Shewhart-type control charts for variation in phase I data analysis

Control charts for variation play a key role in the overall statistical process control (SPC) regime. We study the popular Shewhart-type S2, S and R control charts when the mean and the variance of a normally distributed process are both unknown and are estimated from m independent samples (subgroups) each of size n. This is the Phase I setting. Current uses of these charts do not recognize that in this setting the signalling events are statistically dependent and that m comparisons are made with the same control limits simultaneously. These are important issues because they affect the design and the performance of the control charts. The proposed methodology addresses these issues (which leads to working with the joint distribution of a set of dependent random variables) by calculating the correct control limits, so that the false alarm probability (FAP), defined as the probability of at least one false alarm, is at most equal to some given nominal value FAP0. To aid practical implementation, tables are provided for the charting constants for each Phase I chart, for an FAP0 of 0.01 and 0.05, respectively. An illustrative example is given.