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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 $M$ (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)


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

© 1996-9 Eric W. Weisstein