Target grouping "confidence"

[RAfiringline]

I recently got interested in seeing what kind of groups I could get from a new AR15 I bought, and I've been reading a few threads where accuracy and groups are a common point of discussion.

People write about groups they got, but rarely say how many rounds made up the groups, much less how many times they got that appx. group w/ x number of rounds.

Going to the range, and getting a g" group w/ 3 rounds just doesn't seem like a result that should give much confidence re. the rifle- ammo- shooter combination.

In statistics, you generally want at least 30 trials to have much confidence that the results you're seeing are really valid. When I tested my AR, I used 30 rounds, but I broke it into 6 5-shot targets since I also wanted to tune up the zero (maybe not the best practice when trying to arrive at an moa). Then I averaged the results across those 6 targets and called that the rifle's moa, with that ammo.

I know factors like barrel heating, wind shifts and all that enter into it the longer a shooting session goes on, but has anybody ever made a sensible "protocol" for how to determine a valid moa, for a given rifle- ammo- shooter combination?

[MrBorland]

I recall Bart B. has some good thoughts on the subject. Maybe he'll chime in.

I agree those exceptional 3-shot groups are usually just that, but they make us feel good, so they get stuck on the fridge and posted on-line for bragging rights. I've got my share. But I also shoot enough rounds to know what size group is pretty typical, and if I had to guess, for me, it's in the range of 10-20+ 5-shot groups before I'm confident enough to say "this load, this rifle = X MOA" and subsequent groups consistently bear that out.

[AllenJ]

I'm sure someone out there has made a "sensible protocol", I just don't know that many people that would use it!

Bart B has posted on this subject and had a lot of good things to say about it in my opinion. It has changed some of my thinking in terms of group size and what I'll call a MOA group without quantifying the number of shots in it.

[WWWJD]

Good groups: Good

Repeatability: Good

Hitting your mark 1st shot cold bore: WINNING.

[Unclenick]

Board member Denton Bramwell is a physicist who makes his living at statistical manufacturing and quality control. He told me that a sample size of 30 was a common standard before Fisher wrote the details of his analysis of variance (ANOVA) in 1925 and came up with the z-distribution and other tools. Afterward a lot changed. Denton said feels he gets enough information from a sample size of 15 at this point in time, given the tools available to work with it.

What a 30 sample group will do is give you a visual appearance of both sides of a bell curve, so it invokes more confidence in how well the mean estimates the mean of a very large population.

The problem with group size is, the larger the sample, the larger the group. So when you say how many MOA your gun can group into, if you don't also say how many shots it will group into that number of MOA, the statistic is meaningless. Groups get bigger because a larger random sample increases the number of chances lower probability points of impact have to occur. I have a plot of that below, using an average 3-shot group size of 1.00. 1.00 "what" doesn't matter—moa or inches at a particular range, or whatever. A real statistician may come by and find fault with it because of an assumptions I made, but the shape of the curve will be about right. It shows an 18 shot random group being twice the size of a three shot group. But watch out on that score, because it is the average 18 shot group that will be twice as big as the average 3-shot group. Individual groups can vary quite a bit. Especially the 3-shot ones.

Note that the sample size scale (x axis) is logarithmic because as the sample size get larger you are asking more and more rare events to add to group size and the bell curves tails accelerate rareness exponentially with sample size. 18 shots may double the average size 3-shot group, but, tripling it requires about 300 shot averages.



In general, a single 3-shot group indicates the next 19 out of 20 shots (95% confidence) will be somewhere within 2.5 times bigger and 60% smaller than your current group. For a 15-shot group its about ±15%. For a 30-shot group it's about ±10%. If other groups tend to all be the same size and not change that much, the source of error may not be truly random, but a bias of some kind. That can easily make the distribution non-normal so everything in my plot above then becomes invalid until you account for the bias. If a group is truly random, as ammunition test groups from machine rest barrels tend to be, only then does that plot hold.



Below is a computer generated 1000 shot 2 moa group together with three targets showing the first three, second three, and third three shots of the 1000. A fellow firing three, 3-shot groups like that from a gun could be forgiven for thinking he has a sub-moa rifle. And he does most of the time he fires three shot groups, even though the gun would be 2 moa with 1000 shots. And human nature being what it is, if the fellow gets a 3-shot group that includes some of the more rare holes further from the center, he'll assume he did something wrong because that's not how he believes the gun shoots. Well, it is how it shoots. Just not most of the time.



