The X control chart for monitoring process shifts in mean and variance

Control charts are widely used in statistical process control (SPC) to monitor the quality of products or production processes. When dealing with a variable (e.g., the diameter of a shaft, the hardness of a component surface), it is necessary to monitor both its mean and variability (Montgomery 2009 [Montgomery, D.C., 2009. Introduction to statistical quality control. New York: John Wiley & Sons.]). This article studies and compares the overall performances of the X chart and the 3-CUSUM chart for this purpose. The latter is a combined scheme incorporating three individual CUSUM charts and is considered as the most effective scheme for detecting mean shift δμ and/or standard deviation shift δσ in current SPC literature. The results of the performance studies reveal two interesting findings: (1) the best sample size n for an chart is always n = 1, in other words, the simplest X chart (i.e., the chart with n = 1) is the most effective chart for detecting δμ and/or δσ; (2) the simplest X chart often outperforms the 3-CUSUM chart from an overall viewpoint unless the latter is redesigned by a difficult optimisation procedure. However, even the optimal 3-CUSUM chart is only slightly more effective than the X chart unless the process shift domain is quite small. Since the X chart is very simple to understand, implement and design, it may be more suitable in many SPC applications, in which both the mean and variance of a variable need to be monitored.