Anomalies in Vote Counts and Their Effects on Election 2020

November 30, 2020

A vote analysis (with thanks to Rand Paul):

Executive Summary

In the early hours of November 4th, 2020, Democratic candidate Joe Biden received several major “vote spikes” that substantially — and decisively — improved his electoral position in Michigan, Wisconsin, and Georgia. Much skepticism and uncertainty surrounds these “vote spikes.” Critics point to suspicious vote counting practices, extreme differences between the two major candidates’ vote counts, and the timing of the vote updates, among other factors, to cast doubt on the legitimacy of some of these spikes. While data analysis cannot on its own demonstrate fraud or systemic issues, it can point us to statistically anomalous cases that invite further scrutiny.

This is one such case: Our analysis finds that a few key vote updates in competitive states were unusually large in size and had an unusually high Biden-to-Trump ratio. We demonstrate the results differ enough from expected results to be cause for concern.

With this report, we rely only on publicly available data from the New York Times to identify and analyze statistical anomalies in key states. Looking at 8,954 individual vote updates (differences in vote totals for each candidate between successive changes to the running vote totals, colloquially also referred to as “dumps” or “batches”), we discover a remarkably consistent mathematical property: there is a clear inverse relationship between difference in candidates’ vote counts and and the ratio of the vote counts. (In other words, it’s not surprising to see vote updates with large margins, and it’s not surprising to see vote updates with very large ratios of support between the candidates, but it is surprising to see vote updates which are both).

The significance of this property will be further explained in later sections of this report. Nearly every vote update, across states of all sizes and political leanings follow this statistical pattern. A very small number, however, are especially aberrant. Of the seven vote updates which follow the pattern the least, four individual vote updates — two in Michigan, one in Wisconsin, and one in Georgia — were particularly anomalous and influential with respect to this property and all occurred within the same five hour window.

In particular, we are able to quantify the extent of compliance with this property and discover that, of the 8,954 vote updates used in the analysis, these four decisive updates were the 1st, 2nd, 4th, and 7th most anomalous updates in the entire data set. Not only does each of these vote updates not follow the generally observed pattern, but the anomalous behavior of these updates is particularly extreme. That is, these vote updates are outliers of the outliers.

The four vote updates in question are:

An update in Michigan listed as of 6:31AM Eastern Time on November 4th, 2020, which shows 141,258 votes for Joe Biden and 5,968 votes for Donald Trump
An update in Wisconsin listed as 3:42AM Central Time on November 4th, 2020, which shows 143,379 votes for Joe Biden and 25,163 votes for Donald Trump
A vote update in Georgia listed at 1:34AM Eastern Time on November 4th, 2020, which shows 136,155 votes for Joe Biden and 29,115 votes for Donald Trump
An update in Michigan listed as of 3:50AM Eastern Time on November 4th, 2020, which shows 54,497 votes for Joe Biden and 4,718 votes for Donald Trump

This report predicts what these vote updates would have looked like, had they followed the same pattern as the vast majority of the 8,950 others. We find that the extents of the respective anomalies here are more than the margin of victory in all three states — Michigan, Wisconsin, and Georgia — which collectively represent forty-two electoral votes.

Extensive mathematical detail is provided and the data and the code (for the data-curation, data transformation, plotting, and modeling) are all attached in the appendix to this document[1].

Background

Late on Election Night 2020, President Donald J. Trump had a lead of around 100,000 votes in Wisconsin, a lead of around 300,000 votes in Michigan, and a lead of around 700,000 votes in Pennsylvania. Back-of-the-envelope calculations showed that in order to overtake President Trump, Joe Biden would have to substantially improve his performance in the remaining precincts — many of which were in heavily blue areas like Detroit, Milwaukee, and Philadelphia.

On Election Night, conflicting news reports came in that various precincts were stopping their count for the evening, sending election officials home, or re-starting their counts. There remains a large amount of confusion to this day about the extent to which various precincts stopped counting, as well as the extent to which any state election laws or rules were broken by sending election officials home prematurely. Whatever the case is, various precincts in Wisconsin, Michigan, and Pennsylvania continued to report numbers throughout the night.

By the early hours of the following morning, Wisconsin had flipped blue, as did Michigan soon after. A few days later, Georgia and Pennsylvania followed suit. Given the uncertain context, many American observers and commentators were immediately uncomfortable or skeptical of these trends.

For context, using publicly available data from the New York Times, here is a visualization of the number of votes by candidate in Michigan from the beginning of election night to 7pm Eastern Standard Time (EST) on November 4th, 2020:

As this graph shows, Joe Biden overtook President Trump’s lead through a small number of vote updates which broke overwhelmingly for Biden in Michigan in the early hours of the morning of November 4th.

