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?

## Best Answer

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).