What's a shot group?

you have a sampling of values.

you can't use that sample to predict what another such sampling WILL BE. that is, you can't be 100.000% sure.

but, you can use the average and standard deviation, to figure out what is the most likely distribution, at a certain degree of certainty.

that's called a "confidence interval". when added to both sides of the average, that is called the "confidence limits" (CL).

you calculate the confidence interval with the following:

Z * s / sqrt(n)

where
Z is a number from a table of what confidence you want to have
s = standard deviation
n = number of samples in the group

the table is:
80%, 1.282
90%, 1.645
95%, 1.960
99%, 2.576
99.5%, 2.807

the full table is much bigger than that, but this is just to give you an idea. far and away the 95% confidence level is suitable for most purposes. that's what most people use.

as you can see, a larger confidence interval results in a larger Z value. the spread will have to be bigger if you want to be more confident.

so for example, you have a group of 5 shots:

1.24"
1.86"
0.83"
2.37"
1.45"

the average is 1.55", the standard deviation is 0.59".
for 95% confidence, Z= 1.960
then from the equation above,

1.960 * 0.59 / sqrt(5) = 0.52"

therefore, given the sampling you have observed, 95% of the shots will be 1.55" plus or minus 0.52".

do you want to be sure? then use 99.9% confidence. Z= 3.291 and the CL is +/- 0.87

do you want to be sure sure? absolutely positively sure? then use 99.999% confidence. Z= 4.417 and the CL is +/- 1.16"

at that point, you can be 99.999% confident that you won't have a shot more than 2.71"

In a scientific study, a control group is used to establish a cause-and-effect relationship by isolating the effect of an independent variable. Researchers change the independent variable in the treatment group and keep it constant in the control group. Then they compare the results of these groups.

I apologize for the interchangeability of the word group here.
Once I assemble a given set of components, I fire a test in a given firearm, with specific conditions. Three, five or whatever group of shots; sometimes only one. As in gathering cold bore shot data. (Documented with a shooter's log)
When I am satisfied with those results, it becomes my control group. Being able to duplicate those results at a later time, and the occasional failure to do so, helps me sort out variables, such as wind, light, shooter fatigue or component differences, such as lot numbers, brass dimension changes, or firearm wear.
I have driven myself to the point of distraction doing this with some rifles, but it is a worthwhile process.
At some point, having the shots cluster around the point of aim, be it a front sight, or crosshairs can be more easily tolerated when the job at hand is in perspective. I am good when a shot group is 1 1/2 or 2 inches when shooting large game. Not so much when the target is prairie dogs at 300 yards, or a benchrest target.

With that, I am not sure if I have joined the discussion or caused grounds for another argument. IMO nothing can be 100% where a human being is involved as the operating system.

I just received this timely PRB email today.

https://precisionrifleblog.com/2020/12/12/measuring-group-size-statistics-for-shooters/

Varminterror said:

So many statements about statistical validity made in this thread, and frankly, they all imply the one making the claim fails to understand statistics.

Hint: confidence interval is dependent upon the population. Saying 4 is good enough or 10 is good enough might be wholly meaningless. We have empirical results based on VERY specific rifle types and conditions (benchrest rifle results, as an example) and we have mathematical simulation results, but we don’t truly have a consistent system of multivariate boundaries which universally apply to all shooters with all rifles. I can know, based on literally hundreds of repetitions of groups fired with the same rifle at the same targets under the same conditions, any given 3 shot group has never demonstrated, and in all reasonable probability WILL NEVER demonstrate the extreme size of any given single 10 shot group, and equally, have never and will never see any of these 10 shot groups fall smaller than the average - let alone the raw smallest of the 3 shot groups. Possibilities and probabilities are very different things.

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If the only thing that matters is extreme spread then you have a good point. I have actually done the shooting and measured the result of the difference in extreme spread in large vs. small shot groups and found no significant difference if the results are recorded on a single target using a common poa. I have done this just a few times so it is possible that the results are misleading, but I doubt it. However, extreme spread is only one of the things that can be learned from the data as I am sure that you have learned from your reading. Perhaps we are looking at this from different perspectives. Some of the well meaning statements made in this thread may not be mathematically sound but to assume and imply that there are no people here that understand statistics is a faulty assumption indeed.

I think that we are ruining this thread with discussions on statistics when the question is one of a more practical nature. Precision limits, confidence levels, sample sizes, tainted results, etc. are likely not what was sought when the question was posed.

To the OP: My experience tells me that shooting multiple five shot groups (or three shot groups) and aggregating the results will tell you what you need to know.

