Monitoring the process mean when standards are unknown : a classic problem revisited

Abstract

One of the most common applications in statistical process monitoring is the use of control charts to monitor a process mean. In practice, this is often done with a Shewhart X chart along with a Shewhart R (or an S) chart. Thus two charts are typically used together, as a scheme, each using the 3-sigma limits. Moreover, the process mean and standard deviation are often unknown and need to be estimated before monitoring can begin. We show that there are three major issues with this monitoring scheme described in most textbooks. The first issue is not accounting for the effects of parameter estimation, which is known to degrade chart performance. The second issue is the implicit assumption that the charting statistics are both normally distributed and, accordingly, using the 3-sigma limits. The third issue is multiple testing, since two charts are used, in this scheme, at the same time. We illustrate the deleterious effects of these issues on the in-control properties of the (X,R) charting scheme and present a method for finding the correct charting constants taking proper account of these issues. Tables of the new charting constants are provided for some commonly used nominal in-control average run-length (ICARL0) values and different sample sizes. This will aid in implementing the (X,R) charting scheme correctly in practice. Examples are given along with a summary and some conclusions.