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.
Interestingly, when group size is defined as the mean distance between pairs of shots in a group, the distribution is more normal with skewness of ~0.28.
Gotchya, that makes sense. Earlier in the thread when you mentioned "the mean distance between pairs of shots in a group" I didn't entirely get what that meant. It's the mean separation between all the pairs of shots in the group. I can see how that's similar to mean radius in that they're both accounting for every shot in the group rather than just the two furthest shots. I just downloaded OnTarget to play around with it a little bit. Like I mentioned earlier, I know statistics isn't my strong suit but sometimes digging into a subject with something that's interesting is a good way to learn more about it.
Yeah, you're off to a great start.
I'll point out a couple more interesting results from the simulations. Each shot (100,000 shots total per simulation) was modelled as landing a certain distance from POA following a Gaussian distribution with SD of 1.34 MOA from center. Using 3-shot groups, the mean group size (largest separation between any two shots in the group) was 0.72 MOA with an SD of 0.27 MOA. Using 10 shots, the mean was 1.14 MOA and the SD 0.22. Using 20-shot groups, mean was 1.34 MOA and SD was 0.20 MOA. The mean separation between pairs of shots in a group was the same for all simulations at 0.53 MOA, but the SD obviously decreases as the number of shots per group increases (0.19 MOA for 3-shot groups and 0.06 for 20-shot groups).
The more shots in your groups, the more closely the resulting group size (when defined as the mean distance between all pairs of shots in the group) follows a Gaussian distribution.
