Ex:            Using a spreadsheet or numerical program with observations drawn from a standard gaussian (normal) distribution, (i.e., μ = 0 and σ2 = 1), calculate the estimated standard deviation, , 15 times based on ranges of 12 samples of 20 observations each. Also, calculate the sample mean and sample standard deviation for these 15 estimated standard deviations.

Sol'n:       We use a Matlab® program, (see StatsControlChartXbarEx1.m file), to perform calculations based on the following equations from [1]:

                            

where   k    ≡  total number of samples being considered

                     i     ≡  index designating sample

                     xij  ≡  observations available for sample i

                     d2  ≡  number of observations in each sample

                       ≡  estimated value of σ

Results for one run of the program are as follows:

        = 0.969    = 1.032      = 1.057    = 0.974   = 1.072

        = 1.084    = 1.044      = 1.102    = 1.047    = 0.982

        = 1.025  = 1.026    = 1.028  = 1.016  = 0.994

The calculated mean of the calculated 's is within a few percent of the true σ, and the standard deviation of the calculated 's is only a few percent of the true σ:

Ref:    [1] Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and Keying Ye, Probability and Statistics for Engineers and Scientists, 8th Ed., Upper Saddle River, NJ: Prentice Hall, 2007.