Laphroaig said:

I just received this timely PRB email today.

https://precisionrifleblog.com/2020/12/12/measuring-group-size-statistics-for-shooters/

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That article is really good. By reading through this thread, and reading that article in full, one can get a pretty good understanding of what you should be trying to achieve by shooting groups, recording your information, handloading new loads, and trying agin.

Statistics was invented to provide employment for those of us that are handy with numbers, but don't have enough personality to go into accounting. With that, I hope you'll pardon my unusual enthusiasm for this topic.

The esteemed Mr. Litz, quoted by Laphroaig, makes an arguement that is conceptually correct, but mathematically wrong:

“Measuring the extreme spread of shot groups is quick and easy, but it’s not actually a very good measure of dispersion. What do I mean by a good measure? A good measure should give you useful information, which is information you can use to make good decisions. When you look at the extreme spread of a 5-shot group, that measurement is determined by only 2 out of the 5 shots. In other words, only 40% of the shots are considered in the measurement. Even worse, for a 10-shot group, a center-to-center measurement is only using information from 20% of the total shots. Since the extreme spread, center-to-center measurement, is determined by only a small portion of the total shots available, it’s just sort of an indicator of precision.” – Bryan Litz

What shooters call ES, or extreme spread, the rest of the world calls range.

If you compute a standard deviation, you have taken into account the difference between each point and the mean. It contains all the information we have about dispersion (lack of precision). And Mr. Litz is correct that that by going to range we give up some of that information. The question is, how much do we give up?

If you have only two points in a subgroup, the range contains 100% of the information, just as standard deviation does. If you have five points, the range contains 90% of the available information. When you get to 10 points, the information content drops to 60%. So the rule is, for subgroups of five or fewer, range contains practically all the available information.

Since subgroups of five or fewer contain practically all the available information, there is no good reason to search for more complex solutions, and you can safely ignore everything in the article that follows the assertion quoted above.

Technically, five three-shot groups are slightly better than three five-shot groups, but the difference is small. In practice, the average of three five-shot groups will get you your long term average five-shot group size within plus or minus 25% or so. And that is good enough for practical decision making. To cut the error to plus or minus 12.5%, you have to shoot 12 groups, so three is a sensible operating point.

Seamaster31 said:

If the only thing that matters is extreme spread then you have a good point.

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Re-read my post. I shared a confidence interval analogy for a sample set of extreme spreads, but the fact remains - the confidence interval of a sample set is dependent upon the sample set. Whether your sample set is a series of extreme spread measurements, the radius from centroid, deviation from POA, or whatever quantitative measure you choose, the confidence interval for the sample set remains dependent upon the sample set. An incredibly consistent and repeating sample set with a smaller range and variance will always require less samples than a set with high variance to establish a more robust confidence interval.

As a rule of thumb, shooters and reloader’s are terrible at stats. We see so often guys talking about, “I have an SD of 8 and an ES of 18”... If they did understand the nature of their data, they’d know their results reveal their dataset is insufficient, even without ever considering a T test or Z test.

Hookeye said:

Some people think stats are cool.
Of course they may not be observing of all inputs/variables.

IMHO just shoot 5 shots, letting things cool a little between shots (you and the rifle).
Repeat 2 or 3 times for verification.

If the same, good to go.

Unless you are in some long string match and have to learn where accuracy starts to degrade, it aint worth it.

Of course you should shoot for cold shot vs warm, and know if your rig needs a fouler or two.

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Agreed. I had an ultra light hunting rifle that would not group above two rounds. The third usually opened the group. However the cold bore shot and second shot would consistently land next to each other touching. For practical purposes, it grouped amazingly

WrongHanded said:

Perhaps a better way to do it would be to measure the distance from POA for each shots, and then work out the average of those numbers.

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This theory is only applicable if we know the POA perfectly represents the centroid of the theoretical infinite group. Such in almost all real groups, it’s not applicable, but especially not so when group spread is similar to the adjustment increment of the optic.

Varminterror said:

This theory is only applicable if we know the POA perfectly represents the centroid of the theoretical infinite group. Such in almost all real groups, it’s not applicable, but especially not so when group spread is similar to the adjustment increment of the optic.

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That is something I considered. Though I suppose we could use the group dispersion itself to locate the center of the group, and work from that.

The point is, how tightly a group is configured may be more important than measuring the distance between the two shots that are farthest from one another.

