Statistics Associated with Normal Samples  • 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.   "Statistics Associated with Normal Samples" from The Wolfram Demonstrations Project http://demonstrations.wolfram.com/StatisticsAssociatedWithNormalSamples/ Contributed by: Pierre-Emmanuel Robert |
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