The situation in Wisconsin is even more stark: a single update to the vote count brought Biden from trailing by over 100,000 votes into the lead. Here is the comparable graph, over the same time range, for Wisconsin, with the x-axis (time) expressed in Central Standard Time (CST):

Various versions of these graphs spurred online discourse. While some commentators provided relatively partisan analysis, others merely expressed surprise at the near-vertical leaps in some of these vote updates. Is it likely this phenomenon would arise organically? In an attempt to address this question, this report assesses how extreme and unusual these spikes are with respect to both other vote updates in the states of Michigan, Wisconsin, and Georgia, as well as those around the nation.

Through several investigative mechanisms, we find these four vote updates to be extraordinarily anomalous. While these alone do not prove the existence of fraud or systemic issue, it invites further scrutiny.

The Concept, the Intuition, and the Measurement

Data analysis relies on recognizing and evaluating patterns in data. When we find anomalous data, that is often an indication of underlying differences. This is why in this report we focus on these four vote updates.

There are also a number of general intuitions upon which we draw to direct our research. In general, the larger the sample size, the smaller we expect the deviation from the population average to be. While anomalous vote ratios may occur, the statistical chance of anomalous margins goes down as the size of the sample (or vote update) goes up.

The basic intuition is: big margins are one thing, and so are super-skewed results, but it’s weird to have them both at the same time, as they generally become inversely related as either value increases.

We will demonstrate below that the data overwhelmingly follow this intuition, but that four key vote updates identified by this report cut against this intuition.

In particular, we will show the existence of a very strong inverse relationship within vote updates, across all states and times, between the difference of votes for Joe Biden and Donald Trump (often referred to as the “Biden-Trump margin”) and the the ratio of Joe Biden’s votes to Donald Trump’s votes (often referred to as the “Biden:Trump ratio”). As described in more detail in the next section, we take the natural logarithm of the ratios so that they are symmetric, i.e. so that we are not treating the two candidates differently when graphing and analyzing. These values are often referred to as “Biden:Trump log-ratio.” Since the logarithm is an order-preserving transformation — i.e. if x is bigger than y, then log(x) will be bigger than log(y), and vice versa — we sometimes use them interchangeably when precision is not required.

At any geographical level, we can test the assumption of an inverse relationship between vote update size and the extremity of the ratio between the candidates’ votes, and, as we will see here, the relationship is extremely strong. Across states red and blue, where turnout is high and low, there is an obvious inverse relationship between the two.

Measuring This Relationship Between The Candidate’s Margin and their Ratio

Let us now attempt to quantify the nature of the inverse relationship in the context of a particular state. First we take our data set of running vote totals[2] for each state, and, for each state, calculate the vote differential for each candidate between updates. This produces a sequence of vote differences, the sum of which, within any given state, is the total.

To begin, we consider each sequential update in the state of Michigan where the vote totals for both Trump and Biden are greater than zero[3]. For each of these, we compute two values:

The difference between the number of votes for Biden and the number of votes for Trump — the “margin”
The logarithm[4] of the ratio between the number of votes for Biden and the number of votes for Trump — the “log-ratio”

Note: both of these metrics are symmetrical. If we let f1 be the first metric and f2 the second, the reader will note that, for any positive numbers (X, Y):

And that:

In other words, given X for Biden and Y for Trump, either metric will produce a score which is the opposite of what it would produce if the update instead had Y votes for Biden and X for Trump. This property is extremely useful, and will come in handy during the statistical analysis.

Readers might ask: Why are you measuring the ratio? Why not measure the difference between the vote proportions (or, equivalently, their percentages). The answer to this lies in what we are looking for, i.e. evidence of fraud or foul play which manifests in extremely unusual outcomes. In particular, ratios are almost never used in expressing vote counts (one typically hears of percentages or, when a race is close, numbers) and so anyone committing fraud and looking to “cover their tracks” is more likely to be “gaming” the metrics they’re used to, and much more likely to leave tells in metrics they’re not considering.

This obscures critical differences between the two statistics.

