Robustness of symmetric X charts to nonnormality and control-limit estimation

We study the mean and variance of the run length of the X-chart as a function of the underlying distribution skewness and kurtosis (indicators of nonnormality) and the number n of Phase-I data points for estimating the control limits. Our results show that the nonnormality has significant and nonmonotonic effects on the X-chart performance. Our analytical and simulation results show that (1) the mean and variance of the run length decrease in n; (2) the in-control mean run length is infinity for all bounded distributions whenever the control limits are two or more estimated standard deviations from the known mean; and (3) for unbounded distributions, using n ≥500 Phase-I data points provides run-length distributions close to those obtained with known control limits.