Another thing to notice is the green X's that are the center of each 3-shot group. See how they wander around. The yellow cross is the center when all 1000 shots are included, so it's a good estimate of where an infinitely large sample would be centered, and since you never know which one of the future possible shots is coming up next, the yellow cross is where you want your scope zeroed for the best odds of hitting what you shoot at. But can you find that yellow cross from the three green X's? Not really. You get closer if you combine all nine shots, though it's still going to be low and slightly left. But the more shots you include the closer you tend to get, randomly. You could shoot a couple more 3-shot groups, and there's a chance they'll be high and right and tend to pull the center in a little better.

That moving around of the center of the the center is evaluated by a statistic called the standard error. It is the standard deviation of the standard deviation among multiple 3-shot groups. It tells you how much the mean will vary around the center, as if each group center represented a single combined bullet hole. It is equal to the population standard deviation divided by the square root of the sample size, so it's a number that gets smaller with group size, but shrinks slowly as group size grows, just like the growth in group size got slower as sample size went up in the first graph. 12-shot groups will wander around half as much as 3-shot groups do because the square root of 12 is twice the square root of 3. Your 30-shot groups, on average, will have their centers wander around 3.16 times less than the 3-shot groups do. So this is a measure of how well your group represents the perfect mean value of a theoretical infinitely large number of shots.

There are different strategies for getting a group size number that doesn't tend to change as much with sample size. Such strategies simply discount the influence of the outer-most shots with their lower probability. One commonly used by the military is the mean radius, but in test specifications a particular number of shots is given for it to be good with, as it will grow some with sample size; just not as much as group size does.

The method that should theoretically transcend sample size is the radial standard deviation, but the transcendence is limited by how well it you can estimate population mean and standard deviation from your sample size. Remember, standard deviation taken from a sample is always an estimate of the standard deviation for the whole population of shots ever to be fired in that rifle. The limitations are that real guns wear, so their precision changes over time, and more importantly, small sample sizes doesn't give you a very good estimates if what some theoretical infinitely large population would be like. Which of the those three shot groups I put up in the second image would you try to make these infinite group mean and SD estimates from?

So, if you use small samples you really have to use strategies for determining standard deviation that are more realistic than the usual sample standard deviation calculation can make with small values. Student's T-distribution can be used to get the standard error influence included. Square root bias compensation for small samples has to be used, and so on. Even with those enhancements, you'll want to use several small groups and not just one.

The bottom line is that there is no such thing as an X-moa rifle without a shot count specification. You just have a rifle that averages X moa for Y shots, or Q moa for R shots, etc. If you can accurately estimate radial standard deviation you will have a number that theoretically remains constant regardless of group size, but it depends on the quality of the estimate.

Again, I'm not a real statistician and someone who is should be better able to address the tools I mentioned and whether or not I'm even citing them entirely correctly. I just started from one basic statistics class in college, and then did a bit of further reading. So, caveat emptor, but this is what is happening as far as I can tell.

[FrankenMauser]

I, personally, shoot at least 5-shot groups now. I grew up with the standard 3-shot group, but found that mentality to be fatally flawed in my own testing.

It is VERY easy to cherry-pick a great 3-shot group from an assortment of good-to-mediocre (or terrible) 3-shot groups, and then proceed with load development (or bragging) under false pretenses.
But, adding just 2 more shots to the group equalizes things quite a bit.

Increasing to 10-shot groups tells the story in even more detail.

About 10 years ago, there was a magazine article about the statistical relevance of various common group sizes, including 2-shot, 3-shot, 5-shot, 7-shot, and 10-shot groups. As part of the article, the author fired several 20-shot and 100-shot groups, to see how well the smaller sample sizes represented the whole. I didn't find the article particularly interesting, due to the author using a fairly mediocre load (3-4 MoA) for testing, but it did get me curious about some of my own loads.