Ratios demonstrate an important property: the farther ahead a candidate is, the harder it is to move the next 1 percent ahead. They reflect the relative difficulty of each marginal vote as the pool of remaining votes decreases.As a candidate approaches 0% or 100% of the vote, the rates at which the ratio of that candidate’s votes to the other candidate’s votes converge to zero or infinity are very different.
Ratios allow us to spot a potential sign of fraud: unusually low ratios between the losing (major) candidate and other, less well-known candidates. Because those who watch and participate in elections tend not to think in these terms, if there is fraud, they’re much less likely to have covered their tracks in this respect. A tin-pot-dictator style election where the favored candidate gets 99% of the vote is obviously suspect, but less attention is often paid to details like whether the ratio between the most popular losing candidate and long-shot third-party candidates actually makes sense[5]. Looking at metrics which are less popular in practical use will be tremendously helpful here, as we will see.

To illustrate this, let us consider a sequence of two hypothetical elections between Tom and Harry. Imagine that the first time around, Tom wins with 55% of the vote to Harry’s 45%. Four year later, Harry is the challenger and Tom improves his margin to 60% of the vote. There are many ways that this can happen; winning over new voters, Harry’s previous supporters no longer voting, Harry’s supporters switching to Tom, or some combination of any of the above. Let’s consider merely the last case for the moment. For Tom to get from 55% to 60%, he must convert one out of every nine, or just over 11%, of Harry’s supporters. This may not be easy, but is hardly outside the realm of possibility.

Now consider another hypothetical election in a heavily partisan electorate, between Alice and Bob. In the first election, Alice gets 90% and Bob gets 10%. In order for Alice to achieve the same absolute percentage increase as Tom, i.e. 5%, she must convert 5% among a population of 10%. In other words, she must convert one out of every two supporters of Bob. For reasons outside the scope of this paper, this may not be 4.5 times as difficult as a candidate getting from 55% to 60% of the total vote, but it is without question much harder. A useful example of this is this is San Francisco, CA, which, despite being one of the bluest cities in America numerically and culturally, is one where Democratic Presidential candidates consistently get about 90% of the vote but never seem to crack 95%. There are Republicans in San Francisco, however few of them, and converting half of them is a tall order. This makes ratios a useful tool in our arsenal for answering questions of the form “how much is too much”?. This allows us to assess the data in a way which we believe is qualitatively different — and qualitatively superior — to the common forms of assessment used by average individuals and the news media.

This election represents an extraordinary and unique opportunity for election integrity analysts and the application of statistical fraud detection research, as it is likely the first national election in American history, at the very least, where the general public has had access to time-series election data. Even well-respected academic papers which study election fraud in other countries[6] seem to mostly study after-the-fact information about final tallies; analysis is done on statistics about voter turnout, digit frequencies, and other information which is available in after-the-fact official numbers. After all, if reports of widespread fraud and corruption ordered from the top in elections in, e.g., Russia, Uganda, Ukraine, Iran, etc., are to be believed, then those governments, which tend to have much more control over what can and cannot be published than our government, are unlikely to want to increase the number of dimensions along which their claim to legitimacy can be audited.

A Look at Michigan

Let us now calculate these two values for each vote update in Michigan where both Biden and Trump have positive values. If it follows the intuition that there as an inverse relationship between the margins of an update and its ratio, we should expect to see a large cluster of data with a few points above, below, to the left and right, and virtually no points in either the top right (which would represent a simultaneously extreme Biden-Trump margin and Biden:Trump ratio) or the bottom left (which is analogous but favorable to Trump).

Here is that distribution, presented as a scatter plot, with the numerical margins as the X-axis and the log-ratios as the Y-axis.

As we can see, most observations follow the basic contour of our hypothesis, i.e. the more extreme an update is in one respect, the less extreme it is in another.

For example, the update at (-3,622, -6.449), has a fairly extreme ratio of Biden:Trump votes — about 1:632 — but is not very large, producing only a margin of -3,622 votes for Biden, which, as we can see, is not terribly extreme in the context of this distribution. Similarly, the point all the way to the left, (-66,456, -0.816), is one where Biden’s margin is a significant -66,546, but where the ratio, of about 1:2.26, is not particularly unusual for a vote update which favors Trump.

We can see this pattern as well in almost every Biden-favoring update as well. For example, the update with the 3rd greatest margin for Biden, at (34,450, 0.296), is 134,326 Biden votes to 99,867 Trump votes, and only has a Biden:Trump ratio of 1.34:1. And the update with the 3rd greatest Biden:Trump ratio, at (6,091, 2.184), in which Biden received 6,863 votes and Trump received 773 votes, has a fairly extreme ratio of 8.884 but only nets Biden 6,091 votes, a relatively small amount compared to what we will examine next.

Two points stand out.

