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Based on information I have read and from this website, sampling distributions do exist for statistic variants of measurements other than the mean. Sample ranges, maxima, minima, variance and proportions also have corresponding sampling distribution.

But what I have noticed while studying parameter estimation is that estimates only ever come commonly in two different basis. It is either based on the sampling distribution of the sample mean or the sample proportion. The variance comes only as sort of a supplement, like for example how the standard deviation of the corresponding sampling distribution of the sample statistic is multiplied to a test statistic to create a error of estimate.

My question is why is this so? It feels like the two quantities, mean and proportion are under a collective umbrella. I don't know how to explain it that well but it feels like both are about position which is why in estimates its either about the mean or the proportion. The variance is about the spread of that given position.

Why is that so? By the way, I also don't like how common statistics reference do not really take the time to emphasize that the mean and proportion have this sort of main spotlight. Do they have a collective term? Also off topic question, do parameter and statistic have a collective term?

Though correct me if I am wrong if there is actually a variance equivalent of the estimators, such as a point estimate and interval estimate where the population and sample variance would be used. To show that it is not only the mean and proportion that can be used as an estimator. But that would create complications in solving for the critical error. How does that work?

So mainly, this is about why mean and proportion are mainly used as estimators but also a question of whether any other quantity like the variance can be used as an estimator and also having equivalents for point estimate, interval estimate, margin of error. A double statistic estimation could also possibly existing. Just like for sample mean having difference of sample means and sample proportion having difference of sample proportions. Though I do not think a difference of variances exist. I only hear of ratio of variances.

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    $\begingroup$ I already explained that a sample proportion is a form of sample mean in my answer to your earlier question. I believe a few simple searches on our site will show you plenty of examples of other statistics being used as estimators (e.g. try variance estimator as a search term); similarly searches will turn up examples of confidence intervals for variance (and other parameters). $\endgroup$
    – Glen_b
    Aug 4, 2020 at 11:55
  • $\begingroup$ As I understand, you set the context to inferring information from a sub-sample to the whole population: You surely can also model variance! Assume a certain mean and a certain distribution (e.g. normal) and model different variances to see which variance best fits your sub-sample. Also the use of differences in variance is more common than you might think: E.g. a t-test compares intra vs inter variance of 2 samples to evaluate a difference in mean. $\endgroup$
    – KaPy3141
    Aug 4, 2020 at 12:50

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Without some context it is difficult to know what kind of answer you need. Your impression might come just from the dominance of means and proportions in elementary applications (and courses ...) There are many other quantities being estimated routinely, such as variances. Some other examples you can use as search terms on this site:

  • correlation, autocorrelation, cross correlation
  • regression coefficients (linear regression)
  • odds or log odds (used for example with logistic regression)
  • rates, rate ratios (RR) (used for example with Poisson regression)
  • median, quantile, ...

which are all in heavy use, in many disciplines. It would be easy to make the list longer. But introductory posts have to start someplace.

Another point: Many (not all) of this statistics can be seen as means of some constructed random variable, so what you learn about inference means will help you later with many other statistics.

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    $\begingroup$ Yes you hit the nail. Those quantities are dominant but as I have learned from other questions on the site, point and interval estimates are not exclusive to means and proportions. Variances can be used for example. It is just that they are not that commonly discussed in courses. $\endgroup$
    – AndroidV11
    Aug 8, 2020 at 12:15

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