Exact expression for the expectation of estimated Cpk based on control chart information and the corresponding process capability control charts

Abstract

Among all the process capability indices, Cpk is the most popular among the practitioners, when the underlying quality characteristic follow normal distribution. Due to its complicated expression, properties of its plug-in estimator have been of particular interest to several researchers both from classical and Bayesian statistical arena. In the present article, we have proposed plug-in estimators of Cpk based on the X − R and X − S chart information under the assumption that each of the samples are drawn independently. We have also derived the expressions for expected values of these estimators and the associated bias. These estimators are found to work similar to the Bayesian-like estimators without depending upon any historical information. The corresponding process capability control charts have also been designed to enable continuous assessment of a process. Finally, a simulated example and two real-life examples have been discussed to supplement the theory developed in this article.