I would like to know whether it is possible to derive a closed-form expression for the sample variance of the $n$-th power of the sample mean $\overline{x}_m$ that was calculated using $m$ sample values $x_i$:
$\mathrm{Var}[\overline{x}_m^n] =\;?$,
where
$\overline{x}_m = \frac{1}{m} \sum_{i=1}^m x_i$.
It is clear that in case of $m=1$, $\overline{x}_1 = x_1$, for which this solution can be used. However, I am interested in a general solution for $m > 1$, preferably for arbitrarily distributed samples, but I expect that to be challenging or even impossible. So, I would also be happy with a solution for normally distributed samples.
Update #1: I think that I have used the term closed-form expression incorrectly and might have been too unprecise in general. Let me clarify: I am basically looking for a formula to calculate a good (preferably unbiased) estimate for $\mathrm{Var}[\overline{x}_m^n]$ based sample values $x_i$.
A comment indicated that for normally distributed $x_i$, $\overline{x}_m$ is still normally distributed and as such, the solution should be trivial. However, I still would like to have a formula to estimate the variance $\mathrm{Var}[\overline{x}_m^n]$based on the given set of samples values $x_i$.
Apologies for the misleading initial question.
