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) over bar 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 Andre (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.