Improved feature-based test statistic for assessing suitability of the preliminary samples for constructing control limits of $\bar{X}$ chart

Abstract

In spite of using a large number of subgroups (m) of small samples (n), the estimated control limits of $\bar{X}$chart in phase I can be erroneous unless the preliminary samples are drawn from a stable process. As a result, the performance of the chart in phase II can be significantly affected. The pattern in the $\bar{X}$chart, exhibited by the plots of the subgroup averages of the preliminary samples, will be different depending on stability and instability of the process while the preliminary samples were collected. Based on this concept, a new feature-based test statistic (FTS) is proposed for evaluating suitability of the preliminary samples for the designing of the $\bar{X}$chart. The FTS, for given m, approximately follows $N[1,{\text{ SD(}}m{)]}$, where SD(m) is a function of m. The goodness of the approximation and effectiveness of the test are evaluated using simulated data. The results show that both are satisfactory for m > =48. The proposed statistic is also quite effective in detecting unstable process condition resulting in a cyclic pattern. The computation of FTS involves some complexities. However, now-a-days computers are widely available and so computation difficulty may not be a problem.