Run Length Distribution and Percentiles: The Shewhart Chart with Unknown Parameters

ABSTRACT A well-known concern is that important information regarding the performance of a control chart may be missed by focusing too much on the average run length (ARL). This is particularly true since the run length distribution is generally highly right-skewed. The entire run length distribution should be examined for a more complete understanding of the chart performance and this could be facilitated by an examination of a number of representative percentiles including the median, and some functions of the percentiles such as the interquartile range. Khoo (2004) studied the percentiles for the Shewhart chart when the mean and the variance of the process are specified (the so-called “standards known” case). In this paper we closely examine the run length distribution and the percentiles of the Shewhart chart in the case when the process mean and variance are both unknown (the so-called “standards unknown” case) and are therefore estimated. The exact run length c.d.f. is evaluated and plotted for a number of subgroups (m) and a subgroup size (n) of five, for a nominal false alarm rate (FAR) of 0.0027. The run length c.d.f. for the standards known case, which is known to be geometric, is included for comparison. Moreover, a number of specific percentiles are calculated and compared to those in the standards known case. It is seen that for small to moderate values of m, the run length distributions neither dominate nor are dominated by the geometric distribution. When parameters are estimated and the process is in-control, for percentiles of order less than approximately 0.82, the cumulative probability of early runs is larger for small to moderate values of m, whereas for percentiles beyond that, the cumulative probability of late runs is smaller than those under the geometric distribution. In the out-of-control case (for a step shift of 0.5) a similar phenomenon is seen around the percentile of order approximately 0.62. For m = 500 and n = 5, the run length distribution in the standards unknown case converges to that in the standards known case, namely the geometric distribution. An alternate chart design criterion, based on the in-control median run length, is proposed.