Originally Posted by brydan
Originally Posted by denton
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..

Originally Posted by denton
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.
That’s essentially what my simulation shows. The resulting distribution depends on how we define “group size,” though for practical purposes, both definitions (maximum separation between any two shots in a group, and the mean separation between all pairs of shots in the group) result in distributions that very closely approximate a Normal (Gaussian) distribution. In terms of accurately predicting real-world behaviour, the simulation suggests that the CLT is closely approximated. If you want an accurate model, calculate the mean separation between shots (or the mean radius, which is essentially the same but uses a fixed point of reference) for each group, and expect your calculations to approximately follow a Normal distribution.