Quality control using S 2 or S charts for subgroups of varying sample sizes and their exact average run lengths

This article generalizes the historical S 2 chart (of equal sample sizes) to subgroups of differing sample sizes. As a sequel to a previous article, Section 2 compares all estimators of process standard deviation for more than M> 1 subgroups of equal sample sizes from a normal universe. Section 3 provides the general probability control limits (CONLs) for the S 2 chart of unequal sample sizes. Section 4 uses the chi‐square distribution to obtain the exact α ‐level CONLs for the S chart. Section 5 proves that the average run length (ARL) of S and S 2 charts are identical and tabulates the ARL for the most common sample sizes n = 2, 3, … , and 12. Further, as expected, the ARL of S charts always perform better than those of the corresponding R charts.