Constructing Control Charts With Average Run Length Constraints

In many statistics courses for engineering majors, students learn how to construct control charts for monitoring quality levels of manufacturing processes. However, the students generally just learn how to use the standard “three-sigma” approach, where control limits are established at three standard deviations above and below the average value. Often, no details are given as to how the sample size and control limit choices ultimately determine the performance of the control chart. In this paper, we will demonstrate how with some basic knowledge of geometric, normal, and chi-square random variables, a student can learn to construct X-bar and S control charts that will have specified properties in terms of performance. In evaluating control charts, one is usually concerned with the false alarm rate (how frequently does the chart erroneously signal if the monitored process is on target?) and the detection rate (how quickly does the chart signal if the monitored process is not on target?). Using the simple tools proposed in this paper, the designer of a control chart can determine the sample size and control limits required to establish a desired false alarm rate and a desired detection rate for some specific out-of-control state. Teaching the process control material in this fashion connects the probability material learned in the early part of the course with countless real-world applications, making the probability material much more accessible and relevant to the students.