Percentiles of the Run-Length Distribution of the Group Runs Xbar Chart

Nowadays, quality is essential in every field. Producing quality products or services can help to increase thecompetitiveness, reputation and customer loyalty of a company as this ensures that the products or servicesmeet customers’ expectations. However, for a product or service to have a satisfactory quality level, theunderlying process manufacturing the product or providing the service should be in statistical control. TheGroup Runs (GR) 𝑋 chart is proposed as an improvement of the synthetic 𝑋 chart for the detection ofprocess mean shifts. The GR 𝑋 chart integrates the Shewhart 𝑋 sub-chart and an extended version of theconforming run length (CRL) sub-chart. To have a better understanding of the behavior of the GR chart, inthis paper, we study the percentage points (or percentiles) of the run-length distribution of the GR chart,instead of depending entirely on the average run length (ARL) criterion. Before computing the percentiles ofthe GR 𝑋 chart, we need to obtain the optimal parameters of the GR 𝑋 chart. This can be achieved by writinga computer program to search for a set of optimal L and k values that minimize the out-of-control ARL,while maintaining the in-control ARL at a desired value. Then, the percentiles of the run-length distributionof the GR 𝑋 chart can be investigated using this set of L and k values.