Let us first consider the less extreme of these, i.e. the point at (49,779, 2.447). This point, representing a vote update which went 54,497 for Biden and 4,718 for Trump and arrived at 3:50am ET on November 4th 2020, is both the second-largest vote margin of Biden’s, at 49,779, and also has the second largest Biden:Trump ratio at 11.55:1. As we can see and as was described above, the update with the next largest margin was an update with merely 7,776 votes, while this update had over 7 times as many votes and broke more heavily for Biden.

The oddness of the update described above pales in comparison to that of the update in the top right corner, however. That update, at (135,290, 3.164), represents the vote update described at the top of this report, and is responsible for the extremely noticeable spike which nearly eliminated Trump’s lead in one shot. It arrived at 6:31am ET on November 4th, and went 141,258 for Biden to 5,968 for Trump — representing both the largest vote margin for Biden of any of the 502 updates we have here, at 135,290, while also representing, by a factor of more than 2, the largest Biden:Trump ratio, at a whopping 23.67:1 (the log of which is 3.16). As we will see when comparing with other states, by our metric this is the single most anomalous point in the nation.

This update is also particularly interesting for another reason: there are 2,546 non-two-party votes, while Donald Trump only has 5,968. Here is a histogram of vote-total-weighted Other:Trump ratios[7]:

As we see, when we weight by the number of votes in any given update, this update is particularly anomalous. The next closest vote-weighted Other:Trump ratio is less than two-thirds of this one, and the median — 137.56 — is smaller by a factor of about 464.5. For such a large batch of votes to be counted while also showing such an exceptionally poor performance of Trump relative to the non-two-party vote is clearly very surprising.

In particular, it calls into serious question the veracity of this vote update, and is perhaps some of the strongest direct evidence of fraud in this entire report. Someone looking to fraudulently improve Joe Biden’s margins relative to Donald Trump is likely to be focused on covering their tracks by keeping Joe Biden’s share of the update at a reasonable value. 95% might seem plausible, but 99.9% at this scale becomes prima facie implausible to any honest observer. One effective way of achieving the desired goal of decreasing Donald Trump’s lead at this point would have been to suppress the Trump vote while artificially inflating the non-two-party vote in an attempt to disguise just how Biden-favoring this update actually was. Indeed, this is precisely the reason this report uses ratios — because they are a metric virtually never used for any practical purpose in discussing election results, someone committing fraud is far less likely to consider how unusual a ratio might look. In particular, because the non-two-party candidates received far less media attention than in the 2016 Presidential election, and the Green Party candidate was even successfully sued off of the ballot in one or more states, it is hard to believe that this vote update only favored Trump over the non-two-party vote by less than a factor of 2.5, when the statewide ratio was over 31[8].

Absent a compelling explanation of why this particular update — at such a crucial time, in a crucial state, which improved Biden’s standing in the state so dramatically — also had non-two-party votes performing so unusually relative to Trump votes, it seems unlikely that this vote update reflects an honest accounting of the legitimate votes.

Subsequent sections of this report quantify how extreme it is in other respects and consider the implications if it had been slightly less extreme.

A Look at Wisconsin

Here is the analogous graph for Wisconsin.

The patterns in this graph are somewhat more bizarre. The updates favoring Trump (i.e. those to the left of zero on the x-axis) exhibit an inverse relationship between the margin of victory in a Trump-favoring update and the ratio between Trump and Biden votes. For example, the update at, (-5,433, -4.564), which is the most extreme in the state in terms of ratio, is from an unusually Trump-favoring batch of ballots which went 5,490 for Trump to 57 for Biden, i.e. a Trump:Biden ratio of about 96:1 for Trump. This number itself is quite large, but, critically, it is not anomalous with respect to the shape of the distribution. The tell-tale sign of oddity here is not extremity with respect to either value, but co-extremity.

Biden’s distribution looks slightly odd here, but there is one point which especially stands out, i.e. the one in the top right, at (118,215, 1.74). This was the vote update which arrived at 3:42am CST on November 4th, and went 143,379 for Biden to 25,163 for Trump[9], giving a margin of 118,215 and a Biden:Trump ratio of about 5.7:1 — about 3 times larger than the update with the next largest margin (which was 39,499). At the same time, only one update — one with a mere 6,435 votes (i.e. about a factor of 18 fewer than the update in question) which went 3,037 for Biden to 495 for Trump — has a larger ratio, at around 6.14:1.

A Look at Georgia:

This one seems only slightly more anomalous than other such graphs, but, as we will see, actually contains two of the nine most anomalous vote updates in our combined distribution of 8.954 vote updates. In particular, the point at (136,155, 1.543), representing a vote update which arrived at 1:34am EST on November 4th, is the update with the largest margin of all of the updates in Georgia — it also has the 10th largest Biden:Trump ratio. There are a few smaller updates with more extreme ratios, but, as we will detail later in this report, this point is in fact unusual.

