The Annals of Mathematical Statistics

The Distribution of the Mean

E. L. Welker

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Both population and sample mean distributions can be represented or approximated by Pearson curves if the first four moments of the population are finite. Using the $\alpha^2_3, \delta$ chart of Craig [2] to determine the Pearson curve type for the population, an analogous $\bar \alpha^2_3, \delta$ chart is derived for the distribution of the mean. This defines a one to one transformation of $\alpha^2_3, \delta$ into $\bar \alpha^2_3, \bar \delta$. The properties of this transformation are used to discuss the approach to normality of the distribution of the mean as dictated by the central limit theorem. This is facilitated by superposing on the $\alpha^2_3, \delta$ chart the $\bar \alpha^2_3, \bar \delta$ charts for samples of 2, 5, and 10.

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Ann. Math. Statist., Volume 18, Number 1 (1947), 111-117.

First available in Project Euclid: 28 April 2007

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Welker, E. L. The Distribution of the Mean. Ann. Math. Statist. 18 (1947), no. 1, 111--117. doi:10.1214/aoms/1177730498.

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