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?