Generalization of the Run Rules for the Shewhart Control Charts

Summary

It is well-known that the Shewhart control charts are useful to detect large shifts of a process mean, but it is insensitive for small shifts and/or other types of variation. We extend the Shewhart’s three sigma rule and propose two new rules based on successive observations. One is that a signal occurs when m successive observations exceed k ₁ sigma control limit. The other is that a signal occurs when m − 1 of m successive observations exceed k ₂ sigma control limit. The original Shewhart control chart is included in the first generalized rule as m = 1. The performance of the proposed rules is evaluated under several out-of-control situations by both the average run length and the standard deviation of the run length. These rules are more powerful than Shewhart’s three sigma rule at detecting moderate step shifts.