Run-length distribution for control charts with runs and scans rules

In order to increase the power of the classical Shewhart control charts for detecting small shift, several supplementary rules based on runs and scans were introduced by the Western Electric Company in 1956. In this article we introduce a new method for computing the runlength distribution for a Shewhart chart with runs and scans rules. Our method yields an exact expression for the run-length generating function. We can then use either one of two techniques for extracting the probability function. One leads to recursive formulas and the other to non-recursive formulas. We investigate the performance of some popular runs and scans rules and show that the run-length distribution is highly skewed. Comparing the entire distributions of different rules, rather than simply the widely-used expectations (ARLs), leads to important new conclusions on the advantages of applying each of these rules vs. using a simple chart. Finally, we introduce a Web application that incorporates these theoretical results into a simple and practical tool that can be used by practitioners.