Since then, I've fired several 20-shot, and 50-shot groups, and even a 100-shot group, with a few of my rifles.
Not only does it give you great insight into the consistency and predictability of a given load, but it helps you, as a shooter, get a much better feel for when it was YOU that screwed the pooch. You get better at calling the impact, calling flyers, or admitting that you simply screwed up.
You also see clusters within the group, that, with further testing, may help you eliminate some of the randomness.

As time goes on, and I want to know the truth more, I find my sample sizes increasing. A few weeks ago, I was testing a new load for my 6x45mm AR. I fired 5-shot groups for pressure and function testing, but for accuracy I fired primarily 10-shot groups.

The bottom line for me comes after I've become familiar with the rifle. If it's showing 0.687" at 100 yards with 10-shots, and it FEELS right, then I have confidence in it. But, if I'm not confident in my performance behind the trigger, then none of the holes in that target mean anything....

[kraigwy]

The more shots the merrier regarding getting a zero.

The RO mentioned 30 rounds, that would be good, shoot 30 rounds at the target, then divide the target into 4 equal quarters. Count the rounds in each quarter.

Adjust and do it again until you have an equal number in each quarter.

--------------------------------------------------------------------------------

The best advice I can give for seeing how the rifle/ammo/shooter shoots accuracy wise it to:

Set out a target and leave it.
In the morning shoot one round at the target and quit.
At noon shoot one more round at the target and quit.
At night, shoot one more round and quit.
Keep doing this for about a week.

At the end of the week you should have 21 rounds on the target.

Now measure your group to determine what you true accuracy potential is.

[RAfiringline]

>The best advice I can give for seeing how the rifle/ammo/shooter shoots accuracy wise it to:
Set out a target and leave it.
In the morning shoot one round at the target and quit.
At noon shoot one more round at the target and quit.
At night, shoot one more round and quit.
Keep doing this for about a week.<

Yeah, determining a cold barrel moa, versus a cold, warm, hot barrel range.

[Palmetto-Pride]

This great advice for a hunting rifle or a tactical sniper rifle where that cold bore first shot is critical, but for a target rifle that gets shot multiple times per range session most people probably tend to zero there rifle to a hot bore. You leave the range and your zeroed to your hot bore you come back and your cold bore zero has shifted.......most people start turning those turrets because they think there scope is off. As they start shooting and there bore heats up again they usually end up turning those turrets right back to where they left them the last time they left with a hot bore......

[precision_shooter]

I do 5-shot groups and never really counted the number of groups I shoot. It's spread across many many range sessions. My rule of thumb: 3-shots measures the rifle, 4-shots measures the load/cartridge, 5-shots + measures the one pulling the trigger. If it's not shooting good in 3-4 shots, the 5th, 6th, 7th won't make it any better...

[hooligan1]

I agree with everybody here really, but I test and prove loads like this;
1.Load four different BTO, measurements with the manuals most accurate load with the powder I want to use. In these loads I use 3-shot test to determine if the BTO is even close to decent.
2. After determining a BTO measurement, I will then and only then, load ten rounds and try to prove the measurement is the one.
3.When BTO is determined to be decent, I will add powder in 1grain increments until group spreads then I go back and find the most consistent of the lot.
I've found that it's faster then testing powder weights then finding BTO, and that saves time and money for new components....sometimes I can't match the components of a Book load, so I experiment til I'm happy to stick with the load.



Sometimes I find a super nice load after firing only 13 rounds.

[RAfiringline]

>You leave the range and your zeroed to your hot bore you come back and your cold bore zero has shifted.......most people start turning those turrets because they think there scope is off. As they start shooting and there bore heats up again they usually end up turning those turrets right back to where they left them the last time they left with a hot bore<

...I've probably been there, I've probably done that...

[RAfiringline]

Here's another statistical treatment in which the writer thinks a 10 round group is the way to go.

Statistics, Shooting and the Myth of the Three Shot Group

"Again, we find that the actual distribution of shots is easily twice the size suggested by our 3-shot groups. When using a 10-shot group of the same extreme spread (like we did before), however, we find that the uncertainty in the mean value drops off very quickly. Compare the following figure to one at the top of this page:"

https://precisionrifle.files.wordpre...shot-group.pdf

[Unclenick]

I should have thought to mention before that I'd run into a site called statshooting.com. A number of interesting papers are available there to read.

