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Statistics for differentiating between groups?

The issue here is that the distance from center (I.e., the radius) does not follow a normal distribution and thus the t test is invalid. The distance from center follows something called a Nakagami-q aka Hoyt distribution, and it's unclear to me if there are established hypothesis tests for such a distribution.
Fair point as to question if mean radius is a normal distribution (have read different views on this) but using group size reduces statistical power as you ignore a lot of data points internal to the extreme paints. Mean radius is a better approach to low sample group analysis. Litz has a bunch of good stuff on this issue.
 
Fair point as to question if mean radius is a normal distribution (have read different views on this) but using group size reduces statistical power as you ignore a lot of data points internal to the extreme paints. Mean radius is a better approach to low sample group analysis. Litz has a bunch of good stuff on this issue.
Radius is certainly not normally distributed.

To avoid confusion, it's worth carefully stating that the mean radius of a single group does not have a distribution–its a single number. The radius, which we can compute for every shot, does have a distribution—the Hoyt distribution.

Incidentally, looking for a good link to support that claim led me to a possible answer to my question.



It will take me a bit to digest this though.
 
I know, but I would expect a group size to vary in a way that’s fairly close to normal. At least no reason to expect a left/right skew.

The reality is, you’re picking the tiniest of nits here. If the response variable is group size, I think the method I described would get you to an answer. Can always revert to nonparametric if you’re that worried about normality.

You seem trained to enough on the subject to answer this without our help.
Thinking more on this, you're certainly correct that mean of the group size across many different groups follows a normal distribution. That's the Central Limit Theorem.

Still, testing ammo based on a method that relies on the limiting behavior of the CLT is, uh, too rich for my blood.

That aside, the method you described is valid. I should have seen that sooner. Sorry.



EDIT: I've been playing around with this method (simulated in python, since I'm not filthy rich), and whether or not the group size converges to normal with repeated samples seems to depend heavily on whether or not we're talking about a 2, 3, 5, etc shot group. Even after 10k iterations, the smaller the size of the group, the more right-skewed the distribution is.
 
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I love statistics, and made a fair piece of money with queueing formulas and similar fun stuff.

Having said that, shoot two, or, at most, three shots from a cold bore, and measure that. All that matters for hunting.

My Model 70 .300 Win Mag that Kirby Allen built puts the first 2 touching. Why should I wear out my barrel shooting 20 round groups? That just sounds crazy. Bench rest, maybe. But this is HuntTalk, not BenchTalk.
 
Are you trying to evaluate to where you can pick the proper load to win a bench rest competition or hit an animals vitals?

If benchrest - the guys that do well dont' seem to believe in statistical validity anyway :ROFLMAO:
If hunting - it doesn't matter below a decent base level of precision unless you're already better at field shooting than 99.99% of hunters anyway.

Not that I haven't wasted thousands of rounds and $ chasing these rabbit holes, i just wish i wouldn't have.
 
Not that I haven't wasted thousands of rounds and $ chasing these rabbit holes, i just wish i wouldn't have.

Most shooters have, myself included. This is exactly the issue that's motivating me to put all this info together (I am a research economist, after all).

My goal is to write up a statistically valid method of how to tell if your ammo/load is good enough, and how to tell if you're chasing randomness down a rabbit hole.
 
Couple things.
1) @madtom is correct on method. You need enough observations to produce a normal curve if you want to use parametric statistics. Nonparametric methods are useless, or close, for answering the question that I perceive you're asking. Using group size as your variable is sorta questionable statistically speaking, because those are grouped individual observations.
2) Scope of inference must be well defined and the numerous confounding variables are your enemy.
3) @Carl sent this to me years ago. Houston Warehouse Tests The full article is pretty interesting. It's worth reading for perspective on said variables and just what you can consider attributing to what.

If it's all an exercise in curiosity, there are worse ways to occupy the mind. But yeah if it's functionally a useful test it's pretty much been hashed out by people with more degrees and components at their disposal than most any lay person.
 
Most shooters have, myself included. This is exactly the issue that's motivating me to put all this info together (I am a research economist, after all).

My goal is to write up a statistically valid method of how to tell if your ammo/load is good enough, and how to tell if you're chasing randomness down a rabbit hole.

