After the 2020 election was called for former Democratic Party candidate Joe Biden, “Benford’s Law” — a mathematical principle applied occasionally to the count of votes in any given election — became a hot topic on social media forums like Gab and Parler, where fans of Republican incumbent Donald Trump made excuses for and spread conspiracy theories about his resounding and decisive defeat.
One of those conspiracy theories revolved around a misreading of Benford’s Law, which is defined as follows (without mention of elections, election fraud, or Joe Biden):
[Benford’s Law is] the principle that in any large, randomly produced set of natural numbers, such as tables of logarithms or corporate sales statistics, around 30 percent will begin with the digit 1, 18 percent with 2, and so on, with the smallest percentage beginning with 9. The law is applied in analyzing the validity of statistics and financial records.
For the week ending November 9 2020, Google Trends indicated a spike in searches not only for Benford’s Law, but for “Benford’s Law election,” “Benford’s Law Joe Biden,” “Benford’s Law voter fraud,” and “Benford’s Law Biden votes.” On 4Chan’s divided /pol/, false claims Twitter “banned” any mention of Benford’s Law proliferated:
BREAKING: TWITTER BANS DISCUSSION OF BENFORD’S LAW
TWITTER AUTOMATICALLY BANS YOU IF YOU RETWEET THIS POST
— Jim Hoft (@gatewaypundit) November 8, 2020
GO AHEAD AND TRY IT FOR YOURSELF
THEY ARE RUNNING SCARED LOL
It should be noted that the anonymous user linked above provided a link to a tweet that was in no way banned or restricted due to its mention of Benford’s Law, and its originator (chronic disinformation purveyor Jim Hoft, who runs the Twitter account and right-wing blog @gatewaypundit) bears, for some reason, a blue checkmark.
Hoft’s claim demonstrated significant engagement, all unencumbered by any of Twitter’s “disputed claim” or “this claim about election fraud” flags:
Hoft linked to a November 7 2020 GNews.org item with the headline “Joe Biden’s votes violate Benford’s Law (Mathematics),” which began with a cascade of pseudointellectual lies:
As the vote counting for the 2020 Presidential Election continues, various facts suggest rampant frauds in Joe Biden’s votes. So does mathematics in terms of the votes from precincts.
Benford’s law or the first-digit law, is used to check if a set of numbers are naturally occurring or manually fabricated. It has been applied to detect the voting frauds in Iranian 2009 election and various other applications including forensic investigations.
UPDATE: Facebook and Twitter Suspend Accounts That Posted on Benford’s Law Showing Biden’s Implausible Vote Totals — LABELING IT “SEXUAL EXPLOITATION”
This part appears to be legitimate; we are contacting Facebook for details. However, that warning appears to be more a reaction to previous claims made by Hoft and his ilk. We have contacted Facebook for comment.
In that November 8 2020 post, Hoft primarily accused Facebook and Twitter of censoring shares of his tweet and the GNews.org post, writing:
We have heard from many readers who told us once they retweeted this tweet or tried to post it on Facebook their account was suspended! The social media giants are preventing Americans from posting this mathematical evidence that proves Joe Biden’s numbers violate the Benford Law of normal distributions!
In a prescient November 6 2020 analysis by the Election Integrity Partnership (“Vote Data Patterns Used to Delegitimize the Election Results”), claims about Benford’s Law were one of several topics discussed in relation to potential efforts to delegitimize the final vote tallies with social media-enabled election interference.
The Election Integrity Project also displayed two charts, one called “Vote Tallies Projected against Benford’s Law,” and the other, “Final Vote Tallies Projected against Benford’s Law.” Contrasting the two, they wrote:
The figure above [“Vote Tallies Projected against Benford’s Law”] shows the leading digit of reported vote tallies across select counties. For instance, the final tally in Dane County, Wisconsin was 338,946. This would count for one county in the 3 column. But why would anyone care to look at this kind of frequency distribution? Data forensic experts use these distributions to investigate fraud. They look at whether empirical distributions of leading digits deviate from a special distribution described by Benford’s Law. The law posits that leading digits of numbers are more likely to be smaller numbers (e.g., 1) than larger numbers (e.g., 9).