A Short Survey of Other States

We now turn to other states, particularly those with similar characteristics (e.g. a swing or blue state where one or two urban cores offsets an otherwise very Republican population). These help us establish an initial baseline of what these distributions should look like within any state before we begin comparing updates directly across states.

Pennsylvania:

The inverse relationship is immediately visible here. We have points near the bottom (representing high Trump:Biden vote ratios), a few points far to the left (representing high Trump – Biden values), and a couple (much farther) off to the right, representing a high Biden-Trump margin, but which are not particularly extreme in terms of their Biden:Trump ratio.

Minnesota:

While there is an update which is more extreme in terms of how large the Trump:Biden ratio is, and several updates with extremely large Trump-Biden margins, we see the basic shape remains the same.

New York:

The vote margins for each update are clustered fairly heavily around zero, while the few updates which have exceptionally large margins for either candidate have ratios which are not nearly as extreme as those of many other updates.

Consolidating, Comparing, and Measuring

Having taken a brief tour of states with similar characteristics, i.e. where Joe Biden is currently in the lead and the Democratic vote comes overwhelmingly from a single urban area (or perhaps two, in the case of Pennsylvania), we can see that the Michigan and Wisconsin graphs both look unusual. In order to more rigorously assess the extent to which this is actually anomalous, it is necessary to accommodate the reality that the typical Biden-Trump margin and Biden:Trump ratio will vary substantially between states. If we merely take these values as they are, then most of the differences between, e.g., Alabama and California would likely just be artifacts of the massive discrepancies between how the candidates each performed in these states.

To achieve this, we can use a data transformation process called standardization. This is a process by which, for a series of numerical data, the mean of the data is subtracted from each point, and then the result is divided by the standard deviation. This will produce a series of distributions which permit an apples-to-apples comparison of these values (i.e. per-vote-update Biden-Trump margin and Biden:Trump log-ratio) between states which are both very different in size and lean very differently, politically. Data standardization is a very common technique in machine learning for training models on data sets with very different numerical magnitudes and means[10], as it provides precisely the functionality we need here.

We can thus standardize each individual (margin, log-ratio) point within its state[11], and plot it as we did before. Here is what that graph looks like. The values for Michigan are in red, those for Wisconsin are green, and the values for all other states are blue:

Out of these 8,954 vote updates across the country, we can see how overwhelming the pattern is. In particular, we see that — with a few notable exceptions — as one value grows more extreme in any direction, the other tends to become less extreme.

This brings us to the visually identifiable exceptions.

Directing our attention to the points on the far right end of the distribution, i.e. those which have the most extreme Biden-Trump margin with respect to their state, we immediately see one point from Michigan, which is quite far above where the shape of the plot would otherwise predict it being. This, the point at (15.494, 4.989), is the vote update which arrived at 6:31am EST on November 4th, went 141,257 to Biden and 5,968 to Trump. Recall: this update had both the largest margin (135,290) of any of the 574 updates[12] in Michigan, by about 85,000 votes and a factor of about 2.7 over that of the update with the next-largest update, (5.679, 3.912) — which, critically (and surprisingly, vis a vis what this distribution shows), was both the second largest in terms of Biden-Trump margin and Biden:Trump ratio[13]. It also had the largest Biden:Trump ratio (roughly 23.69:1), by more than a factor of 2 over that of the update with the next-largest Biden:Trump ratio. The visual discrepancy between that update and the overwhelming pattern followed by the other updates is glaring, and we will shortly quantify just how extreme

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One Response

Bill Corden says:

November 30, 2020 at 8:33 pm

All of the leftie news outlets smugly report that recounts have confirmed certain states, they smugly report that judges have kicked out appeals, but not one of them reports on the real anomalies that have surfaced.
If you go down to your bank and withdraw $500, the teller counts it, then you count it and and confirm that there appears to be $500 in the bundle.
Say you spend some of that money and the clerk in the store points out that the two of the twenties you submitted are counterfeit? Taking your money back to the bank just for recounting doesn’t do anything to address the forgeries. The bank has to examine each bill, determine which ones are good and which ones are bad and reimburse you for the difference.

So, with the votes. It’s not enough to do a simple recount, because nine times out of ten it will come back just about the same result.
If, however you cull the questionable votes and examine them for veracity then you’re going to have a completely different tally. Whether President Trump and his lawyers can get their appeals to the levels needed, is an increasingly fading light but he must fight to the last breath while he still has the reins of power.

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