Comparisons of the symmetric and asymmetric control limits for X¯ and R charts

Though both symmetric and asymmetric control limits can be applied to X¯ and R   charts, no well-controlled comparison of the resulting performance has been conducted. Symmetric limits such as 3-sigma limits are the customary choice. However, previous researchers have proposed asymmetric control limits as a more appropriate choice than symmetric limits for skewed distributions. This paper examines the relative performance of symmetric and asymmetric limits. It compares the out-of-control average run length (ARL) for symmetric and asymmetric limits for fixed values of the in-control ARL. Two testing examples are employed: the exponential and Johnson unbounded distributions. The results of the performance comparison are mixed. For both X¯ and R   charts, the impact of the control limit choice (symmetric or asymmetric) depends on two factors: the skewness of the charting statistic (sample mean X¯ for the X¯ chart and sample range R for the R   chart) and the shift direction. When the charting statistic has a right-skewed distribution, symmetric limits perform better if the monitored process property (the mean for the X¯ chart and the standard deviation for the R chart) shifts upward, but worse when it shifts downward. When the charting statistic has a left-skewed distribution, the outcome is reversed. Although neither type of control limits dominates even for a skewed population, the asymmetric limits are more robust to the shift in the process mean or variation. The effect of the sample size is also discussed.