The Operating Characteristic of the Control Chart for Sample Means

Abstract

In this paper we derive the operating characteristic of the control chart for sample means when process standards are unspecified. Under the null hypothesis the distribution of the process is $N(\mu, \sigma^2)$, where $\mu$ and $\sigma$ are fixed but unknown. Under the alternative the process mean is a random variable with a $N(\mu, \theta^2\sigma^2)$ distribution. Exact results are obtained for cases ranging from two samples of size 2 to four samples of 10. Bounds on the operating characteristic are obtained in particular cases ranging from five samples of 5 to 25 samples of 10.