Thank you for the very clear and precise answer! I think we have a few shortcomings at my end. I really appreciate your help, and I am trying my best to understand.
As an example, for another set of data than the original in this thread, I have compared the output of out statistical software with the output of R using the qcc-library. I would like to ask you to observe the control limits. In the output (see picture 1) from our statistical software, the control limits are on either side of the target value, even though this puts them both below the X-bar-bar. I was under the impression that one would calculate the control limits from the x-bar-bar (X-bar-bar +/- (A2*R-bar). Thus, the control limits should be above and below X-bar-bar - not the target.
When doing the same data in R (see picture 2), the control limits are above and below the center line (X-bar-bar). This seems right to me. Maybe I am missing something important, that makes me draw the wrong conclusion. Something just seems wrong with the T, X-bar-bar, UCL and LCL in picture 1.
Picture 1. Output from our statistical software
Picture 2. Output from R