There is No ‘Undeniable’ Evidence of Election Theft

For a few days now, a number of conservative online sources have been hard at working pushing the narrative that Democrats and Joe Biden have stolen the 2020 election. One of the foremost pieces of alleged evidence they’ve so far produced is a comparison of vote tallies with Benford’s Law. This argument has already made its way to the Washington Examiner, where it’s being reported that Trump campaign staff themselves are touting it as proof of election theft. However, the primary, most popular source for these claims appears to be an article for The Red Elephants, boldly declaring There is Undeniable Mathematical Evidence the Election is Being Stolen.

In this article, the focus will mainly be kept on the allegations concerning Benford’s Law and the election. Because of that, this will be a bit of a long one and will get technical in places, but I believe it’s worth the read and could (hopefully) be enlightening. One reason for choosing this focus is that many of the claims made in the Red Elephants piece fall somewhere in between speculation and the simple presentation of figures without a clear argument connecting them to fraud. An easy example comes early on in the discussion of absentee voting in Wisconsin and Michigan.

The author of the article notes that Election Day data in Wisconsin — the source they cite is NBC News — showed “Republicans led Mail-in Ballots requested 43% to 35%,” as well as in the number of mail-in ballots returned. A similar claim is made regarding Michigan. What is conveniently left out of all this, though, are the 22% of ballots requested and returned in Wisconsin that listed Other for party affiliation, or the 20% of those in Michigan. As Twitter user Shylock Holmes remarks in the article, it’s reasonable to infer that those who identify as Democrat will likely vote for a Democrat, but inferring who the share of votes listed for Other will go to is a far more hazardous guess.

This kind of rush to judgment with a heavy dose of speculation characterizes the entire piece. Another section that discusses the “Massive Enthusiasm Gap” assumes that variations in the enthusiasm voters feel for their candidate is some reliable predictor of who turns out to vote. When these differences are measured in clear and contrasting terms, perhaps there is something to this analysis, but the poll the article relies on uses terms like “Extremely,” “Very,” and “Moderately” to measure voter enthusiasm. The author seems to only want to look at that first category, where Trump’s advantage is “double-digit,” yet once again the bigger picture — the fact that enthusiasm for both candidates fell overwhelmingly into one of those three categories as opposed to “Slightly” — is ignored.

I think this goes a long way to explaining why a point-by-point rebuttal of the Red Elephants essay is unnecessary. The majority of the evidence it purports to lay out does not come close to approaching the level of either mathematical or undeniable proof teased in the title. Additionally, if their own claim about Benford’s Law serving as a useful tool for determining the legitimacy of an election is true, then a closer look at this argument could suggest something to us about the likelihood of these other ‘anomalies,’ oddities, and surprises being due to anything more than chance.

Put simply, Benford’s Law is a statistical rule positing that the leading digits of numbers are more likely to be smaller than larger. In sets of numbers that follow the rule, the number 1 often appears as the leading digit, while higher numbers going up to 9 become less and less likely. The Wikipedia entry for this law gives the example of the heights of the 58 tallest structures in the world. The leading digit for most of these structures is, as expected, 1. From there, the other numbers sequentially decrease in their probability. Sets of data that do not comply with Benford’s Law are said to violate it.

There is more to this rule that we’ll get to in a moment, but it’s worth pausing for a second to note that the article we’re examining here, which displays a publication date of November 5th 2020, has actually received a number of edits and alterations over the last few days. Making use of the Internet Archive’s Wayback Machine, we can see that the entire section on Benford’s Law is not original to the article and only appears after November 7th. Further changes have also been made, making it especially ironic that the November 7th version of the page saw fit to draw special attention to an edit on Wikipedia’s entry for Benford’s Law that calls into question the reliability of the law when applied to elections (seen below).

This is relevant because the article’s claims and graphs involving Benford’s Law seem to keep changing almost every day — a strange thing for undeniable mathematical evidence to do. The first version (Nov. 7th) of the section we’ll be looking at, linked to above, contains a handful of graphs depicting the leading digits of precinct vote counts for Milwaukee, WI, Allegheny, PA, and Chicago. The latest version of the page, as of Nov. 10th, shows the first two of these, but omits Chicago.

A 2005 report by The Carter Center, Observing the Venezuela Presidential Recall Referendum, addresses the issue of using Benford’s Law to evaluate elections for fraud. The report considers appeals made to this statistical rule over a constitutional referendum in Venezuela following the election of Hugo Chavez in 1998. For our purposes, one passage in particular is worth quoting from page 134:

The panel believes that there are many reasons to doubt the applicability of Benford’s Law to election returns. In particular, Benford’s Law is characteristic for scale-invariant data, while election machines are allocated to maintain a relatively constant number of voters per machine. [Henry] Brady finds, for example, that the first digit of precinct-level electoral data for Cook County, the city of Chicago, and Broward County,Fla., depart significantly from Benford’s Law, primarily because of the relatively constant number of voters in voting precincts.

