C-chart, X-chart, and the Katz Family of Distributions

In statistical process control, the primary method used to monitor the number of nonconformities is the c-chart. The conventional c-chart is based on the assumption that the occurrence of nonconformities in samples is well modeled by a Poisson distribution. When the Poisson assumption is not met, the X-chart (individuals chart) is often used as an alternative charting scheme in practice. In this article I investigate the relative merits of the c-chart compared to the X-chart for the Katz family covering equi-, under-, and over-dispersed distributions relative to the Poisson distribution. The need to use an X-chart rather than using the c-chart depends upon whether or not the ratio of the in-control variance to the in-control mean is close to unity. The X-chart, which incorporates the information on this ratio, can lead to significant improvements under certain circumstances. Both the 3-sigma cand X-charts fail in providing reliable information on the status of the process with a small in-control process mean when a downward mean shift occurs. In these cases, charts based on probability limits are much more appropriate.