We're estimating mean, variance, proportion or compare samples with them. I understand for mean and variance. But why is it required for proportions? 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?

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)
