Mean Distribution

For an infinite population with Mean $\mu$ , Variance $\sigma^2$ , Skewness $\gamma_1$ , and Kurtosis $\gamma_2$ , the corresponding quantities for the distribution of means are

$\displaystyle \mu_{\bar x}$	$\textstyle =$	$\displaystyle \mu$	(1)
$\displaystyle {\sigma_{\bar x}}^2$	$\textstyle =$	$\displaystyle {\sigma^2\over N}$	(2)
$\displaystyle \gamma_{1,\bar x}$	$\textstyle =$	$\displaystyle {\gamma_1\over \sqrt{N}}$	(3)
$\displaystyle \gamma_{2,\bar x}$	$\textstyle =$	$\displaystyle {\gamma_2\over N}.$	(4)

For a population of

(Kenney and Keeping 1962, p. 181),

$\displaystyle \mu_{\bar x}^{(M)}$	$\textstyle =$	$\displaystyle \mu$	(5)
$\displaystyle {\sigma^2}^{(M)}$	$\textstyle =$	$\displaystyle {\sigma^2\over N} {M-N\over M-1}.$	(6)

References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.