Guaranteed In-Control Performance for the Shewhart X and X Control Charts

When in-control parameters are unknown, they have to be estimated using a reference sample. Due to the use of different reference samples in phase I, the control chart performance in phase II will vary across practitioners. This variation is especially large for small sample sizes. To prevent low in-control average run lengths, new corrections for Shewhart control charts are proposed that guarantee a minimum in-control performance with a specified probability. However, a minimum in-control performance guarantee generally lowers the out-of-control performance. To balance the trade off between in-control and out-of control performance, the minimum performance threshold and specified probability can be adjusted as desired. The corrections are given in a closed form so that the bootstrap method, which has recently been suggested, is no longer required. The performance of our proposed correction is illustrated by simulating some practical situations. Furthermore, a comparison is made with tolerance intervals and self-starting control charts.