The Probability of an Out of Control Signal from Nelson’s Supplementary Zig-Zag Test

Nelson’s ’supplementary runs’ tests are widely used to augment the standard ’out of control’ test for an ¯x control chart, or a chart with individual values, to determine if any special causes exist. The fourth of Nelson’s tests gives an out-of-control signal when fourteen points in a row follow a zig-zag pattern (alternating up and down); it is thus a signal that the process has negative autocorrelation. Using a recursive formula, the exact probability of a zig-zag sequence of length 14 is calculated for an in control process. This value does not appear in the SQC literature, but can be simply determined from results of Andr´e (1879, 1881, 1883), rediscovered by Entringer, (1966), which long precede the development of SQC. Two curious properties, relating the probabilities of zig-zag sequences of successive lengths, are also demonstrated. AMS Subject Classification: 60-01 and 62P30.