I apologize for my sophomoric response, but I want to make sure you understand the use of Shewhart's control chart method (X-MR charts are not diagnostic charts). The purpose, as originally intended by Shewhart, was to compare sources of variation (components of variation) to determine which source has the greatest effect on the response variable (in control chart language, is the greatest source within subgroup or between subgroup?). Of course, you are not limited to two layers of a sampling plan and each layer is correlated with a specific set of x's or components of variation. The sampling plan, which will be evaluated by control charts, is of utmost importance. As you change subgroup sizes, you change what sources of variation are captured within subgroup. As you change the frequency of taking the samples, you change the sources of variation captured (that can be influential) between subgroup. Once the greatest source has been identified, continue to use sampling and control charts (or other tools like DOE) to further disaggregate the components and focus on the set of variables that should be investigated to reduce variation.
In order to accomplish the purpose, Shewhart first suggested the basis for comparison must be evaluated for consistency. In other words, are the within subgroup sources of variation stable/consistent? (is the range chart "in-control"?) If not, you should seek to understand why. If those sources are consistent, then a comparison can be made. The X-bar chart is a comparison chart. It compares the sources of variation changing between subgroup (visualized as the averages plotted on the x-bar chart, FYI, they are biased to the between subgroup sources as a function of averaging) to the sources of variation captured within subgroup (as visualized by the control limits on the x-bar chart, A2*R-bar). If the averages are within the control limits, then the within subgroup sources dominate, if there are signals of averages varying more than the within sources (e.g., points out-of-control), then the between sources dominate. Rational subgrouping and sampling strategies are a key to this methodology working.
“The engineer who is successful in dividing his data initially into rational subgroups based on rational theories is therefore inherently better off in the long run. . .” Shewhart
The Shewhart control limits were empirically derived. There is no normality assumption for the use of control charts! They were set to be a guide, possibly conservative, to decision making regarding which sources of variation are most influential. There are, of course, the Western Electric patterns that can assist in the interpretation of "out-of-control" conditions which can help in the interpretation of which sources dominate. Ultimately, you will be looking at the data graphically and deciding which sources you believe to dominate (with the assistance of statistics when those decisions are not obvious).
Your supposition of "the control limits, target and sigma would be set based upon an in control time period..."is non-sensical from a Shewhart control chart methodology approach.
Please read:
Shewhart, Walter A. (1931) “Economic Control of Quality of Manufactured Product”, D. Van Nostrand Co., NY
Wheeler, Donald (2015) “Rational Sampling”, Quality Digest
Wheeler, Donald (2015) “Rational Subgrouping”, Quality Digest
"All models are wrong, some are useful" G.E.P. Box
