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Independence of Sample Mean and StdDev

See the attached .pdf file that shows the derivation of the proof.
The independence of the sample mean and sample variance is an important feature of the X-bar and R chart.  It means that the R chart can contain information about the process that is not also included in the X-bar chart.  In contrast, the XmR chart does not share this feature.  The information in the mR (moving range) chart is completely dependent on the information in the XmR (individuals) chart.