[Art Eatman]

I grew up with the five-shot group as a standard. It takes a string of sub-MOA 5-shot groups before I feel safe in calling the rifle/ammo a sub-MOA package.

Once I've established a base line for a particular rifle, I use three-shot groups to test a sequence* of new loadings, for sight-in of a new scope, or just for a check before going hunting.

* Different powder charges with a particular bullet or different bullets with an "old standard" powder charge.

[22-rimfire]

If I am checking zero on a scope, three shots are enough for me with centerfire. IF I am confident with my shooting, I will adjust a scope after one shot. This is for hunting rifles mostly and ones that I seldom have any desire to shoot. Check rifle, check zero and mechanics... head to the woods. But I do know my limitations and won't take a 300 yd shot. Never had a 300 yd shot in my life except on wood chucks and that takes more practice and more rounds.

With 22LR, I shoot 5-shot groups. I would rather shoot two 5-shot groups than one 10 shot group simply because I am often blowing out my aiming point and I begin to aim at bullet holes.

[Unclenick]

22-rimfire,

That's a good post. It brings up the basic point that there is no such thing as a statistically insignificant group. It's just the confidence you can have in how well a group foretells future performance that increases with the number of shots. If the confidence from 3 shots is adequate for your purpose, there is no need to shoot a group that is larger.

It works out that, group size and standard error included, a 3-shot group gives you 95% confidence that future shots won't impact outside a circle about 3 times bigger than that single group. So, if you get a single 3-shot group that is 1 moa, want to stay in a 10" kill zone, and you zero at 200 yards, your 10 inch point blank range with a perfect hold and trigger control is still going to be all the way to almost 250 yards with a lot of common .308 loads, about 19 times out of 20. And about 14 out of 20 will group to about 1.5 moa.

[Mobuck]

I normally use 3 shot groups. Three shots from a hunting rifle is enough to determine either POI or accuracy in a hunting situation. With a sporter weight rifle every shot increases the "human factor" of the group size since those type stocks aren't designed to sit on the bags uniformly.
Another thing to consider is what you're trying to accomplish. Are you testing ammo, rifle, or shooter? Truthfully, an AR isn't the easiest rifle to shoot. When I find a load that is getting cloverleaf 3 shot groups, I'm done.
Regarding cold bore zero: I have a 100 yard range right outside my yard gate. I check cold bore zero by going to the bench and firing ONE shot per day leaving the rifle(s) in outside dry storage overnight. After the third day, I'm pretty sure where those cold bore shots will land. Alternately, I fire one shot and drag a bore snake through the barrel before letting it set overnight. Again, after 3 days, I know where the first shot will land. Then I can fire the rest of a 3 or 5 shot group and determine the difference (if there is any).

[Rob62]

Lots of really good info here including some that just went over my head.

To judge a rifles accuracy I only use 5 shot groups as a baseline. The next thing to know (for me) is where the cold shot will go. My thought is that it does me no good to have a rifle that shoots 5 - shot, .534" groups at 100yds if I can't get the cold shot to my point of aim.

Regards,

Rob

[Unclenick]

Originally Posted by Mobuck

After the third day, I'm pretty sure where those cold bore shots will land.

The difference between that and the statistical approach, is the latter allows you to quantify "pretty sure" to some specific number of times out of ten or 100 or whatever your interest might be. Like probability statistics, feeling "pretty sure" is actually a form of estimate. It never guarantees what any particular shot might do, but it tells you whether or not the odds of it doing what you want seem to be in your favor. The human brain can sometimes make very good estimates. Other times it can be fooled, but I think that varies by individual experience. On the plus side, the brain tends to fool itself more about the validity of less frequent outliers than it does about the larger two thirds or so of shots.

The one shot per day approach is not convenient for those of us who have to drive to the range. For a hunting rifle, I always shoot three in rapid succession onto targets one, two, and three on the target backer. Then I clean the gun while it cools and do it again. When I've got five holes on each target, Excel scatter plots will give me a trendline so I can see how my cold shot and two follow-ups tend to move. I usually do this from prone, and repeat the exercise from sitting to see if that shifts anything. The bench has its uses, but if the gun is to be worked in field positions, that is the way to test it for sight adjustment, I think.

