Sigma in the control chart.
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Dear Professor Mean, I ran a control chart in SPSS for individual values, and the control limits don't correspond with what I would expect from the descriptive procedure that I ran first. In particular, the value of sigma in the control chart appears to be an approximation of what I computed earlier. Why would SPSS use a different calculation for sigma?
The use of the word "sigma" in quality control circles is different than how it is used in other circles. So the sigma limits that you get in your control chart are different than what you would get in any other descriptive statistics procedure. In particular, sigma represents neither the sample standard deviation, nor the standard error of the mean.
The reason the sigma in a control chart differs from any of the numbers you might get in a descriptive summary is that:
1. In a control chart, sigma represents short term variation only. If sigma included long term variation, then it would be too big. It would include things like a slow drift or trend that would inflate the limits and make the control chart useless.
2. In a control chart, sigma is corrected for bias. The correction factor is more complicated than the "n-1" factor which makes the variance (but not the standard deviation) unbiased.
3. In this particular control chart, the moving range is used to approximate sigma. The moving range is not used either directly or indirectly in other descriptive summaries. Keep in mind also that a moving range is only meaningful if there is a natural time sequence to your data (as would have to be the case for a control chart). Many of the data sets that might be summarized in a descriptive procedure would not have a natural time sequence.
Further reading
Donald Wheeler has authored or co-authored three excellent books that explain this material far better than I can.