For interval or ratio data (stuff you can measure with a ruler, meter, etc.), we use the T Test to see if two groups of data really have different means, vs. the difference being easily explained by random variation. The T Test tests a difference. Because of the CLT, the T Test is robust to non-normality as long as you have decent sample sizes.
When you get to measures of dispersion, we use the F Test (or one of its cousins). The F Test tests the ratio of two variances (variance = standard deviation squared, a measure of dispersion). Because the CLT doesn't work here, the F Test is sensitive to non-normality.
Group size is a measure of dispersion. So, again, there is no CLT.
It's possible to simplify things by reducing the problem to one dimension. Think of the target in terms of r and theta, rather than x and y. We really don't care about theta most of the time. We just care about how far the bullet missed. So just do stats on r, and you can take a mean and a standard deviation. Now the stats are better behaved. You can simply say that 95% of shots will fall within plus and minus 2 standard deviations, and that works.
I don't think many folks will do standard deviations in the field. Something simpler is needed.
Group size, mean distance from center, and all the rest all contain the same information, wearing different shirts. There is no need for anything beyond group size and standard deviation. For 5 shots, group size is 90% as good as standard deviation.
So for ranges, you can pull out some exotic tools like ANOMR, or you can just punt and do the simulation. Then you can sort the resulting simulation numbers, and note the upper and lower 2.5% points, and you have your 95% Prediction Interval.
It's been fun.... not many are interested in this esoterica. Hope I have shed a little light on the subject.
I appreciate the insight and explanation. Even if I'm not actually using those other tests that gives a basic understanding of what can be measured and how they can be used. Sometimes just having some intuition for how thing's work helps to understand what we're seeing at the range.
