The fundamental issue is that all measures of dispersion are differences between data points. Data points behave as we have come to expect. Differences do not..
Things are simpler if you simply worry about how far each shot is from the center of the group, but that is more complex than anyone is going to do in the field. For groups with 5 shots, group size has practically all the statistical strength of standard deviation.
After reading through this thread a few times I think I'm getting a better handle on this particular concept. So if I'm understanding this correctly (I suck at statistics so my terminology/understanding is probably off), we can't treat group size measurements as independent variables because of the underlying dispersion that those numbers are based on. Since they're not independent variables, the CLT doesn't apply. Therefore if we try to treat those group size measurements as independent variables, the results are unlikely/less likely to accurately model real world behavior. Am I on the right track with that?
On the second part, if instead of using group size measurements, we use the location of each individual shot, those are independent variables, therefore the CLT applies, and we can assume a normal distribution and analysis. Something like that? If so, would I be correct to assume that if I want to get a more accurate model of how my system actually performs within say 20 shots, it would be more accurate to use mean radius than shooting 4-5shot groups?
I appreciate everyone's input in this thread. It's interesting stuff to think about.
