Small Sample Uni- and Multivariate Control Charts for Means

Several characteristics, say d of them, are measured on the output of a process. Such characteristics may or may not be correlated. A case has been made that one should not monitor such processes by keeping separate control charts on each of the d characteristics. What is often suggested is a single control chart based on Hotelling’s T . See Alt (1985) and Murphy (1987) on this point and for further entries into the literature. It is usually assumed that the observations come in subgroups of size k each and each such subgroup is used to compute an estimated covariance matrix in order to capture the local variation pattern. These covariance matrices are then averaged in order to obtain a more stable estimate to be used in the formation of the Hotelling T 2 criterion. Often subgroups are of size k = 1 or the variation within a subgroup is not representative of the between subgroup variation, i.e., we may be dealing with batch effects. In this latter case it would make more sense to average the observations in each subgroup and let the variation of these averages speak for themselves, i.e., we will again deal with k = 1. It is this latter situation (subgroups of size k = 1) that we address here and we follow the univariate strategy of an X-chart. Such charts, also called charts for individual measurement, are discussed in Montgomery (1991), chapter 6-4. Since an X-chart is typically based on substantially fewer data points, say n = 20, the variance estimate obtained from the moving range formula is not yet very stable. This should be viewed in contrast to the fairly stable estimate based on n = 20 subgroups of size k = 5. In that case we would have 20 · 4 = 80 degrees of freedom in estimating σ as opposed to 19 degrees of freedom when the group size is k = 1. Here the 19 degrees of freedom are based on using the sample variance of all 20 observations. This however has the drawback of completely ignoring the time order of the observations and any trend in the observations could be mistaken for natural variability. In the univariate situation such trends would easily be visible on the control chart but not so in a multivariate situation. For this reason one prefers σ estimates that are based on local variation, such as the moving range or the moving squared range. The latter is more easily adapted to the multivariate situation and was proposed by Holmes and Mergen (1993), however without allowing for the degree of freedom loss due to the overlap of the ranges. We will show that with n = 20 the moving squared range estimate of σ has only roughly 13 degrees of freedom, since the local variabilty estimates use overlapping data values. For the same reason the moving range estimate would presumably have similarly reduced degrees of freedom, but here the effect is more difficult to assess analytically. This reduction in degrees of freedom is still better than the 10 degrees of freedom one gets when using the 10 nonoverlapping, consecutive data pairs to estimate σ. To account for this small sample instability in the σ estimates we propose to use control limits based on the Student-t distribution (F distribution in the multivariate case) with appropriately adjusted degrees of freedom.