “an observation about the frequency distribution of leading digits in many real-life sets of numerical data”. Basically, in a large number of sets of data, the leading digit of the numbers is not equally to be any of the nine digits. It’s more likely to be 1 than any other number. This law works well enough that deviations from it are considered red flags, and in financial statements, can trigger an audit by the IRS.

This video looks at the counts of the first digits of precinct totals in a number of areas, and compares those counts with the number expected given Benford’s law.  The charts it present show that most of the data sets conform pretty closely with this law, until we get to Biden’s vote total counts in certain areas.

Fortunately, the video’s description contains links to the actual data, so I was able to download the spreadsheets and do some more analysis.

The chi-square test is used to compare the difference between observed values and expected values in data. For example, Benford’s law predicts that for a number of sets of data, the leading digit should be 1 about 30% of the time. 

After a bit of fiddling, I was able to extract the leading digit of the counts, count those up, and compare them with the theoretical distribution. 

I looked at the numbers for Allegheny County, PA, and Fulton County, GA.

In Allegheny County, Trump’s vote totals fit pretty closely with Benford’s law. The value of the chi-squared statistic is 5.80, with a p value (the likelihood this value occurs by chance) of 0.669. 

In Biden’s case, the chi-squared statistic is 190.5, which has a p value of 5.73e-37.  This means the likelihood this distribution is due to chance is one in 1,745,000,000,000,000,000,000,000,000,000,000,000

Fulton county, GA has numbers that hew more closely to Benford’s law, and the chi-square statistics are much less extreme.

For Donald Trump’s vote totals, the chi-square statistic is 4.00, with a p value of 0.857. 

For Biden’s vote totals, chi-square = 15.50, with a p value of 0.050.

Trump’s vote totals are pretty much dead normal, and Biden’s teeter on the edge of statistical significance. 

I think it’s safe to call “shenanigans” for Allegheny county.