Symmetric (X)-charts : Sensitivity to nonnormality and control-limit estimation

We study the classical symmetric (X)-chart with control limits set k standard deviations from the known in-control mean. The standard deviation is estimated with in-control data, in what we refer to as Phase-I. We consider three performance measures: the average run length (ARL), the standard deviation of the conditional average run length (SDARL), and the corresponding coefficient of variation. Modeling the (X) data as independent and identically distributed, with marginal distributions chosen from the Johnson family, we investigate in-control and out-of-control sensitivities to three factors: the third and fourth standardized moments of the (X) data distribution and the number of Phase-I observations. Considering both bounded and unbounded data distributions, our analytical, numerical, and Monte Carlo simulation results show that nonnormality has a substantial effect on all three performance measures; and the effects are nonmonotonic in both skewness and kurtosis. We show that all three performance measures are flawed when estimating the standard deviation. In particular, we show that ARL and SDARL values increase, eventually becoming infinite, as the number of Phase-I observations decreases, even in cases where run length is finite with probability one. We show analytically that, for bounded data distributions with any finite shift, estimating the standard deviation sometimes results in infinite ARL and SDARL values.