# Expected standard deviation of probabilistic sample

Let's say I have a collection of $$n$$ individuals, each individual is associated with a single value $$x_i$$, with $$i = 1, 2, 3, ... n$$. I now gather a sample of $$s$$ individuals from this collection, without replacement. $$s > 1$$. Each individual has a certain probability $$p_i$$ of being included in my sample, and I know this probability. My sample of $$s$$ individuals will have a certain standard deviation with respect to $$x$$.

My question: is there a way to calculate the expected standard deviation of $$x$$ of my sample, given: all individuals' values $$x$$, all individual's probabilities $$p$$, and my sample size $$s$$? If I were to randomly draw my sample infinitely many times, what would the average sample standard deviation be?

• newonlinecourses.science.psu.edu/stat506/node/16 – whuber Sep 24 '19 at 11:30
• As far as I understand, the notes on the Horvitz-Thompson estimator (and Hansen, Hurwitz for cases with replacement) give me the variance of my total estimate/mean estimate, over samples. I am looking for the expected variance within my samples. – d0d0 Sep 24 '19 at 14:23
• Your "collection" is the population. This is a direct application of the HT estimator to that population with your given inclusion probabilities. – whuber Sep 24 '19 at 17:36