*Norman Fenton and Martin Neil*

UPDATED 19 Nov 2020

There has been much discussion about whether statistical analysis alone can establish if there was fraud in the US election. While the claim that Benford's Law provides proof of fraud is easily dismissed (see below) there are certainly other examples of statistical anomalies and sampling anomalies that cannot be. The findings here are certaintly damning and any self-respecting statistician should be looking at these - it is curious that almost no academic statisticians are. If there is no causal explanation (such as fraud or something else like systemic bugs in the Dominion counting software) for the many observed statistical anomalies, then they would have to suspend their belief in much of classical statistics. <5 Dec update: see this video and my response)https://twitter.com/profnfenton/status/1335359470013206528?s=20 https://twitter.com/profnfenton/status/1335359470013206528?s=20 https://twitter.com/profnfenton/status/1335359470013206528?s=20 Consider, for example, the following data from Justin Hart that tracks votes in Wisconsin and Michigan respectively :

Similarly, in all of the key swing states (as shown below) Trump was well ahead when the voting was (unusually) stopped in the early hours of 4 Nov, with most of the votes counted; the fact that he ended up losing in * each *case means that in each case the remaining votes cannot have come (in a purely statistical sense) from the 'same population'. It is argued that it was indeed a 'different' population comprising primarily postal votes heavily favouring Biden; but the question statisticians need to address is whether there was also a difference in the population of early postal votes (which were already counted) and late postal votes sufficient to explain the reversed outcomes in each case.

Other claims to consider:

- The claim that the first digit total votes for Biden in Chicago districts defy Benford’s Law and hence must be fraudulent. This video by Matt Parker provides a description of Benford’s law along with an explanation of why it is not relevant in this case. In other words, the ‘statistical analysis’ based on Benford’s Law does not establish fraud.
- The claim that large batches of postal votes (in one case a batch
of over 23,000) all of which were votes for Biden must represent fraud because
it is a statistical impossibility otherwise. This claim only works if ‘fraud’ or ‘luck’
were the
possible causal explanation for 23,000 consecutive votes all cast for Biden. If these were the only possible causal explanations then this would certainly prove fraud even if the ballots came from a district where, say 90% of people really are Biden supporters. The probability that all 23,000 votes would be for Biden purely by chance given that each vote has a 90% probability of being for Biden is 0.9 to the power of 23,000. That is a number much smaller than 1 divided by the number of atoms in the observable universe. People saying it is ‘as likely as being struck by lightning’ are massively understating how unlikely it would be. More like being struck by lightning on several consecutive days. However, the argument is flawed if there is another plausible causal explanation for the bag contents other than fraud and luck. For example, it may be possible that these ballots were part of a set that had already been counted and sorted. Or, perhaps, this was a deliberate hoax or set-up. So, the focus needs to be on whether any of these alternative explanations is feasible rather than on the statistical analysis. The statistical analysis only proves that the batch*only*have come from a random set of ballots.*cannot* - The claim that an unusually high pattern of people voting 'Republican but not Trump' compared to 'straight Republican' in districts in key swing states proves fraud. Assuming the data here are accurate, there could still be causal explanations other than fraud, including the possibility that these are the areas where the Republicans are 'never Trumpers'.
- The claim that the sudden large swings to Biden which started happening in key swing states after the counting stopped at 3.00am on election night (as in this analysis by an anonymous data scientist) proves fraud. Assuming the data here are accurate, this does indeed look like convincing evidence of fraud. However, because it is at a State level, there could still be a causal explanation other than fraud. For example, it may be possible that large numbers of ballots that came in late were primarily from Biden-supporting areas.

If there really was fraud, then (ignoring the possibility of automated counting machine fraud) the
simplest and most efficient way of identifying it statistically would be a
variation of what was done by the anonymous data scientist above but ** at a
much more local/granular level** and focused

**. In other words, districts sufficiently small such that there is less chance of a systematic or random interference in the natural process by which ballots are collected (no mail sorting, no sorting at the centre into for/against bundles etc. i.e. the “draws” come naturally as close as possible on a per household basis). The more granular we get, the closer we are at detecting anomalies that are not explainable by anything other than fraud. If there is some model of causal interference, then the normal and hypothesized abnormal process need to be tested against each other i.e. against patterns from previous elections.**

*only on postal ballots*We hypothesise that districts with a total of no more than 5,000 postal votes may be a suitable level of granularity to analyse. In other words at this level of granularity there seems to be no reasonable causal explanation for the distribution of votes in postal ballots counted before and after the 3.00am ‘cut off’ point on election night to be significantly different.

So, let’s consider a hypothetical
example of how we would undertake the necessary analysis if we had the relevant
district level data. Consider, a district with say 4,000 postal ballots. Suppose
that 2000 ballots are counted before the cut-off and candidate *A* has,
say, 55% of these (i.e. 1100). Using a Bayesian analysis**
(which assumes that the ‘true proportion’ of people favouring candidate *A*
before we see any votes cast can be anything between 0 and 100%) observing the
1100 votes out of 2000 means that we can update the ‘true proportion’ of people
favouring *A* as shown in Figure 1. Specifically, the true proportion is
still quite uncertain – but there is
only a 5% chance it will be less than 53.2 and only a 5% chance it will be more
than 56.8.

As also shown in the Figure, we
can use this revised probability distribution of the true proportion of *A*
voters to predict the expected number of ballots for *A* out of 2000
counted after the cut-off – since we are assuming that these come from the same
population of voters. This also enables us to calculate how unlikely any
observed ‘swing’ is.

Figure 1 Bayesian analysis

For example, we can calculate the
probability that there will be a swing of more than 5% between the before and after proportion of
votes for *A* (i.e at least 1200
votes after, meaning 60% after compared to 55% before) as shown in Figure 1.
The probability is extremely low (0.078% is less than 1 in 1000). Even a swing of just 2% in favour of *A*
is unlikely (less than 10% probability).

Now, assuming we have the before
and after ballot count data for a large number of districts in the same state –
say 100, then if there is just one district with a swing larger than 5% to
candidate *A*, this would not be so unusual that it cannot have happened
by chance. There is a probability of about 7% that at least one district would
have such a swing without some other causal explanation.

If, however, 4 out of the 100 had
swings of more than 5% and all were in the direction of candidate *A* then
this would also be so unlikely (about 1 in a million probability) that it would
almost certainly require some other causal explanation. The same applies if
there is even a relatively small number of instances of smaller swings all in
the same direction. For example, if there are 10 swings above 2% which are ** all**
in the direction of candidate

*A*then this would also be so unlikely (about 1 in 100,000 probability) that it would require some other causal explanation.

Hence, the data needed to
establish fraud in the swing states are the postal ballots for a reasonable
number of small areas separated into those counted
before and after the night of the election.If anybody has such data we would be happy to analyse it.

*(pdf version of an earlier version of this article can be found here)*

*A*

- The claims of election fraud
- Anomalies in vote count in the US election
- The smoking gun
- US election sampling irregularities
- Why do Biden's votes not follow Benford's Law
- The election battle is just beginning
- "It Defies Logic": Scientist Finds Telltale Signs Of Election Fraud After Analyzing Mail-In Ballot Data
- Dominion voting system leads to voter fraud
- Analysis of Michigan Votes Reveals Unfortunate Truth of U.S. Voting Systems
- Impact of biased news reporting on the election
- Robert Esptein: How Google shifted votes