Self-Starting Cusum Charts for Location and Scale

In some quality control problems, it is not known what the exact process mean and standard deviation are under control but it is desired to determine whether there have been drifts from the conditions obtained at the process start-up. This situation is not well-covered by standard cumulative sum procedures, which generally assume known process parameters. This paper uses the running mean and standard deviation of all observations made on the process since start-up as substitutes for the unknown true values of the process mean and standard deviation. Using some theoretical properties of independence of residuals, two pairs of cusums are set up: one testing for constancy of location of the process, and the other for constancy of the spread. While the process is under control, both these cusum pairs are of approximately normal N(O, 1) quantities (and therefore are well understood), but if the location, the spread or both change, then non-centrality is introduced into one or both of the location and scale cusum pairs, and it drifts out-of-control. It is shown that the procedure performs well in detecting changes in the process, even in comparison with the often utopian situation in which the process mean and variance are known exactly prior to the start of the cusum. precisely if large numbers of spurious signals are to be avoided. While it has usually been assumed in discussion of cusums that the process specification is known exactly, there are many circumstances in which it is to some extent uncertain, and under these conditions the assumption of known mean and variance is inappropriate. A case of particular interest to the author arises in the control of assaying in a chemical laboratory for bias and precision. A good way of doing this is (Mandel 1964) by using reference materials. A reference material is a batch of the same sort of material as is assayed in the laboratory (obtained for example from a previous consignment), portions of which are assayed regularly along with the production material. A depar- ture of the assays of the reference material from their previous mean or standard deviation indicates a change in bias or precision. Note that the control required is simply for any change in the values obtained for the reference material. It is immaterial what the true assay of the reference material (the ideal target process mean) actually is, and the quality control system should not be predicated on the assumption of an exact knowledge of the process mean. This indeterminancy presents