Polls and opinion
surveys often predict results that never happen. Is there a scientific reason that
can explain it? I think so. The problem could be that the mathematical theories
behind the polls are misapplied.

A branch of
statistics is called

**. It was invented to solve the problem of estimating whether the products of a factory are well made or defective, without having to analyze them one by one, which would be too costly.***sample theory*
Let us say,
for example, that a factory produces

**. In theory they should be checked one by one, but since that is impossible, only one part is analyzed. Which part? This is what sample theory tries to solve.***one million screws a day*
Suppose we
analyze just 2000 screws, and find that one of them is defective (0.05%). Can
we extend this result to the million screws and assert that in that population
there will be approximately

**?***500 defective screws*
There is a
theorem of sample theory that computes the confidence we can have in the
assertion that the result of the sample applies to the whole of the population.
Interestingly,

**, with a sample of 2000 “individuals,”***if certain conditions are met***, we can have 95% confidence that the results of the analysis can be extended to the population. In other words, if we analyze 2000 screws, we can have 95% confidence that the result will apply to the entire set of screws, regardless of whether there are one hundred thousand, one million or ten million screws.***regardless of the population size*
Electoral
polls often apply the theorems of sample theory without due consideration. If
we look at the technical data that come with these surveys, we will see that
they often say things like these:

*Size of the population surveyed: 2000 people.*

*Confidence coefficient: 95%.*
But let us
look at the sentence highlighted in red two paragraphs above. What are the
conditions that must be met in order to apply the theorem? Essentially there
are two:

•
The

**must be***population***.***uniform*
•
The

**must be***sample***.***meaningful*
That the
population is uniform means that all the screws must be equivalent in
principle, that

**; such as large screws with small screws.***different sets are not mixed*
That the
sample must be meaningful means that, before extracting the sample, we must

**; otherwise we could take a sample formed exclusively by screws produced by a concrete machine that has a problem, or by a perfect machine, while none of those produced by other machines would be analyzed. In such a case, the results of the analysis could not be extended to the total population with the same confidence.***mix well the million screws*
What
happens when the theorem is applied to a human population to predict the
outcome of an election?

- The most serious problem is that
. We know very well that the votes of some people are worth much more than those of others. In the U.S. elections, for instance, the constituency is the state. Although states with a large population, such as New York or California, may elect more representatives, each candidate requires more votes to be elected than in states with less population, such as Oklahoma.*the population is not uniform* - Whether
depends on the survey being well-designed. For instance, the respondents to a poll should be chosen from all states in proportion to their populations. But that means that, in a sample of 2000 people, there will be very few from Oklahoma. Can the result of the election in that state be predicted with 95% confidence level from such a small sample? The simple and straightforward answer is that it can not.*the sample is significant* - There is an additional problem:
. When a screw is analyzed,*people are not screws*; we can trust that the properties we detect are real, unless we are using defective instruments to measure them. Instead,*it cannot lie*, or they can refuse to tell whom they are going to vote for. Pollsters take this into account, and apply corrections to estimate the possible vote of those who do not want to give their opinion. But can it be held that the degree of confidence is still the same as stated by the theorem? The simple answer is again negative.*people can lie*

In
conclusion: the technical data usually coming with polls and surveys are
fictitious, based on a poor application of the theorems. The confidence coefficient
they provide is enormously exaggerated. Although they say that its value is 95%,
it can be not greater than 50. Can we wonder that the surveys are frequently wrong?
The odd thing is, sometimes they hit the mark.

**Manuel Alfonseca**

**Happy Christmas and New Year**

**We'll meet again in January**

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