Studies on the effect of parameter estimation on control-chart performance have mostly focused on the marginal (unconditional) run length (RL) distribution and some associated characteristics. However, once process parameters are estimated from an in-control (IC) reference sample, the RL follows its conditional distribution given the parameter estimates. With this in mind, our focus is the conditional RL distribution of the S and S2 charts. First, we concentrate on the IC conditional RL distribution, which is geometric with parameter (probability of success) αTRUE, the unknown attained false-alarm probability, given the estimate of the process standard deviation. We obtain and examine the distribution of αTRUE as a function of the number and the size of the reference samples. We consider a one-sided prediction interval for αTRUE and, for several sample sizes, obtain the minimum number of reference sample observations that guarantees, with a specified probability, that αTRUE will not exceed the nominal value of the false-alarm probability, αNOM, by more than a prespecified percentage. Next, we argue that (and demonstrate why), in the particular case of S and S2 charts, there is no practical need of a similar analysis of the effects of parameter estimation on their out-of-control performance, and draw some conclusions in terms of guidance for the user.
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