If 8 of 10 shots are in one hole, and the other two are 1.5" apart, is that better or worse than a 10 shot group of 1.25" where most shots are 0.5" or more from the center of the group?

I suppose it depends on what a person expects from their rifle/load combination.

Most hunting rifles could not shoot a 25X1 continuous group without heating up and go off point of aim. But these same rifles could shoot 5X5 groups with a cooling period between each of the five and retain point of aim.

WrongHanded said:

If 8 of 10 shots are in one hole, and the other two are 1.5" apart, is that better or worse than a 10 shot group of 1.25" where most shots are 0.5" or more from the center of the group?

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We have statistical metrics as standards for predictability which let us compare sets like this. The set determines the stats - so comparing the two, we’ll have range, averages, (a centroid), standard deviation, variance, and ability to determine confidence intervals to determine the relative stability of the two sets.

What’s disappointing is that MOST shooters and reloaders will share your implied sentiment that the group made up by a cluster plus a couple of flyers is inherently better than the more evenly distributed group. But in reality, statistically, this little hypothetical comparison demonstrates a common trap into which too many shooters fall. Guys will chase solutions to those flyers in the first group and give up on the second, even if the true consistency, the true predictability of the second is better.

Varminterror said:

What’s disappointing is that MOST shooters and reloaders ........Guys will chase solutions to those flyers in the first group and give up on the second, even if the true consistency, the true predictability of the second is better.

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Its been stated in this thread and in the linked article that a lot of people shoot maybe 5 rounds and circle the 3 they like for their sub MOA group, thus discounting the flyer(s) as “Whoops, that was me” or “Wind push”, etc.

I agree with the “discounting” statements, based on my interactions with other shooters. At the competition level, I tend to believe the shooter, however.

3Crows said:

Most hunting rifles could not shoot a 25X1 continuous group without heating up and go off point of aim. But these same rifles could shoot 5X5 groups with a cooling period between each of the five and retain point of aim.

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The question of the thread is inherent to this hypothetical proposal:

You’re asserting that dividing into multiple groups implies shooting slower, on a cooler barrel, which you assert would reduce POI shift (forgiving the fact you mistyped point of AIM instead of POI).

If the shooter fired the 25 shots, 5x5 vs 1x25, in the same amount of time, with the same temperature conditions of the barrel, which groups will be smaller? All of the 5shot groups? The Average of the 5 shot groups, or the 25 round group?

Varminterror said:

What’s disappointing is that MOST shooters and reloaders will share your implied sentiment that the group made up by a cluster plus a couple of flyers is inherently better than the more evenly distributed group. But in reality, statistically, this little hypothetical comparison demonstrates a common trap into which too many shooters fall. Guys will chase solutions to those flyers in the first group and give up on the second, even if the true consistency, the true predictability of the second is better.

Click to expand...

From my own limited experience, flyers are often my fault. So whilst I don't simply discount them (unless I'm confident that it truly was my fault), I do take into consideration the possibility that I may be responsible. From that point, I will commit to further testing with that load.

So I understand what you're saying. But in my personal experience (which I also understand is vastly outweighed by your own), the overall group has some merit to me, not just the distance between the two shots spread farthest from one another.

@WrongHanded - I tend to consider myself responsible for every round that leaves my barrel. “Missing onto target” or landing a flyer into my group (bad reticle picture at the break, but the flyer lands within the group) does happen. We all want to think accidents only elicit error, but in reality, bad breaks can work in any direction, so half of the time, they’re working towards the “good,” not always towards the “bad.”

As I have mentioned many times in this thread, there are variables acting on our group sizes which do not follow “normal” Gaussian distributions. A beautifully tight group can be wrecked by an anomalous poor trigger pull, or breaking the shoot with an incorrect reticle picture (POA, hold, what have you). At the end of the day, outside of scoring for competition, shooting for groups is a means of data acquisition for root cause analysis. If the RCA reveals a loose trigger finger, great - demonstrate it, eliminate it, and you have a new data set without the anomalous influence. If I call a flyer in a group during testing, I shoot it again. I also shoot 10-20 shot groups most often as comparison - I know my group size MUST grow shooting more shots, so if I see a flyer in 3 out of 5 groups of 5 shots, and all of them are within the bounds of my 10 shot or 15 shot group, I admit to myself, either I’m shooting like crap - which is part of my normal error probability then - or those “flyers” actually aren’t flyers at all and actually do represent my load+rifle+shooter system.

