The Difficulty in Designing Shewhart X and X Control Charts with Estimated Parameters

The performance of the Shewhart X̄ control chart with estimated in-control parameters has been discussed a number of times in the literature. Previous studies showed that at least 400/(n – 1) phase I samples, where n > 1 is the sample size, are required so that the chart performs on average as if the in-control process parameter values were known. This recommendation was based on the in-control expected average run length (ARL) performance. The reliance on the expected ARL metric, however, averages across the practitioner-to-practitioner variability. This variability occurs due to the different historical data sets practitioners use, which results in varying parameter estimates, control limits, and in-control ARL values. In our article, we show that taking this type of variability into consideration leads to far larger amounts of phase I data than what was previously recommended. This is to ensure low levels of variation in the in-control ARL values among practitioners. The standard deviation of the ARL (SDARL) metric is used to evaluate performance for various amounts of phase I data. We show that no realistic phase I sample size is sufficient to have confidence that the attained in-control ARL is close to the desired value. We additionally investigate the effect of different process standard deviation estimators on the X̄-chart performance, showing that it is best to use a biased estimator. We also study the design of the X-chart for the case n = 1, drawing similar conclusions regarding the amount of phase I data. An alternative approach to designing control charts is recommended.