Department of Statistics Unitmark
Dietrich College of Humanities and Social Sciences

Run Length Distribution for Control Charts with Runs Rules

Publication Date

July, 2001

Publication Type

Tech Report


Galit Shmueli and Ayala Cohen


In order to increase the power of the classical Shewhart control charts for detecting small shift, several supplementary rules based on runs were introduced by the Western Electric Company in 1956. In this paper we introduce a new method for computing the run length distribution for a Shewhart chart with runs 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 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.