A Control Chart Based on Cumulative Scores

SUMMARY Process control schemes using a simple scoring system are presented for controlling the mean of a normal distribution in the one and two sided case. The average run lengths of the schemes are simpler to compute than those of cumulative sum schemes, and for small deviations in the process mean the schemes are considerably more sensitive than Shewhart schemes. Some examples and comparisons are given. corrective action is indicated whenever a sample mean falls outside "action limits" placed at juo ? k2 /v/n. Page (1955) has suggested the inclusion of "warning limits" at juo ? k 1 a/v/n (k 1 < k2), with the rule that the process is halted if r consecutive points fall between the warning and action limits, or a single point falls outside the action limits. In the cusum scheme (Page, 1954) for detecting increases in the mean from its target value, the cumulative sums of the differences between the sample means and some reference value K, (Xi -K), are plotted against sample number. If the cusum becomes negative, the cumulation is restarted, but if it reaches some value H (the decision interval), then corrective action is indicated. For two-sided control, it is necessary to test also for decreases in the mean, and this is done by operating a second cusum with reference value and decision interval - K and - H respectively. The schemes proposed in this paper assign a score of -1, + 1 or 0 to the sample means according to whether they are "extreme negative", "extreme positive" or otherwise. In the two- sided case corrective action is indicated when the modulus of the cumulative score reaches some fixed value, which amounts to operating a cusum on the scores with zero reference value. In the one-sided case, a new decision rule is proposed. Like the Shewhart scheme, both schemes have the attractive property that the average run length, ARL (u), can be expressed as a simple function of the tail areas of the quality distribution, and the basic Shewhart scheme is in fact a special case. The advantage of these schemes is that they can be more sensitive to small deviations in the process mean than Shewhart schemes, at the expense of some efficiency for