A single X chart outperforming the joint (X)over-bar &R and (X)over-bar &S charts for monitoring mean and variance

The Shewhart (X) over bar & R and (X) over bar & S control charts have traditionally been used for detecting mean shift and standard deviation shift (sigma). This article studies and compares the overall performance of the (X) over bar chart with that of the (X) over bar & R and (X) over bar & 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) over bar chart with a sample size n=1) is always the optimal version of the (X) over bar chart for detecting and (sigma). Moreover, the single X chart even outperforms the joint (X) over bar & R and (X) over bar & S charts in overall detection effectiveness. On average, the X chart is more effective than the (X) over bar & R and (X) over bar & 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.