The Exact Run Length Distribution and Design of the S2 Chart When the in-Control Variance Is Estimated

When monitoring the process variability, it is a common practice that a Phase I data set is used to estimate the unknown in-control process standard deviation σ0 or variance [Formula: see text] to set up the control limits, then monitoring proceeds. Once the process is considered to be in-control, the estimated control limits are assumed as fixed. This practice ignores the effect of estimating the unknown in-control process variance [Formula: see text]. In this paper, we derive the exact run length distribution of the S2 control chart when the in-control process variance [Formula: see text] is estimated and find that m = 200 or more Phase I samples are needed to neglect the effect of using estimated control limits. New control limits when m is small are also derived.