Statistics: Proof of Malfeasance in Reporting of Election Totals?

From Dr. Roy Spencer’s Weblog

November ninth, 2020 by Roy W. Spencer, Ph. D.

You might need seen experiences within the final a number of days concerning proof of fraud in poll totals reported within the presidential election. There’s a statistical relationship generally known as “Benford’s Regulation” which states that for a lot of real-world distributions of numbers, the frequency distribution of the primary digit of these numbers follows an everyday sample. It has been utilized by the IRS and monetary establishments to detect fraud.

It ought to be emphasised that such statistical evaluation can not show fraud. However given cautious evaluation together with the likelihood of getting outcomes considerably completely different from what’s theoretically-expected, I feel it’s a great tool. Its utility is very elevated if there’s little or no proof of fraud for one candidate, however sturdy proof of fraud from one other candidate, throughout a number of cities or a number of states.

From Wikipedia:

“Benford’s regulation, additionally known as the Newcomb-Benford regulation, the regulation of anomalous numbers, or the first-digit regulation, is an commentary concerning the frequency distribution of main digits in lots of real-life units of numerical information. The regulation states that in lots of naturally occurring collections of numbers, the main digit is more likely to be small. For instance, in units that obey the regulation, the #1 seems because the main vital digit about 30% of the time, whereas 9 seems because the main vital digit lower than 5% of the time. If the digits have been distributed uniformly, they’d every happen about 11.1% of the time. Benford’s regulation additionally makes predictions concerning the distribution of second digits, third digits, digit combos, and so forth.”

For instance, right here’s one broadly circulating plot (from Github) of outcomes from Milwaukee’s precincts, displaying the Benford-type plots for Trump versus Biden vote totals.

Fig. 1. Benford-type evaluation of Milwaukee precinct voting information, displaying a big departure of the voting information (blue bars) from the anticipated relationship (pink line) for Biden votes, however settlement for the Trump votes. That is for 475 voting precincts. (This isn’t my evaluation, and I would not have entry to the underlying information to examine it).

The departure from statistical expectations within the Biden vote counts is what is predicted when some semi-arbitrary numbers, presumably sufficiently small to not be simply observed, are added to a few of the precinct totals. (I verified this with simulations utilizing 100,000 random however log-normally distributed numbers, the place I then added 1,2,Three, and so forth. votes to particular person precinct totals). The frequency of low digit values are lowered, whereas the frequency of the upper digit values are raised.

Since I just like the evaluation of enormous quantities of information, I believed I might look into this subject with some voting information. Sadly, I can not discover any precinct-level information for the overall election. So, I as an alternative checked out some 2020 presidential major information, since these are posted at state authorities web sites. Up to now I’ve solely seemed on the information from Philadelphia, which has a LOT (6,812) of precincts (really, “wards” and “divisions” inside these wards). I didn’t observe the first election outcomes from Philadelphia, and I’ve no preconceived notions of what the outcomes would possibly appear like; these have been simply the primary information I discovered on the internet.

Outcomes for the Presidential Main in Philadelphia

I analyzed the outcomes for four candidates with probably the most major votes in Philadelphia: Biden, Sanders, Trump, and Gabbard (information obtainable right here).

Benford’s Regulation solely applies properly to information that that covers no less than 2-Three orders of magnitude (say, from zero to within the a whole lot or 1000’s). Within the case of a candidate who acquired only a few votes, an adjustment to Benford’s relationship is required.

Essentially the most logical method to do that (for me) was to generate an artificial set of 100,000 random, however log-normally distributed numbers starting from zero and up, however adjusted till the imply and customary deviation of the info matched the voting information for every candidate individually. (The significance of utilizing a log-normal distribution was instructed to me by a statistician, Mathew Crawford, who works on this space). Then, you are able to do the Benford evaluation (frequency of the first digits of these numbers) to see what’s theoretically-expected, after which evaluate to the precise voting information.

Donald Trump Outcomes

First, let’s take a look at the evaluation for Donald Trump throughout the 2020 presidential major in Philadelphia (Fig. 2). Word that the Trump votes agree very properly with the theoretically-expected frequencies (purple line). The classical Benford Regulation values (inexperienced line) are fairly completely different as a result of the vary of votes for Trump solely went as much as 124 votes, with a mean of solely Three.1 votes for Trump per precinct.

So, within the case of Donald Trump major votes in Philadelphia, the outcomes are extraordinarily shut to what’s anticipated for log-normally distributed vote totals.

Fig. 2. Benford-type evaluation of the variety of Trump votes throughout 6,812 Philadelphia precincts. The classical Benford Regulation anticipated distribution of the first digits within the vote complete is in inexperienced. The adjusted Benford Regulation outcomes based mostly upon 100,000 random however log-normally distributed vote values having the identical imply and customary deviation because the vote information in in purple. The precise outcomes from the vote information are in black.

Tulsi Gabbard Outcomes

Subsequent, let’s take a look at what occurs when even fewer votes are solid for a candidate, on this case Tulsi Gabbard (Fig. Three). On this case the variety of votes was so small that I couldn’t even get the artificial log-normal distribution to match the noticed precinct imply (zero.65 votes) and customary deviation (1.29 votes). So, I would not have excessive confidence that the purple line is an effective expectation of the Gabbard outcomes. (This, in fact, won’t be an issue with main candidates).

Fig. Three. As in Fig. 2, however for Tulsi Gabbard.

Joe Biden Outcomes

The outcomes for Joe Biden within the Philadelphia major vote present some proof for a departure of the reported votes (black line) from principle (purple line) within the course of inflated votes, however I would want to launch into an evaluation of the arrogance limits; it could possibly be the noticed departure is inside what is predicted given random variations on this variety of information (N=6,812).

Fig. four. As in Fig. 2, however for Joe Biden.

Bernie Sanders Outcomes

Essentially the most fascinating outcomes are for Bernie Sanders (Fig. 5.), the place we see the most important departure of the voting information (black line) from theoretical expectations (purple line). However as an alternative of lowered frequency of low digits, and elevated frequency of upper digits, we see simply the alternative.

Is that this proof of fraud within the type of votes subtracted from Sanders’ totals? I don’t know… I’m simply presenting the outcomes.

Fig. 5. As in Fig 2, however for Bernie Sanders.


It seems that a Benford’s Regulation- sort of study could be helpful for locating proof of fraudulently inflated (or perhaps lowered?) voter totals. Cautious confidence stage calculations would must be carried out, nevertheless, so one may say whether or not the departures from what’s theoretically anticipated are bigger than, say, 95% or 99% of what could be anticipated from simply random variations within the reported totals.

I need to emphasize that my conclusions are based mostly upon evaluation of those information over solely a single weekend. There are individuals who do that stuff for a dwelling. I’d be glad to be corrected on any factors I’ve made. A part of my cause for this publish is to introduce folks to what’s concerned in these calculations, after understanding it myself, since it’s now a part of the general public debate over the 2020 presidential election outcomes.

[CR note here is the actual title of Dr Spencer’s article. I modified it to reduce social media censorship.]

Benford’s Regulation: Proof of Fraud in Reporting of Voter Precinct Totals?

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