My point is that you can't tell how accurate a given gun is (mechanically) with only a 3-5 shot group any more than you can accurately calculate the average height of the kids in a high school by pulling 3 of them at random and measuring.
The sample size is just too small.
If you want to demonstrate accuracy (in a statistically valid fashion) with only 3 shots, then they essentially have to all be single hole or darn close.
Follow me through here with an example (slightly different methodology, but you'll see it doesn't matter).
Let's say you shoot three shots and measure the distances from a center pinpoint which was your point of aim. Your results:
Shot 1: .5"
Shot 2: .25"
Shot 3: .25"
One would be tempted to say that you have an average error of .33", so not too bad. But these three shots are just a sample of thousands of possible locations and outcomes. What we're trying to discover is not how good a single group is, but what is the likely error of EVERY shot? What would a 3-shot group that seems pretty good tell us about what we could expect a 100-shot group to look like?
If you analyze this using statistics that account for the small sample, you find that the true average of all rounds (based on this sample) could be anywhere from 0 (an infinitely small error-- perfect accuracy) and 0.69". All values between 0 and 0.69" are equally likely (95%) to be the true average of all the rounds you will shoot. Think of this as the center of your group relative to the center of the target. Would you adjust your sights or not? How can you tell if you are dead center or off by over a half inch? You can't.
Then there's the spread within the group- or the average distance from the center of the group to each actual hole. This average error ranges from 0.075 (3/40ths of an inch) up to almost a full inch.
So here's what your three shot group tells you about what 100 rounds might look like:
-- There's a 95% chance that the 100 round group might be centered on the bullseye, or off .69"
-- There's a 95% chance that the 100 round group would have 68 rounds within 0.08" or within 0.9"
--There's a 95% chance that the 100 round group would have 95 rounds within 0.16" or within 1.8"
-- There's a 95% chance that the 100 round group would have 99 rounds within 0.24" or within 2.7"
So, the reality is that your 3-shot group indicated you MIGHT have a gun capable of 1/4" groups all day long, OR you might have a gun that shoots almost 3" groups all day at the same distance. They are equally likely.
Where I come from a 3" group and a 1/4" are different enough that I'd say that if my "testing" says they are equally likely, then I need a better test.
More info here: 68-95-99.7 rule - Wikipedia, the free encyclopedia
I get your need to use stats. But your missing the point behind why most guys measure groups. Why? Because if you shoot 100 shot groups with some larger calibers you could ruin the barrel from overheating for one reason.
When i shoot groups its usually to test ammo...no need to shoot more than 5 of each type ammo to tell me what i need to know.
I also shoot all my loads thru a chronograph and use a Kestrel weather meter. I record, group size, load data, velocity of each round, and all the weather details in a Shooting Log Book. So i can look back thru my Log Book and tell you how my guns performed with a given load and weather condictions...much more useful info for me.