I’ve written many times about Benford’s Law, the non-intuitive discovery that certain types of data sets have an odd property: The first digit of their data is more likely to be small than large – in a classic Bendford’s dataset, numbers will start with a “1” about 30% of the time.
I talked about this in my Zoom session at the annual meeting of the N.H. Joint Engineering Society, where I noted that the state’s database of daily COVID cases didn’t have a great Benford’s Law distribution – there weren’t nearly enough numbers starting with “1”. But as I said at the time, my math chops aren’t sufficient for me to determine whether that anomaly means anything.
Which brings us to the presidential election. One portion of the conspiracy-theory flailing by fans of Donald Trump invokes Bendford’s Law to say that Joe Biden’s vote totals are somehow suspect. I first encountered this in the Talk page of the wikipedia article about Benford’s Law, where the section titled “Election Day edits” goes on and on and on.
It’s a complicated argument, easy to seem convincing to us laymen, so I was delighted to see it taken up by Matt Parker, a math-themed comedian whose videos are wonderful. You should watch his whole 17-minute YouTube video on the discussion here if for no other reason than to hear his Australian accent.
For the impatient, here is the main point: Benford’s Law is not a good test for election results because it only holds if your data covers at least a couple orders of magnitude. In vote tallies, that often doesn’t apply. And it definitely doesn’t apply here: The conspiracy theory looked only at precincts in Chicago, which by definition are all roughly the same size.
Furthermore, Chicago is overwhelmingly Democratic so most votes went to Biden, which further undoes a Benford’s Law distribution for him. By contrast, Trump’s precinct-level tallies were more spread out (in some precincts he got less than 10 votes), and that explains why Trump’s data from Chicago fits Benford’s distribution. If you did this same test in an area that was overwhelmingly Republican, it would show Biden’s results matching Benford’s Law but Trump’s results not matching it.
But don’t believe me – watch the video. Parker is great.