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My question is about the expected value of the variance of a collection of samples. For example, say I have a computer simulation to generate observations from a distribution with a set population mean and variance. If I run this simulation $x$ number of times to produce a collection of sample means and variances, what would be the expected value of the mean and variance of this collection? Would it just be the sample variance and sample mean? It seems counterintuitive to not depend on the number of simulations.

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closed as unclear what you're asking by Michael Chernick, Juho Kokkala, kjetil b halvorsen, Peter Flom Nov 20 '17 at 11:51

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ It is unclear whether by "expected value of the mean and variance of this collection" A) Population parameters are fixed and the expectation is taken over the randomness in the simulation or B) (Bayesian posterior) expectation of the population parameters conditional . The title sounds like you are talking about (A) but how could the expected value of the sample mean be the sample mean when the sample mean is not known? For B you would need to define the population parameters as random variables (i.e. prior distributions) $\endgroup$ – Juho Kokkala Nov 20 '17 at 7:07
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If the population mean and variance exist ($\mu$ and $\sigma^2$), then under random sampling of the population:

  • the expected value of the sample mean is $\mu$ and

  • the expected value of the Bessel-corrected sample variance is $\sigma^2$ (i.e. the version with the $n-1$ denominator), as long as $n$ is at least $2$.

These results don't depend on $n$.

However, if you're asking about the sampling distribution of the sample mean of sample means and the sample means of sample variances (where you take $m$ samples of size $n$) then those are sample averages of sampling statistics and those second-stage averages will of course have their variance affected by sample size.

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