Comparisons of the symmetric and asymmetric control limits for [X] charts

This work compares the X macr chart performance for the symmetric and asymmetric limits. The Shewhart X macr control chart is useful in detecting a shift in the process mean. A conventional way of setting the control limits is using a set of symmetric limits, e.g., the 3-sigma limits. Despite literature of constructing asymmetric control limits for skewed distributions exists, none has compared these two kinds of limits on an equal basis. We compare the out-of-control ARL (average run length) for symmetric and asymmetric limits while keeping their in-control ARL values the same. Two testing examples are used: gamma and Johnson unbounded distributions. Our empirical study shows that if the quality characteristic has a right skewed distribution, the symmetric limits perform better than the asymmetric limits when the process mean shifts to a larger value and worse otherwise. Analogously, if the skewness is negative, the symmetric limits perform better than the asymmetric limits when the mean shifts to a smaller value and worse otherwise. None fully dominates the other. Therefore, even when the quality characteristic has a skewed distribution, the choice of symmetric or asymmetric limits depends on the directions of the shift and skewness.