Home Forums Reloading Big Game Rifles Group statistics

Forgive me for being slow on the uptake, but by the data distribution do you mean the distribution of the group sizes?

Sorry for the confusion. Yes, exactly.

OK, then if I remember the CLT correctly that's not what it says is converging to normal as the number of samples increases.

Holy schidt, you guys still talking statistics? Go out and shoot and really learn something..

Yeah, it’s referring to the sampling of the underlying shot data regardless of the probability distribution of that data, but curiously the group size samples are behaving similarly.

LOL, I shoot plenty and learn from both theory and practice.

Just for fun, I ran this video through the Planet 4MOA translator I've built on ChatGPT. I won't bore you with the details, but the AI's conclusion:

If a guy like me keeps shooting long enuf, there's going to be a distribution of good and bad groups. With all the minute-of-pie-plate groups I shot while living on Planet 4MOA, I'm going to be shooting bugnuts for the rest of my life. It's all statistical!

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.

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.

https://grtools.de/doku.php

GRT has an excellent group calculator built in and it’s 100% free. I looked at the OnTarget at one time from the link on accurate shooter, it worked well but I dumped it due to the fact it’s shareware/free to try, not free as represented. May be worth 12 bucks but I despise the intentional misrepresentation.

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.

Seems like you're getting the hang of it.

Let me try rephrasing a bit for more clarity:

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.

Thanks for the link, that'll keep me busy tinkering for awhile

That's interesting and makes sense intuitively. With the 3 shot there's a high degree of variability (high SD) because the group size is much smaller than the "true precision" I guess you could say of that system. The 20 shot is a lot closer to the true group size so it's larger in size but with less variability.

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.

I was thinking about Bryan's statement on that podcast that group size SD tends to be about 30% of the mean group size, in his experience, so I ran some simulations based on my model. I modelled a rifle/load that averages 0.915 MOA for 5-shot groups and one that averages 0.306 MOA. I used 100,000 shots total, but explored what happens if those 100,000 shots are divided up into 3-,4-,5-,7-,10-,20-,35-, and 50-shot groups. Then I looked at the average group size, and the group size SD, in addition to the kurtosis and skewness of the mean shot separation within a given group (average distance between a pair of shots for every possible combination of shots within a group), for all the groups in each data set. Some interesting trends emerged.

Just using visual inspection, the group size SD seems to have an exponential dependence on the number of shots in each group, asymptotically approaching 11% (where SD is 11% of the group size mean), and the skewness follows a similar trend. Kurtosis doesn't seem to have a strong correlation with the number of shots in each group. Interestingly, these trends seem to hold true regardless of the level of precision of the rifle/load, whether 0.915 or 0.306 MOA on average. So to Bryan's point, it seems that SD is about 25-37% of mean group size for 3-5-shot groups, but drops to 11-20% for 7-shot groups or larger. That gives us a good idea of how much dispersion we can expect from shot to shot, whether we have a 1 MOA rifle/load or something approaching BR standards at 0.3 MOA.

0.915 MOA rifle/load


0.306 MOA rifle/load