The Mean and Standard Deviation of the Run Length Distribution of X̄ Charts When Control Limits Are Estimated

Control charts with estimated control limits are widely used in practice. Common practice in control chart theory is to estimate the control limits using data from the process and once the process is determined to be in control to treat the resulting control limits as though fixed. Little is known about the run length distributions of these charts when the fact that control limits are estimated is taken into account. For example, no calculation has ever been done to find the mean and standard deviation of the run length distribution of X̄ charts when process mean μ and standard deviation σ are estimated. In this paper, we derive and evaluate the mean and standard deviation of the X̄ charts when control limits are estimated in three different ways. The results are then used to discuss the inadequacy of the widely followed empirical rules for choosing the number of samples m and the sample size n.