Run sum X¯ control chart with estimated process parameters

The run sum (Formula presented.) control chart is usually investigated under the assumption of known process parameters. In practice, process parameters are rarely known and they need to be estimated from an in-control Phase I dataset. However, different practitioners use different numbers of Phase I samples to estimate the process parameters. As a result, the commonly used performance measure, ie, the average run length becomes a random variable. In this study, we present a run sum (Formula presented.) control chart with estimated process parameters and use the standard deviation of the average run length to evaluate the average of the average run length performance of the run sum (Formula presented.) chart when process parameters are estimated. Based on the standard deviation of the average run length criterion, the number of Phase I samples required by the estimated process parameter–based run sum (Formula presented.) chart to have an average of the average run length performance close to that obtained under the assumption of known process parameters is recommended.