[4runnerman]

I do mine different than most here. I take a 2 ft by 4 ft chunk of cardboard.
My targets are 1/2 inch circles with a small black dot in the middle. I will set it out 100 yards ( 223 only). I will fire 2 rounds in to each target. Both rounds need to touch each other and need to be in or tear the black dot. The issue with firing 20 or more rounds into one target is ( as others have said) You shoot the bullseye out and now you are putting bullets through a hole in the paper. You don't know where they are hitting, In essence- you are wasting bullets for a target that you don't know where they are going. When I can line 10 targets across and put 2 rounds in each one-In the black or tear it, then I am done.
Groups are great, but only if they hit the bullseye.

[overthere]

10 shot groups while doing load development but the real test (for the rifle/load) is 20 shots for record on a calm day.

I used to do 5 shot groups but start shooting 10 and what might start out looking like a so-so group could end up being the best of the bunch, as opposed to a 5-shot 'all touching' deteriorating into mediocrity.

I'd shoot 20 shot groups if it wasn't for time and cost (components and barrel life).

[FrankenMauser]

The issue with firing 20 or more rounds into one target is ( as others have said) You shoot the bullseye out and now you are putting bullets through a hole in the paper. You don't know where they are hitting, In essence- you are wasting bullets for a target that you don't know where they are going.

There's a really simple solution to that "problem" ... and a lot of people do it.

It is quite common for people to sight in high or low, and use a different point of aim than the actual center of the target.
This style of benchrest target is a perfect example: EZBRT

A simple square without the bullseye would be just as effective, but a lot of us are lazy and use what's easy.


I, for example, use similar benchrest targets for a lot of load development and practice with my hunting rifles. ...But I turn the targets upside down at 100 yards. That gives me a point of aim below the bull. For "flat-shooting" cartridges, the impact will be pretty close to the center of the bull. For some other cartridges, it may be a little lower (.458 SOCOM) or much higher (.35 Whelen).
Regardless... the point of aim does not get shot out, but all of the shots are still on a clean area of the target, and I get my groups.

Here's an example with a varmint cartridge (so POI is closer), during load development.

(If you wondering about the semi-photoshopped look... its from scotch tape being put over the holes - front and back.)

---
And, although I'm sure many people will find it a little strange, sometimes when shooting at 50 yards or ~200 yards, I turn the targets sideways, because many of my rifles will shoot out the aim point at that range. Turning the target sideways puts the bull on the left side, point of aim on the right side; and I adjust the zero to move the point of impact horizontally (be it 2 clicks, 4 clicks, 8 clicks, etc.).

[Mozella]

I too was struggling to find increased accuracy for my bolt action .223 until I stumbled upon this new and exciting way to evaluate groups. It's done wonders for improving my accuracy.

I shoot five shot groups and then toss out the two worst shots, leaving me with a great looking three shot group, at least most of the time. Then I cherry pick the best of these groups and destroy the bad ones. Finally, I brag long and loud about the good ones.

This method is cheap, easy, and I guarantee your accuracy will improve if you use it.

Judging from what I read on the Internet, some of you already are using this method.

[Road_Clam]

One of the aspects of successful "real world" recreational shooting IMO is it's all about AVERAGING , and logging your data to obtain both positive and negative trends. I am NO master marksmen, I'm an amateur mid range precision shooter and I've only been hand loading for about 4 years (and still learning). My shot group method consists of shooting two separate 4 shot sessions , and I triangulate the best 3 shots within each session. I shoot a total of 8 shots letting the barrel cool down after the first 4 shot session. I also calculate all my data shooting at 200 meters rather than the standard 100 yds as my groups are more opened up and thus more measurable and it's easier to define vertical stringing. I do not calculate cold bore shots as they are always off a bit. I also use a chronograph and a custom made bench top type rest to keep my rifle as steady as possible. You want to keep shooter mechanical inconsistencies out of the equation as much as possible. I always favor a consistent less accurate average load vs a one time highly accurate load that has a history of wider inconsistencies. So for example if I have a load that consistently shoots .7 MOA vs another load that shot a .22 moa and then shot a 1.8 moa I will always favor the .7 moa load. I have over 280 test sessions logged in my Excel shooting journal and the data is a HUGE help in consistiantly advancing my skills. This post is strictly just my opinions on what seems to work very well for me..