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?

  • $\begingroup$ newonlinecourses.science.psu.edu/stat506/node/16 $\endgroup$ – whuber Sep 24 '19 at 11:30
  • $\begingroup$ 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. $\endgroup$ – d0d0 Sep 24 '19 at 14:23
  • $\begingroup$ Your "collection" is the population. This is a direct application of the HT estimator to that population with your given inclusion probabilities. $\endgroup$ – whuber Sep 24 '19 at 17:36

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