Have you heard of or looked into the WEZ calculator?
 
Let's see you're looking to develop the BEST HUNTING LOAD. Well, load the bullets you want to use, a hunting bullet. Then load what you feel will meet your needs velocity wise. Then load 15 rounds of each load you like. Okay now go to the range set the target at 200 yards. Fire 3 rounds of each load letting your barrel cool between groups. Take the target down noting which load used on each target. Go home and rate each load on group size and poa vs poi. Go back the next day and repeat the 3 shots on a target noting the load info then repeat the next day and the next. Now figure out which load is the most consistent and go with that load.

Your next step is to learn how to hunt by getting close to the animal so you can shoot it at bow range. Just kidding what I mean is put the hunt back into hunting and not make it sniping. But that's just my opinion or you could say my $.02 which in today's world isn't worth much anymore. But most important thing is don't sweat the small stuff and have fun!!!
 
Most shooters have, myself included. This is exactly the issue that's motivating me to put all this info together (I am a research economist, after all).

My goal is to write up a statistically valid method of how to tell if your ammo/load is good enough, and how to tell if you're chasing randomness down a rabbit hole.

For "good enough", you can use the WEZ tool to work towards a scenario where ammo/rifle precision has next to no impact on hit rates compared to human error.
 
Don't forget to remove all the other variables.
Temperature
Clean bore, or you have to have exactly the same fouled barrel each time.
Temperature of barrel.
Wind.
Absolutely no movement during the shot.
Use the same brass, each one fired the same time. Measure each one to ensure they are exactly the same.
I'm sure I'm forgetting a bunch.

Or as someone said. Fire off 20 rounds one after another. Do that with each different load. Aim at a pie plate at 200 yards and see which one has the best overall grouping.

I've been going down this rabbit hole of late with my 308. I'm shooting at 400 meters with my loads. I've decided if I can keep it in an 8 inch circle at that distance I'm good.

223 is next.
 
I would start with shooting more rounds per group. 5 shots per group doesn't tell you much. Some people suggest 20-30
No. 2 rounds per group. Lots and lots of 2-round groups. Then a simple t-test. Very easy to do, very simple analysis. I wrote an essay on this once.
 
No. 2 rounds per group. Lots and lots of 2-round groups. Then a simple t-test. Very easy to do, very simple analysis. I wrote an essay on this once.

I'd love to read that essay, if you still have a copy. I've forgotten how to do a proper power calculation to figure out how many samples need to be taken for the CLT to apply in any given case, but I remember the rule of thumb was thirty samples. So, we're talking about 60 rounds per load in this case, right?
 
I'd love to read that essay, if you still have a copy. I've forgotten how to do a proper power calculation to figure out how many samples need to be taken for the CLT to apply in any given case, but I remember the rule of thumb was thirty samples. So, we're talking about 60 rounds per load in this case, right?
It's a PDF. If you send an email to [email protected], I will send you one back.

Sample size is highly dependent on variance and desired effect size to detect. Rules of thumb are very, hmm, imprecise, to put it kindly.
 
Wilcoxon rank sum test, but you need more groups.
I finally came back around to this, and I'm curious what you mean by needing more groups. Can you elaborate?

So far the best option I can see for comparing two groups of n shots is to measure the radius of each shot and then compare the two distributions with a Wicoxon Rank Sum test, which seems to be a popular recommendation for comparing means of small-sample non-normal distributions. But I can't find anywhere that someone has applied that test to two groups. In point of fact, I can't find any examples whatsoever of someone applying a theoretically valid statistical test to see if one group is different from another.
 
I finally came back around to this, and I'm curious what you mean by needing more groups. Can you elaborate?

So far the best option I can see for comparing two groups of n shots is to measure the radius of each shot and then compare the two distributions with a Wicoxon Rank Sum test, which seems to be a popular recommendation for comparing means of small-sample non-normal distributions. But I can't find anywhere that someone has applied that test to two groups. In point of fact, I can't find any examples whatsoever of someone applying a theoretically valid statistical test to see if one group is different from another.
Forget about testing groups. You are only interested in testing loads. The difference might sound a bit trivial and semantic, but it is not.
 

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