What are the skewness and kurtosis of the sample mean? Consider a sequence of IID random variables $X_1,X_2,X_3,...$ from a common distribution with mean $\mu$, variance $\sigma^2$, skewness $\gamma$ and kurtosis $\kappa$ (all finite).  The mean and variance of the sample mean are well-known:
$$\mathbb{E}(\bar{X}_n) = \mu
\quad \quad \quad \quad \quad 
\mathbb{V}(\bar{X}_n) = \frac{\sigma^2}{n}.$$
What are the corresponding formulae for the skewness and kurtosis of the sample mean?


What are the first four central moments of the sample mean?

Computer algebra systems are particularly adept at this sort of algebra munching. Here I am using the mathStatica package for Mathematica where $s_r = \sum _{i=1}^n X^r$. The first say 7 central moments of the sample mean $\frac{s_1}{n}$ are:

where:
$\mu_r$ denotes the $r^\text{th}$ central moment of the population.
The calculation takes just .1 of a second for all of them to be calculated live.
