The purpose of process data analysis is to answer the question, "May the observed data fluctuations reasonably be attributed to common-cause variation?" The principal use of the Shewhart control chart is to guide efforts for process improvement by assisting in the discovery of peculiarities in the data suggesting nonrandom, special-cause influence. Process data tend to wander somewhat over time, so the long-term standard-deviation estimate is greater than the short-term estimate. The time-ordered Xbar & R chart capitalizes on this by using small subgroups of data to obtain a smaller within-subgroup estimate of sigma than might otherwise have been obtained. This is accomplished by taking observations that are homogeneous (as much alike as possible). The result is that the variation within the subgroups is minimized and the variation between the subgroups is maximized, giving the tightest possible control limits and the greatest opportunity for discovering nonrandom special-cause variation.

Hence, with a single-stream process (i.e., no parallel paths such as a multiple-cavity die), a good sampling plan would be to take five consecutive pieces to make each subgroup. Subgroups should be taken at those times when expert knowledge of the process gives the greatest concern for manufacturing problems such as: 

 …