But I understand the point you’re chasing - and again, the stats really don’t change. You can establish a sample set centroid (as I’ve mentioned multiple times in this thread), then measure a mean radius from centroid, but at the end of the day, variability is variability. 8 of 10 in a bunch with two bad flyers can remain to have a higher standard deviation and/or higher variance than a more evenly distributed group. But for some reason we all pretend flyers are ALWAYS flyers, even when they happen to every group we shoot...

Ru4real said:

that a lot of people shoot maybe

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I would not say "a lot of people" in reference to shooters; in reference to internet bravos, perhaps--and yes, that's splitting hairs.
I like to think that there is a core of "us" who accept that we shoot as we shoot. That there's always room for improvement. That there's little to be gained in claiming fractional MOA when shooting Turkish 8mm through a worn out Kar98K. (Ducks fanboi brickbats.)

Or, maybe that's just me--shooting 2X makes me want to shoot 3X the next time (or the next volley). I suspect that we in shooting wind up with issues like many in golf do. It's not about slice or hook--it's about consistency.

Back to OP, many of the European battle rifles were test fired for acceptance by using mass groups. The Germans put about 40 rounds into a target, and 50% had to fit into a given size box. The Brits had a similar system for their rifles (except they used a circle, IIRC). So, people buying rifles in the millions were willing to accept them based on groups of 10, 20, or more.

Ok, it's not internet-cool to brag that you can beat the 4 MOA factory acceptance on your milsurp (and a sizeable number will group far tighter, over a bench, than their factory acceptance.

So, Your Mileage May Vary. (There's Stat joke in there, but it's not germane, or Tito [])

If you have to resort to graduate level statistics to determine how accurate your rifle is, you might not be a redneck.

Seriously though, unless you are a bench rest competitor or super long range shooter, it probably doesn't matter all that much. It is an interesting topic though and a big focus of many informal shooters/handloaders. But unless fired from a machine rest, you are testing the rifle/shooter system. Some shooters are better than others so results may vary. It doesn't take much to have a shot go astray.

A few years ago, a friend was preparing for his first elk hunt. He was really straining at the minutiae: bedding, velocity, group sizes between various bullets and powders, trajectory, etc. Data analysis (no real stats experience, though) was a big part of his hyper focus.

The third time he spent an entire Monday lunch fussing over his weekend bench loading and shooting results, I finally said, “Dude, do you know how big a cow elk’s heart and lungs are?” “Well, no.” “Dude, they are big. Like, bigger than a yearling steer big. Most elk are killed at less than 200 yards, and their lungs are bigger on each side than a basketball, and the heart is bigger than a softball. You’re not talking about shooting a prairie dog in the eyeball at 500 yards. You’re talking about punching through a bunch of bone and popping a basketball at less than 200 yards. The extreme accuracy you are trying to work up won’t matter a bit if you can’t shoot a basketball from field positions, on demand, after running 100-400 yards. *Any* of your loads are way more accurate than that, in the stock your rifle is in right now, bedded the way it is right now. Which load you use needs to have a bullet that will hold together while it punches through bone and pop a basketball at impact velocity. The rest of what you’re talking about doesn’t even matter.”

He started laughing once I was about halfway done with my rant. And started running twice a week.

3Crows said:

Most hunting rifles could not shoot a 25X1 continuous group without heating up and go off point of aim. But these same rifles could shoot 5X5 groups with a cooling period between each of the five and retain point of aim.

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I cannot think of one hunting contour barrel that will hold poi over a twenty five consecutive shot string.

Groupings (3, 5, 10, 50, 100, 1,000, etc) tell you what you and your firearm are doing; with what precision you and your firearm are doing it. Group shooting is familiarity and practice with that firearm, equating to confidence in you and your firearm. In a hunting or defense scenario, group shooting prepares you for that first (and most of the time) only shot - the confidence to put that first shot where it needs to go the first time and hopefully every time.

3Crows said:

Most hunting rifles could not shoot a 25X1 continuous group without heating up and go off point of aim. But these same rifles could shoot 5X5 groups with a cooling period between each of the five and retain point of aim.

Click to expand...

A cooling period between shots will allow conditions to change effecting your group size.
IBS LR competition calls a course of fire 5 shots for record light gun or heavy gun, NBRSA LR course of fire is 5 shots for LG and 10 shots for Heavy gun. weight class is 17 pounds or less lg

South Prairie Jim said:

I cannot think of one hunting contour barrel that will hold poi over a twenty five consecutive shot string.

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Neither can I but I am not going to say there is not one because sure as I do a fella will show up with one that will and pound me over the head with it. I tend to shy from absolutes. But, yes, I agree with you.