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?

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.