In short, leading or first digit data like that shown in the graphs above is not a useful metric for ascertaining election fraud. This is because, as the excerpt explains, there is usually not that much variation in vote share across precincts in smaller areas like cities and counties. Rather than conforming to Benford’s Law, the first digit in these sets of data will usually be determined by the voter size of each precinct. The use of Chicago as a specific example here is telling, too, since it may hint at why this city’s graph was dropped in newer revisions to the page. We’ll come back to that a little later.

Another article published by the Election Integrity Partnership also cautions readers on the use of Benford’s Law to assess election results. The author(s) clarify that there are two important assumptions or conditions behind the law that must be met in order for it to apply:

…the law holds consistently when certain assumptions are met: all numbers must be equally likely to appear (i.e., you can’t only tally batches of 6 votes and expect the totals to start with 7, 8, or 9) and the numbers must span multiple orders of magnitude, such as ranging from 100 to 10,000,000. Violations of these assumptions lead to violations of the law. For vote tallies, all numbers are equally likely, but not all states meet the second assumption. In the state of Nevada, Esmeralda County has around 900 people while Clark County has over 2,250,000 people. In the state of Vermont, the bounds are much narrower.

Data that do not meet these assumptions do not show reliable evidence of fraud. All that may be shown in them is that the rule is being inappropriately applied in this instance. This is essentially another way of putting the findings and conclusions of Brady and the other members of the panel writing for The Carter Center. But it makes the valuable point of underscoring how the rule’s assumptions should not be discarded, particularly where elections are concerned.

There is another way of using Benford’s Law to examine election results, though, which was introduced by Professor Walter Mebane. This method utilizes a second digit test. On the Wikipedia entry for Benford’s Law, Mebane’s work features prominently under the law’s application to election data. The author of the Red Elephants piece also refers to Professor Mebane in introducing a new set of graphs based on this second digit test.

What’s immediately worth noting here is that the numbers along the y-axis in these two graphs do not match. Narrower figures will give the appearance of bigger changes, and the differences are pretty substantial, as one y-axis uses a factor of 2 and the other uses one of 0.5. Adjusting these scales to be equal will flatten the peaks in Biden’s graph to bring it closer to the distribution seen for Trump. This graph comes from the Nov. 7th version of the page and is no longer up. The Wayback Machine shows it was only recently removed, quite possibly because its extremely misleading nature was exposed.

Currently, the only chart of second digit data remaining up is absentee ballots in Allegheny. The section featuring Benford’s Law has been curiously pushed further to the bottom of the page since I first viewed the article. This can be seen from earlier versions stored on the Wayback Machine, too.

Looking beyond some of these graphs, it should be evident the Red Elephants article doesn’t urge much in the way of cautious interpretation of the data. An early line beginning the section was: “Biden pretty clearly fails an accepted test for catching election fraud, used by the State Department and forensic accountants.” This has since been amended to read: “Benford’s law is one accepted test for catching election fraud, used by the State Department and forensic accountants.”

The link embedded for the State Department use of Benford’s Law goes to a paper by Professor Mebane. Amusingly enough, this paper is a response written by Mebane to critics of his use of the law with election results. Even if one accepts the rebuttal, this fact is enough to show that Benford’s Law and the second digit test are not accepted by everyone, including other political scientists and statisticians. The Carter Center report is another example of a critique that casts significant doubt on the reliability of this statistical rule when applied to elections.

The essay wraps up its confused and consistently revised presentation of this ‘undeniable’ mathematical evidence with the following:

Walter Mebane weighed in on the second digit analysis in a paper. He claims that Fulton county Georgia has inconsistent second digits for Democrats, but does not provide a statistical analysis as he claims he is lacking precinct data. Mebane also mentioned Chicago was inconsistent as well. He evaluated Milwaukee County and Allegheny County (not absentee ballots) and concluded that second digit distribution is consistent with Benford’s law.

Mebane’s final verdict regarding the elections in these and other jurisdictions is that analysts “should await the final production of completed vote counts and should draw on additional information about election processes that go beyond mere vote count data.”

This excerpt can be seen as the point where all the dots come together in a few different ways. Mebane’s paper (which I recommend reading for yourself from the link in the quote above) is titled, “Inappropriate Applications of Benford’s Law Regularities to Some Data from the 2020 Presidential Election in the United States.” If this doesn’t exactly sound like the support the Red Elephants author attempts to portray it as, you’d be right to think so.

