Statistics Associated with Normal Samples

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Suppose is a random sample of size from a normal distribution with mean and variance one. This Demonstration shows the sampling distributions of four statistics computed from this sample (blue box) along with the corresponding empirical distribution of the statistic over 2000 random samples. and denote the mean and variance, respectively, of the distribution over which they appear.

Contributed by: Pierre-Emmanuel Robert (March 2011)
Open content licensed under CC BY-NC-SA



• The top left‐plot, included for reference, gives the distribution of the sample mean , with mean and variance for the sampling distribution (blue box). Note that when , the sampling distribution is the same as the parent distribution.

• The top‐right and lower‐left plots show the distributions of the sum of squares and the sum of squared deviations from , that is, , respectively.

• The distribution of the sum of squared deviations from the sample mean is given in the lower‐right plot. The number of degrees of freedom is decreased to , accounting for the fact that the population mean is estimated from the sample mean .

• Finally, if you set the population mean equal to 1, the distribution of switches to a noncentral chi-square distribution, with parameters and (simplified compared to the general case because ). More information: I. Miller and M. Miller, John E. Freund's Mathematical Statistics with Applications, 7th ed., Upper Saddle River, NJ: Prentice Hall, 2004.

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