Like many others, I strongly suspect widespread ballot fraud in the presidential election, as well as miscounting due to software problems in certain jurisdictions. I therefore fully support the legal challenges and recounts now getting underway. However, there is one indicator of fraud, now widely cited by Republicans, in which I have no confidence as applied. It’s a statistical tool based on Benford’s Law, which can serve as a signal of voter fraud. I mentioned it briefly in my last post. At the risk of getting ahead of myself, here’s what I said then:

“… Benford’s Law … is a “forensic” test of fraud based on statistical theory, but I do not trust the form in which it’s been invoked thus far. Violations have been cited in several counties over the past few days. However, a violation of this law obviously doesn’t constitute direct evidence of fraud, and the test is a reliable indicator only when the number of voters in different precincts vary by orders of magnitude (there must be a mix of [numbers in the] 10s, 100s, 1,000s, 10,000s). With precinct sizes, that is often not the case. There is a more reliable form of Bedford’s law, but I have not seen its application to any results in this election.“

The last link above is to a paper by Walter Mebane of the University of Michigan. I’ll refer to his work below, including some post-election tests he’s conducted.

First Digits

Benford’s Law holds that many collections of numbers encountered in nature or human affairs (populations of ant colonies, accounting data) will have a large proportion of leading digits that are low numbers. For example, the number 1 will tend to appear as the leading digit about 30% of the time; the number 2 will be the leading digit about 18% of the time, while the number 9 will be the leading digit less than 5% of the time. The broader the range of the numbers, the more accurately they will conform to Benford’s Law. As I stated above, a range of numbers covering several orders of magnitude will approximate Benford’s Law fairly well, while a range confined to a single order of magnitude generally won’t conform unless its distribution is extremely skewed toward the low end of the range.

What does that have to do with election fraud? If the number of votes across different voting precincts cover several orders of magnitude (for example, single digits, 10s, 100s, and 1,000s), they should conform to Benford’s Law. The distribution of first digits across precincts should look a lot like the chart above. If they don’t conform, it’s an indication that votes may have been altered or added. That’s because Benford’s Law tends to break down when an independent process leads to additive changes to the original numbers (rather than multiplicative changes, such as population growth).

So again, there have been claims that several cities had presidential voting patterns suggesting violations of Benford’s Law for Joe Biden, but not for Donald Trump and other candidates. These were Milwaukee, WI, Chicago, IL, and Allegheny County, PA. Subsequently I saw similar claims about other cities and counties, such as Fulton County, GA.

The chart below shows the results for Milwaukee. I show only three of the candidates’ distributions of first digits, but the other candidates, who garnered relatively few votes, look much like the one on the far right. The chart shows that Joe Biden’s distribution looks nothing like Benford’s Law would suggest, while Trump’s does. The assertion is that Biden’s pattern is a sign of fraudulent voting.

The problem with these claims is that the size of the precincts and variations in votes across wards might not support the validity of Benford’s Law. I looked at the 327 election wards in the City of Milwaukee, which range in size from just a few voters to several thousand, but most have less than 1,000 voters. The average turnout of registered voters across wards was over 78%, and the average number of ballots cast per ward was 757. Biden received almost 80% of the votes in Milwaukee, or about 595 per ward; Trump received an average of 148.

(I should note that in seven wards there were controversial, post-election upward adjustments in the number of registered voters, where voter turnout had originally been calculated as greater than 100%. Needless to say, that is rather suspicious. However, I disclose now that the data were collected after these adjustments were made.)

What’s important in the application of Benford’s Law is the distribution of votes across wards. Biden’s distribution of votes across wards in Milwaukee was concentrated between 186 and 1,196 (the middle 90% of his distribution of ward votes), and again, centered at 595. For Trump, 90% of his ward vote totals were between 14 and 412. It should be no surprise that a large share of Biden’s vote totals would have leading digits of 4, 5, and 6, while Trump had lower leading digits. So the charts of leading digits for Milwaukee are really artifacts of the narrow distributions of ward votes for these candidates. Broader distributions covering several orders of magnitude would provide first-digit analysis more capable of indicating fraud, if it occurred.

Second Digits

The other Benford-type test of fraud mentioned above is based on the second digit of vote totals, and it is not sensitive to the width of the vote distributions. The typical pattern of second digits is much less pronounced than first digits, but there is still a smooth decline from smaller to larger second digits. I found the two charts below on the Golden Age of Gaia site, of all places. They contrast the frequency of second digits from the Biden and Trump vote totals by precinct for ballots in Allegheny County, PA. The usual pattern of second digits is plotted along the orange line, but whoever prepared these charts mislabelled the horizontal axes (they should run from zero to nine).

Joe Biden’s frequencies are irregular, with significant differences for some values of the second digit. Trump’s pattern is more typical. However, I learned today that Walter Mebane had performed a few second-digit tests on Allegheny County and Milwaukee. He calculates an overall test statistic for the full set of second-digit values and finds the statistics for those counties to be within a certain reasonable range, or at least he felt they could be explained by other factors.

Visually, however, there is a sharp contrast between the Biden and Trump charts. And the data has been in flux, so it’s not clear that the charts correspond to exactly the same data tested by Mebane.

In the end, these tests offer no real guidance in this case. All tests of this kind offer circumstantial evidence, at best, and they are invalid under some circumstances. As Mebane said in his 2006 paper:

“… to prevent election fraud, appropriate practices need to be used while the election is being conducted. Insecure or opaque voting technology or election administration procedures should not be used. The election environment should not foment chaos and confusion. Not only should elections be secure and fair, but everyone should know they are secure and fair.“

Chaos and confusion…. yes, that sounds about like the 2020 election environment. Mebane is obviously aware of the limitations of the statistics in which he specializes. Nevertheless, these tests are broadly used in a variety of applications. Crazy results raise suspicions, but sometimes they are not the best leads in pursuing claims of election fraud. There are plenty of other red flags in the present case. The states now in dispute are close, and most of those votes will be subject to recount anyway.