Run length,average run length and false alarm rate of shewhart x-bar chart: exact derivations by conditioning

The effects of estimation of the control limits on the performance of the popular Shewhart X-bar chart are examined via the average run length and the probability of a false alarm, when one or both of the process mean and variance are unknown. Exact expressions for the run length, the average run length (ARL) and the false alarm rate are obtained, in each case, using expectation by conditioning. Applying Jensen's inequality, together with expectation by conditioning, a simple lower bound to the ARL is obtained. This could be useful in designing the charts. The expressions for the exact ARL and the exact probabilities of false alarm are evaluated, using simulations, for various numbers of subgroups and shift sizes. The calculations throw new light on the performance of the Shewhart X-bar chart. Some recommendations are given.