Control Charts for Simultaneous Monitoring of Unknown Mean and Variance of Normally Distributed Processes

Phase II Shewhart-type control charts are considered for simultaneous monitoring of the mean and variance of a normally distributed process when both of these parameters are unknown and are estimated from an in-control (IC) Phase I reference sample. The charts are based on single plotting statistics obtained by suitably modifying two popular existing charts in the literature (the max chart and the distance chart) for the known parameter case. Control limits for the proposed charts are tabulated for practical implementation. Follow-up procedures are considered for both charts for post-signal detection of the type of shift. A comprehensive simulation study is carried out to investigate the IC and out-of-control (OOC) performance properties of the charts in terms of the average, the standard deviation, the median, and some percentiles of the run-length distribution. The modified max chart is found to be preferable for detecting a larger shift in the mean accompanied by a smaller shift in the variance. In all other situations, the modified distance chart displays better performance. A numerical example is given for illustration. Summary and conclusions are offered.