Effect of sample size on the performance of Shewhart control charts

Abstract

The control chart is one of the most powerful techniques in statistical process control (SPC) to monitor processes and ensure quality. The sample size n plays a critical role in the overall performance of any control chart. This article studies the effect of n on the performance of Shewhart control charts, which have traditionally been used for monitoring both the mean and variance of a variable (e.g., the diameter of a shaft and the temperature of a surface). The study is conducted under different combinations of false alarm rate and process shift. The detection speed of the Shewhart charts is evaluated in terms of average extra quadratic loss (AEQL) which is a measure of the overall performance. It is found that n = 2 is the best sample size of the Shewhart $$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{R} $$ and $$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{S} $$ charts. The comparative study reveals that the $$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{R} $$ and $$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{S} $$ charts with n = 2 outperform the $$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{R} $$ and $$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{S} $$ charts with n ≥ 4 by at least 9 and 7 %, respectively, in terms of AEQL. These results contradict the common knowledge in SPC niche that n between 4 and 6 is usually recommended for the $$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{R} $$ and $$ \overset{\_}{\boldsymbol{X}}\&\boldsymbol{S} $$ charts.