Mebane begins with the same critique of first digit precinct data offered from The Carter Center report, noting it’s “widely understood” that this is not useful stuff for identifying election fraud. If that’s still not clear enough, he ends those comments by stating that “first-digit distribution has nothing whatsoever to do with any kind of election fraud.” This makes one wonder why the graphs with leading digit data in Allegheny and Milwaukee have been left up in the face of such unequivocal denunciation.

However, the Red Elephants description of Mebane’s final verdict still sounds pretty dishonest, to put it charitably. The remarks made on “inconsistent” data in Georgia and Chicago give half the story. From Mebane’s paper, we get the real details:

For Fulton County (Figure 4) a few statistics have statistically distinguishable values, but these are probably benign: the DipT results are due to bimodality that is evident in the densities shown in Figure 5 — some precincts lean slighty in favor of Trump/Pence while more overwhelmingly favor Biden/Harris. The P05s value for Dem.Total.Votes might be a concern, but notice that the P05s value for Rep.Total.Votes nearly exceeds the expected value of .2 as well. For Chicago (Figure 6) P05s is slightly elevated for BidenHarris, and C05s is slightly elevated for TrumpPence: the diferences from the expected values of .2 are small.

These inconsistencies, contrary to Red Elephants’ misleading omissions, do not pose problems in Mebane’s analysis. Furthermore, the ones in Chicago show a similar inconsistency on Trump’s side. Yet reading Red Elephants and the mention of inconsistencies in Georgia “for Democrats,” one could well be under the mistaken impression that Chicago’s inconsistencies lean one way, too. Also left out of the article is any discussion of Mebane’s eforensics modeling which he says showed 0.9797 probability of no fraud in Milwaukee and 0.9839 probability of no fraud in Allegheny.

While Professor Mebane does advise that final verdicts should await complete vote counts and utilize further information, the sentence he ends his paper with is far less ambiguous: “To date I’ve not heard of any substantial irregularities having occurred anywhere, and the particular datasets examined in this paper give essentially no evidence that election frauds occurred.”

Consider that Mebane is not only the source dangled around by the article’s author, but a leading authority in his field who is largely responsible for introducing the very kind of statistical test being wielded by Republicans and conservative outlets like Red Elephants to now claim election fraud. If these responses and critiques, including from Professor Mebane himself, are not enough to show that this allegedly “undeniable mathematical evidence” is only so much sound and fury hung on the thinnest of hopes by those who are reluctant to accept their candidate losing fairly, I’m not sure what else would qualify.

It would be almost too easy to point to the numerous, baseless claims made about widespread election fraud by President Trump in the months leading up to the election and conclude that, as always, Trump’s devoted followers took him at his word, lapping up the victim card he dealt them, and will refuse to allow that it be anything other than sheer prophecy. If that seems grossly hyperbolic, then I would suggest this is precisely why claims and behaviors like these are worth digging out, bringing into the light, and challenging. Why even documenting the scrambling, confused, and misleading life cycle of an arguably anti-democratic web page for Red Elephants may be more important than it appears at first glance.


Since the original publication of this article, numerous election officials and expert sources have denied and debunked claims of fraud. A statement issued by the Election Infrastructure Government Co-ordinating Council, which is made up of senior officials from the Department of Homeland Security and the US Election Assistance Commission, announced on November 12th that “There is no evidence that any voting system deleted or lost votes, changed votes, or was in any way compromised.” President Trump fired his own pick for director of the Cybersecurity and Infrastructure Security Agency over alleged inaccuracies in its reporting of election security, even though the agency addresses and refutes many of these specific allegations in its Rumor Control page.

Election officials in every swing state have said no evidence of election fraud has been found. An audit of Georgia’s voting machine discovered no evidence of fraud. A delegation of international observers including 28 experts from 13 different countries witnessed no instances of voter fraud or irregularities, giving a detailed report on the procedures of many states on Election Day. A postal worker in Pennsylvania that claimed to have seen evidence of ballot tampering has since recanted. One Trump lawyer in Pennsylvania admitted to a judge that even among the 592 ballots they were requesting be thrown out, there was no connection to fraud. Multiple law firms have withdrawn support from the president’s efforts to sue over a “rigged election.”

Rather than trading in the speculation and dubious calculations of random figures on Twitter, Facebook, Fox News, and other media platforms, these findings come from the diligent research and knowledge of seasoned experts, as well as from legal minds that know the difference between crying foul to a party of one’s close supporters and honestly meeting a reasonable standard of proof for election fraud. Even before one considers the flimsy arguments made in the Red Elephants piece, these sources constitute part of a growing mountain of evidence disputing accusations of a stolen election — a mountain that is proving increasingly impossible to climb for proponents of this conspiracy theory.

Writer, reader, musician, and graduate in philosophy and religious studies.

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