Armchair investigators during the election have already begun to argue that too many of the submitted vote totals begin with larger single digit numbers (7 or 8 for example), which is being spun as evidence of voter fraud. We caution against this conclusion. Having the distribution of leading digits stray from the expected percentages predicted by Benford’s Law can happen by chance, though it is more common when the law’s assumptions are violated, as they often are with vote tallies. Benford’s Law, and other math-based inquiries, can be used to detect voter fraud, but the vast majority of these violations are not conclusive evidence of fraud.
Returning to our voting tally in Figure 1, you will see that the tallies deviate from the line of expectation. So, does this mean fraud? Does it mean that vote counters were up to something nefarious? In this case, absolutely not. First, the example above is a simulation based on a computer script, rather than one based on real voter data. If we consider the final output of this 72 county simulation, it ends up looking like Figure 2 [“Final Vote Tallies Projected against Benford’s Law”]:
These final results are more predictable and follow the expected counts more closely, but still exhibit expected deviations. These same deviations are occurring in the voting counts currently being reported in the 2020 election. Our aim in this post is to prepare the public and journalists for these misleading arguments and to provide context for the claims already being made online.
In their conclusion, the Election Integrity Partnership noted that claims about Benford’s Law “proving” election fraud were based on early, incomplete data — not to mention a fundamental misunderstanding of how it works. Essentially, claimants citing Benford’s Law were cherry picking early or incomplete results to stake their claim:
At this stage, the assumptions that lead to Benford’s law are violated leading to the patterns generated in the Figure 1 above. Only once all counties have been counted does the distribution approach something consistent with Benford’s law, seen in Figure 2. Even at this stage, the distribution of county sizes still makes it unlikely to exactly match expectations. A more complete model might include non-random voting patterns whereby rural counties lean a different direction than urban ones. This, compared with the relationship between the rate of vote counting and county or precinct size would probably cause more drastic violations of assumptions. As this is a rapid response, incorporating this complexity was impractical.
Unsurprisingly, a spike in interest involving one particular mathematical principle (Benford’s Law) led to drama over on the topic’s Wikipedia page. On the “Talk” page for “Benford’s Law,” one section (“Benford, QAnon, and the 2020 election”) began:
Following the 2020 United States presidential election result, a number of QAnon folks have been promoting a theory on social media that the failure of voting numbers for Biden to match Benford is a demonstration of likely electoral fraud. This is likely why there has been a big increase in interest in this page, and in particular the electoral fraud section. The short answer is no. These claims are baseless, and come from a misapplication of Benford’s law to particular cities in a county, or wards in a city, as opposed to all counties/cities in the US (which is how Benford detected possible fraud in Iran. If you do this analysis in the US you find that yes, all the numbers fit Benford perfectly). Of course, this cannot be posted in the article as it would constitute original research, but it is worth keeping a close eye on the article as there may be misleading edits made in support of the conspiracy theory over the next few days. Awoma (talk) 09:46, 8 November 2020
Application (or misapplication) of Benford’s Law to the 2020 election eventually made an appearance in a massive, regularly updated Twitter thread by political reporter Isaac Saul cataloging disinformation around the results of the 2020 election:
1/ ALRIGHT Y'ALL. ELECTION FRAUD DEBUNKING MEGA THREAD. I’m following claims of fraud and looking into them. I think I’ve solved most now. Nothing is holding up under any scrutiny so far so I’m making a thread to track in one place. Please RT! #ElectionResults2020 #Election2020
— Isaac Saul (@Ike_Saul) November 5, 2020
On November 5 2020, Saul first mentioned Benford’s Law, noting that he was not initially familiar with the principle nor its purported relation to current election fraud claims:
76/ Okay! This was BY FAR the hardest one for me yet. It took some digging, because I wasn't very familiar with Benford's law and this person is not good at Twitter, so the thread is jumbled. But this is going very viral on FB and Twitter: https://t.co/WEbJqRGDjC
— Isaac Saul (@Ike_Saul) November 6, 2020
Saul eventually cited a 2011 paper (“Benford’s Law and the Detection of Election Fraud”) from Political Analysis, vol. 19, no. 3. Its abstract explained:
The proliferation of elections in even those states that are arguably anything but democratic has given rise to a focused interest on developing methods for detecting fraud in the official statistics of a state’s election returns. Among these efforts are those that employ Benford’s Law, with the most common application being an attempt to proclaim some election or another fraud free or replete with fraud. This essay, however, argues that, despite its apparent utility in looking at other phenomena, Benford’s Law is problematical at best as a forensic tool when applied to elections. Looking at simulations designed to model both fair and fraudulent contests as well as data drawn from elections we know, on the basis of other investigations, were either permeated by fraud or unlikely to have experienced any measurable malfeasance, we find that conformity with and deviations from Benford’s Law follow no pattern. It is not simply that the Law occasionally judges a fraudulent election fair or a fair election fraudulent. Its “success rate” either way is essentially equivalent to a toss of a coin, thereby rendering it problematical at best as a forensic tool and wholly misleading at worst.
A 2006 paper [PDF] presented at a political methodology conference addressed the application of Benford’s Law alone to evidence claims of election fraud:
Another important issue concerns whether Benford’s Law should be expected to apply to all the digits in reported vote counts. In particular, for precinct-level data there are good reasons to doubt that the first digits of vote counts will satisfy Benford’s Law. Brady (2005) develops a version of this argument. The basic point is that often precincts are designed to include roughly the same number of voters. If a candidate has roughly the same level of support in all the precincts, which means the candidate’s share of the votes is roughly the same in all the precincts, then the vote counts will have the same first digit in all of the precincts. Imagine a situation where all precincts contain about 1,000 voters each, and a candidate has the support of roughly fifty percent of the voters in every precinct. Then most of the precinct vote totals for the candidate will begin with the digits ‘4’ or ’5.’ This result will hold no matter how mixed the processes may be that get the candidate to roughly fifty percent support in each precinct. For Benford’s Law to be satisfied for the first digits of vote counts clearly depends on the occurrence of a fortuitous distribution of precinct sizes and in the alignment of precinct sizes with each candidate’s support. It is difficult to see how there might be some connection to generally occurring political processes. So we may turn to the second significant digits of the vote counts, for which at least there is no similar knock down contrary argument.
On skeptics.stackexchange.com, one reader asked about the Benford’s Law and Biden votes rumor. Another commenter reiterated that such claims were predicated on cherry-picked early numbers, and promoters of the claim were lying with graphs:
I’ll address just the second charts, because they are straight out of How To Lie With Statistics.
As commenters have noted, the vertical scales are different. Narrow vertical scales make changes look larger. While wide vertical scales smooth out changes. Biden’s graph is using a more narrow scale than Trump’s.
Put them all together in one graph with the same scale and they don’t look so different anymore.
I had to eyeball the numbers from the graphs, but more precise numbers won’t change the outcome. I don’t even know if the numbers are correct. I can say with some certainty that the graphs are deliberately constructed to sell a lie. One or the other scale is a natural choice, either 0 to max or min to max. Someone had to choose to use different vertical axes for each graph.
Rumor’s that Biden’s victory was impossible because it somehow “violated” Benford’s Law gained further traction after Biden’s victory was called on November 7 2020, promoted by disinformation purveyors like Jim Hoft. Under even the slightest scrutiny, the claims dissolved for a number of reasons — such as their basis on early or single-district results, and general existing indications that Benford’s Law was a poor model with which to “prove” election fraud across the board.