A special pattern of consecutive points on a control chart often suggests that the process is out of control due to a special cause. Usually, the probability of such a pattern is not difficult to calculate, but its relation to the rate of a false signal in a control chart is unclear. In this paper we give such a relation for a wide class of patterns that includes six of the eight patterns of special cause discussed in the AT&T Statistical Quality Control Handbook (1958). We derive a surprisingly simple formula for the rate of a false signal in terms of the pattern probability and use this result to improve on the test proposed in the handbook. We also use this formula to give a simple proof of a result of Wolfowitz (1944, Annals of Mathematical Statistics 15: 163–172) and Olmstead (1945, Annals of Mathematical Statistics 17: 24–33) on the monotone sequence. Finally, we give an approximation of the rate of a false signal for another wide class of patterns that covers the two remaining patterns of special cause in the AT&T handbook.

In the proof of Theorem 2.6 in [1], we reduced the comparison of Ii(n) and Ii+1(n) to Ii−k(n − k) and Ii+1(n − k). Then, we mistakenly stated: “Suppose i − k > k, then we repeat the same reduction.” In fact, the reduction can be repeated only if i′ = n − k − i = i − k + 1. Consequently, Theorem 2.6 was proved only for t = 2. For similar reasons, Corollary 2.7 was proved only for t = 2, and Theorem 2.8 only for t = 1.

We study the Birnbaum importance of the consecutive-k-out-of-n line and clear up confusion in some claimed but unproved results. We introduce some new techniques which not only prove these claimed results but also generalize them much further. Finally, we extend our results to the 2-out-of-m-out-of-n line.