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I have learned how to estimate a mean, variance or proportion from a sample.
and also, how to compare those for samples.

I'm understanding well why we might need to estimate or compare means or variance.

But the need of a specific formula for estimating or comparing proportions troubles me.

Please accept my question even if it looks strange:
Are the calculations for proportions some required statistical formulas, that if they weren't here then statistics wouldn't stand, and some things could never be checked or done,

or are these proportions formulas only some convenient ones that have a family spirit with the means ones, and that only start to apply when you want to compare a $\frac{\text{number of elements having a property in your set}}{\text{cardinal of your set}}$ instead of the value of that element?

It really seems to me that observing values or proportions are quite the same thing, and I want to say :

  • querying a sample about its mean or proportion is asking quite the same thing,
  • but querying a sample about its variance is something really different

I must be missing something important if I tend to disparage the observations of proportions and what I can do with them. Can you give me a hint about what I should figure?

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There is indeed, strictly speaking, not a need for a particular formula for proportion. The proportion of observations that are equal to $x_0$ can be calculated by assigning all observation of $x_0$ a value of $1$ and all other observations a value of $0$, and then calculating the mean of this value. The reason there are distinct formulae for proportions is that the general formulae become simpler when only two possible values are possible, and so it's easier to work with the simpler formulae rather than starting with more complicated formulae and doing unnecessary work to simplify them down each time.

BTW, you seem to have had words in your native language (French, I presume?) that have different senses, and have different translations into English depending on the meaning. For instance, you seem to have translated "la fonction" not as the cognate, and more appropriate, word "function", but as "office" (which as I understand it, has more of a "le bureau" meaning).

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A proportion calculated from a sample does have an associated statistical uncertainty, just as a mean or a variance. So your misunderstanding could be about some of these very important concepts: Sample, population, uncertainty.

  • If your only interest is in your sample, and you have no intentions of using it to extrapolate to some larger population, then there is no statistical uncertainty, and you do not need tests or confidence intervals. This applies the same for means, variances, proportions, ...

  • But, more usually, you want to use the sample to learn about some larger population. Then, exactly as you cannot be sure that the mean of the sample is the same as the mean of the population, you cannot be sure the proportion of A's in the sample is the same as the proportion of A's in the population. And variance would be no different.

Then you say:

But the need of a specific formula for estimating or comparing proportions troubles me

Could you tell us why that troubles you? There will indeed not be one formula universally applicable to proportions. It could depend on how you do the sampling (with or without replacement, ...). And, proportion could mean different things, see What is the difference between "count proportions" and "continuous proportions"? and Beta as distribution of proportions (or as continuous Binomial)

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I'm not entirely sure I understand the question, but maybe this answer will help.

With numerical variables, there are two distinct concepts:

  • Quantifying variation among the replicate measurements with a standard deviation (or quartiles...)
  • Quantifying how precisely the mean of the sample estimates the mean of the population (based on sample size and standard deviation). This can be quantified with a standard error of the mean or (better) a confidence interval, and can be generalized to compare two or more means.

With a categorical variable with two outcomes, there really is no need to quantify variation. You just compute #"successes"/total as a proportion. But you still can (should) quantify how precisely you have determined the population proportion (as a confidence interval) and can use that logic to compare groups.

More simply. With a measured variable, you need three parameters to (begin to) describe the distribution. Usually, these are mean, standard deviation and sample size. With a binomial categorical variable, you really only need two parameters, the proportion "success" and the sample size.

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