A SIMPLE RELATION BETWEEN THE PATTERN PROBABILITY AND THE RATE OF FALSE SIGNALS IN CONTROL CHARTS

A special pattern of consecutive points on a control chart often sugge sts 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 uncl ear. In this paper we give such a relation for a wide class of pattern s that includes six of the eight patterns of special cause discussed i n the AT&T Statistical Qualify Control Handbook (1958). We derive a su rprisingly simple formula for the rate of a false signal in terms of t he pattern probability and use this result to improve on the test prop osed 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.