A single X chart outperforming the joint X & R and X & S charts for monitoring mean and variance

The Shewhart X & R and X & S control charts have traditionally been used for detecting mean shift δμ and standard deviation shift δσ. This article studies and compares the overall performance of the X chart with that of the X & R and X & S charts, as well as the X&MR chart. The comparative study led to surprising results that contradict the conventional wisdom in Statistical Process Control (SPC) niche. It is found that the simplest single X chart (i.e., the X chart with a sample size n = 1) is always the optimal version of the X chart for detecting δμ and δσ. Moreover, the single X chart even outperforms the joint X & R and X & S charts in overall detection effectiveness. On average, the X chart is more effective than the X & R and X & S charts by around 5% under different circumstances. Most importantly, the X chart is very simple to understand, implement and design. As a result, it may be highly preferred for many SPC applications, in which both the mean and variance of a variable need to be monitored.