Why do Biden's votes not follow Benford's Law?

Matt Parker did a good video with lots of visuals to explain why Benford's Law works in general, but why it cannot always be simply applied to election results. ▶ this video

I am replying for the sake of the hidden sibling comment: I don't watch any major media coverage of politics, but a mathematician and comedian I follow on YouTube (from a different country who presumably has no horse in the race) has analyzed some of the numerical allegations of fraud: - link - link - ▶ this video

Precints have roughly even populations and therefore typically don't conform to Benford's law. ▶ this video

There were some pretty shit analysises of the 2020 US elections that Matt Parker covered with videos like "Why do Biden's votes not follow Benford's law?"[1] and "Why was Biden's win calculated to be 1 in 1,000,000,000,000,000?"[2]. I don't know about other countries, but the amount of data that every county in every US state produces makes systematic fraud pretty much impossible. If there's literally only 3 numbers produced by the Venezuelan government, you need to be seriously incompetent to have detectable fraud because techniques like Benford's or Zipf's law need lots of individual numbers. [1]: ▶ this video [2]: link

First of all it looks at suspiciously limited data (Oct 26th to Nov 11th) when we know more data is available. This is a known tactic from climate deniers who claim e.g. that data from a single weather station in Canada from October 2006 to June 2014 shows the world is actually cooling. Second of all it makes vague assumptions about distributions without clarifying or even justifying why it should have a different distribution. This is straight from the playbook of Trump’s election denialism where some variation of election results was somehow proof of tampering[1]. With statistics you can find any results you want if you look at limited enough data from a large enough pool, and you can find any abnormality you want by giving unrealistic expectations. This is why you need to clarify which distribution you expect, and—more importantly—justify it by pointing at other examples from similar pools. This is never done here, which is why I call it pseudo-statistical nonesence. Stating it is abnormal is not enough, even when you provide some fancy box plots to do so. 1: ▶ this video

I don't know about that particular analysis, but there've been a number of such claims that don't stand up (mostly because, as you ask, the districts themselves don't follow Benford's law). See, for example, "Inappropriate Applications of Benford’s Law Regularities to Some Data from the 2020 Presidential Election in the United States" by Walter R. Mebane, Jr. [0], and "Why do Biden's votes not follow Benford's Law?" by Matt Parker [1]. This fits the general pattern that there's been a lot of suspicion raised about fraud in the 2020 election, but none of it actually seems to pan out. [0] link [1] ▶ this video

Benford's law doesn't give much of a guarantee except in highly specific scenarios. You can however show it's pretty accurate if you've got a 'smooth' probability distribution spanning multiple orders of magnitude (how much of a guarantee you can give depends on how specific you define what it means to be 'smooth'). The rigged elections example you gave was, I believe, mostly explained by the fact that the number of votes they were looking at were all roughly the same order of magnitude [1]. [1]: ▶ this video

> Can you link me to those majority of cases? link ▶ this video And this third one just shows how much of a joke some of the claims in the lawsuits are: link Like, this is basic high school level statistics they can't even get right.

Precincts in most places are designed to have roughly the same number of voters. Hence, Benford's law does not work for them. Here's a good explanation, using Chicago precinct data [1]. A good rule of thumb is that if Benford's law does not apply to the legal voter population of your precincts/counties/wards/districts/whatever, or when applied to the total number of registered voters in those areas, then it won't apply to the results of a fair election either. (This is why if you ask every adult you know their height in inches and then look at the distribution of first digits, it will look nothing at all like a Benford's law distribution. The heights are going to be the sum of two normal distributions (one for men, one for women)). A good example of that is the county results in Georgia. There is a video going around from a forensic CPA showing the votes by county did not followed Benford's law, but he failed to look at the underlying population. I've written a little web-based simulation using the Georgia data to make it easy to play with the data to get a feel for why Benford's law does not work there [2]. Another way to tell when a deviation from Benford's law is not an indicator of election fraud is to look at what it would take to make the results fit Benford's law. If you can't find any reasonable way to adjust the votes to make it fit, then the lack of fit was not a fraud indicator. (And if you can find such an adjustment, it also needs to stay a Benford fit if you mak