The X control chart for monitoring process shifts in mean and variance - Khoo, Michael B. C. (2013)
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 Ẋ chart is always n = 1, in other words, the simplest X chart (i.e., the Ẋ chart with n = 1) is the most effective Ẋ 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.