Bill Levinson
Industrial Statistician and Trainer
How can you do the process capability study with a measurement system that hasn't been qualified? Also, in theory at least, we're supposed to select parts for the gage study that represent the operating range of the process. According to AIAG "logic" we determine the operating range of the process with an unqualified measurement system. None of it makes any sense, on any level.
It should be safe to assume that the operating range of the process is inside the specification limits, which will (hopefully) not be a wide enough range to bring in issues like linearity. We also hope the specification limits are at least four process standard deviations to either side of the nominal (or else the process is not capable). This could in fact be used as a starting assumption for short-range or startup SPC (when we know nothing about the process standard deviation and have to make assumptions).
Also, regardless of the distribution of the process data (normal or non-normal), it is usually reasonable to believe the measurements will follow the normal distribution. That is, even if the part dimension comes from a non-normal distribution, the measurements we get from that part should be random normal numbers whose mean is the actual part dimension and whose standard deviation is the equipment variation. A possible exception might involve a measurement with a hard lower limit of zero (e.g. impurities or trace elements) and where the gage has a lower detection limit, but that is not the traditional application.* If one is suspicious, a gage study ANOVA will return residuals that can be assessed to test the normality assumption.
If the process can work to different nominals (as might be applicable in short-run SPC, or the kind of SPC that requires the z chart for different part dimensions from the same tool), a separate study might indeed be desirable for each nominal to determine whether there is a linearity issue and also whether the equipment variation depends on the size of the part. This is not something I have ever tried but you do raise an important point.
* I have looked into this issue, however, for process capability studies for processes with lower detection limits, where "zero" means simply less than the LDL; left-censoring techniques borrowed from reliability engineering can be used.