How Many Groups to Measure Accuracy?

[denton]

The question keeps coming up, how many groups do you need to shoot, to know the accuracy of a firearm?

I didn't run the math for all possible scenarios, just for five-shot groups. The calculations also assume that you have removed all causes for things like stringing and shifts: Loose scopes, barrel rubs, etc.

The math is for groups measured using "extreme spread", or center to center distance between the two most distant holes.

For individual five shot groups, a groups as small as half of the long-term average, and as large as 1.5X the long-term average are common. Within that range, a single group is not giving evidence of real change of performance for the rifle, ammunition, or shooter.

If you shoot five five-shot groups, and average the ES, your estimate of group size can be depended on to about plus or minus 25%. If you do 10 groups, the result can be depended on to plus or minus 18%.

Example: You shoot five five-shot groups that average 1". You now know the true long term average from the firearm is between .75" and 1.25".

The actual percentages aren't quite symmetric... the plus percentage isn't quite equal to the minus percentage, but the numbers I've given here are simpler to remember and completely adequate for practical use.

[MarkL]

Thanks, Denton, that's good to know.

Do you have any thoughts on using the Averge Group Radius method? My gut feel is that it would take fewer groups/shots to get a result that is fairly close to the "long term average".

For those that are unfamiliar with AGR, you average the X and Y coordinates of each hole to find the group's center, then average the distance of each hole from that. Therefore, it takes ALL the holes into account, whereas the standard Extreme Spread method only takes the worst two into account.

[denton]

Theoretically, any system that accounts for the distance of all points from some reference, contains all the global variation information that is available. AGR contains all that information, but structures it awkwardly. In theory, it is better than extreme spread.

For five-shot groups the information content of range ("extreme spread") is 90% of theoretical maximum, so the practical difference is not worth the extra effort.

In effect, extreme spread contains practically all the available information, when n is small.

By the time you get to n=10, the effectiveness of extreme spread is down to 60% or so. That is the point where most statisticians kick over to using standard deviation, which has 100% information content.

[Cal Sibley]

Two five shot groups fired behind each other is enough for me. Otherwise I may be factoring in shooter fatigue, changing range conditions, barel heating up and other factors. I'm not a math major and don't try to reach a finite point in measuring. Just one mans opiion. Best wishes.

Cal - Montreal

[Brithunter]

Hi All,

Now I jave an article somewhere which shows how using a "Master target" will provide the true group size of your rifle. You put two target of the same type exactly over one another and then shoot a group on the tope one, then you change the top target and do this at least twice more that session/day.

Now you save the master target and repeat using it with one group over period of weeks until you have shot 10 goups on it using a new target each time over the top. Numbering the top targets as you go. Once you have shot you ten targets the "Master Target" will show the true grouping of you rilfe ammunition combination over an extended period and it seems that what we might call flyers normally come into this larger true group so you do not discount flyers. Of course you have to use the same type of ammunition over this trail to make it mean anything.

[MarkL]

I don't think flyers should be discounted altogether, but I also don't think they should be given more weight than the other shots. That's why I'm dissatisfied with the Extreme Spread method and drawn to the Average Group Radius method. If you shoot a 25-shot group and measure the two widest shots, they will probably be fliers and all the other shots will be ignored.

Suppose you fire 9 shots into one tiny hole and the tenth shot lands 2 inches away. ES says that's a 2 inch group, same as a group in which the shots are randomly distributed over 2 inches. AGR will tell you that the first group is better because the lone flyer only increases the average by about 0.2 inches. For most purposes (e.g. prairie dog hunting), I'd rather have most shots going into one hole with an occasional flyer, than have them randomly distributed.

Suppose you're a bullseye-type shooter, where the target has scoring rings and you add up the number of shots in the 10 ring times 10 plus the number of shots in the 9 ring times 9 and so on. That's actually an approximation of AGR (assuming the point of aim coincides with the center of the group). All shots count with equal weight. One flyer won't completely blow your score if most of the shots hit the center.

Also, I recently learned that another name for AGR is Mean Radius. Googling for MR turns up more hits than AGR. Apparently, MR is used by many military organizations (e.g. the U.S. Army) and their vendors to rate the accuracy of small arms and ammo. I recall reading about a military test in which something like 100 rounds were fired at a target to measure the MR.

That brings up another point about AGR/MR. You only need one group if you fire enough shots. More shots make the result more reliable (as does firing more groups when using ES). If you get a flyer when shooting a group for AGR/MR, there is no need to groan and assume it's a blown group. It will increase the average some, but subsequent shots can shrink it back. If more shots produce more flyers, they will be counted appropriately.

The main problem is that you must be able to distinguish the individual bullet holes. The 100-round test I mentioned above was conducted at long range (600 yards I think) which was probably sufficient to keep the holes from merging. For those of us shooting 10 or 15 shots, I would think 200 or 300 yards would be sufficient. Express the result in MOA so groups fired at different ranges can be compared. If you're getting one ragged hole at those ranges, I don't want to hear any complaints!

[denton]

MarkL

Your points are quite valid.

If you are going to shoot a large number of shots into a single target, then it is very clear that ES is not the way to go. As you point out, one blown shot changes the measurement a lot, where, if you are averaging, one blown shot doesn't change the result much.

The problem with MR is that it is constructed in a way that makes it awkward to do anything with other than a simple comparison, even though it does contain all the available information. Fundamentally, it's what us statisticians would call a "mean absolute deviation".

You still get the effect of averaging if you shoot several five-shot groups. In that case, you're averaging groups of five, instead of individual items. As I said, for groups of size 5, you get practically as much information as you do with the global measures, such as standard deviation.

Whether you should discard fliers or not depends on which question you are trying to answer. If the question is, how does the gun itself perform, and if you know that you flinched, or closed your eyes, then the point can be removed. If the question is, how well does the gun-shooter system perform, then you really flinched or closed your eyes, and the flier stays in.

Some people just routinely remove fliers, and that's a big mistake. Guns do generate fliers, and fliers are part of the characteristics of the gun.

In the end, you should use whatever method most conveniently answers your questions, or that you like. I don't think there is a "right" or "wrong" way, just more or less useful or convenient ways.