Just as in 2016, the 2020 polls underestimated President Trumpâ€™s support. While polling has become more difficult in the age of cell phones and caller-ID, it is mathematically impossible not to conclude that these polls were either deliberately inaccurate or conducted by incompetents.

Polling is based on the mathematical disciplines of statistics and probability. For example, letâ€™s say you flip a coin four times. Although, there is a 50% chance of getting heads per flip, the actual chance that one gets exactly two heads with four flips is only 38%. There is a 12% chance of getting all heads or all tails and a 50% chance of getting three tails or three heads. But the more you flip the coin, the greater the chance that the number of heads will be closer to 50%.

Furthermore, the chance of the number of heads deviating from 50% greatly decreases. For example, if you flip a coin 100 times, there is a 95% chance the number of heads will be between 44% and 56%. But if you flip a coin 1,000 times, there is a 95% chance that the number of heads will be between 47% and 53%.

With more flips, the distribution of the number of heads resembles what the mathematicians call a â€śbell-shaped curveâ€ť that centers around 50%. This not only applies to something as simple as flipping a coin, but to more complex situations such as what team is going to win a sporting event (which is why the bookies always win) and who is going to win an election. When a lot of data is available, a bell-shaped curve is generated.

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This concept is illustrated in the accompanying graphic. But what is fascinating about this graphic is that mathematicians can also tell you the chance that your data may be wrong based on the concept of â€śstandard deviation,â€ť which is represented but the Greek letter sigma on the horizontal line.

While the mathematics behind this is somewhat tedious, the practical application if quite simple. Letâ€™s go back to flipping a coin 1000 times. Using specific formulas, mathematicians can predict that there is a 68% chance (one standard deviation) the number of heads will be between 48.5% to 51.5%. There is a 95% chance (two standard deviations) the number of heads will be between 47% and 53%.

But what is the chance of flipping a coin 1,000 times and getting heads 44% of the time or less â€“ in other words, four standard deviations or more? As you can see from the graphic, this happens only 0.0063% (100% â€“ 99.9937%) or about 1 in 16,000! In other words, if you flipped a coin 1,000 times and only got heads 44% of the time or less â€“ the overwhelming odds are that you are dealing with a coin that was weighted to come up tails more often that heads rather than this happened by random chance.

Now if you have somehow managed to make it this far without dozing off, letâ€™s apply this to polling. Polls do not use the term standard deviation; they use the term â€śmargin of error.â€ť By convention, the margin of error is defined as two standard deviations, or 95%. Thus, if you read a poll that says that Candidate A will garner 55% of the vote and the margin of error is 3%, this means that on election day there is a 95% chance that Candidate A will receive between 52% and 58% of the vote.

The final poll from Quinnipiac University stated Joe Biden would receive 50% and President Trump would receive 39% with a margin of error of 2.4%. The actual result was Biden 51% and Trump 48%. Note that Bidenâ€™s 51% is well within the margin of error.

But look at the result for President Trump. The poll was incorrect by 9%. Dividing 9% by 2.4% is equal to 3.75. Since the poll margin of error is equal to two standard deviations, this means the Quinnipiac pollâ€™s estimate of President Trumpâ€™s support was off by 7.5 standard deviations. Again, look at the graphic. It only goes up to six standard deviations and the chance of that happening (I wonâ€™t bore you with the math) is less than 1 out of 25,000,000! This was a flagrantly flawed poll.

Similar results were generated by other â€śreputableâ€ť polls. The WSJ/NBC poll had Trump 42% with a margin of error of 3.1% (chance of being wrong about 1 in 50,000). The Siena College/NYT poll had Trump at 41% with a margin of error of 3.4% (chance of being wrong about 1 in 50,000). Even the so-called right-wing Fox News poll had Trump at 44% with a margin of error of 2.5% (chance of being wrong about 1 in 1,750,000). Yet all of these polls were not only wrong, they were out of the ballpark.

The narrative being pushed by the media and the pollsters is that better methodology is required. But the overwhelming odds are that these polls had a liberal bias and were issued to shape opinion, rather than reflect it. Expect these â€śreputableâ€ť pollsters to repeat the same scam the next election cycle.

*Joe Bentivegna is an ophthalmologist in Rocky Hill.*