Rimfire Pistol Accuracy

[SteveT]

IMHO if the group size from a good rest is larger than 1/4 - 1/3 the size of the shooters offhand groups, then it is time to work on the gun/ammo combination. At that point the gun errors will make it hard to call shots accurately.

Nonetheless, I have seen a dozen or so "target" 22LR guns, including Rugers. If they can't shoot 2" or better at 50 yards with some SV or target ammo then something is wrong. Usually it is the shooters rest technique or bad ammo. If it is the gun, it is usually a damaged crown. All 22's I've seen can get near 1" groups with some brand/type of ammo. Most can shoot under 1" with something. Some can shoot 1/2" groups (yippee when you find one!). I use 30 shot groups because every gun can produce some 10 shot groups that are much smaller than normal.

As to the statistical discussion, some of it went over my head, but if we are using extreme spread to talk about group sizes then we must add the errors. Sometimes the two errors are going to add together to improve the group but sometimes they will add together to make it bigger. Occasionally they will add together in the worst possible way. If you shoot enough shots offhand and measure the extreme spread it will be the sum of the two errors. Most of the shots will be well inside but you will still get some worst case shots.

Best regards,
Steve Turner

[tenex]

... As to the statistical discussion, some of it went over my head, but if we are using extreme spread to talk about group sizes then we must add the errors. Sometimes the two errors are going to add together to improve the group but sometimes they will add together to make it bigger. Occasionally they will add together in the worst possible way. If you shoot enough shots offhand and measure the extreme spread it will be the sum of the two errors. Most of the shots will be well inside but you will still get some worst case shots.

Best regards,
Steve Turner

Hi Steve,
This is exactly why I like to calculate the average of the extreme spread of several 10 shot groups. Here's how it works out...

first, some assumptions:
1. the probability distribution of the groups follow a bivariate normal distribution (Gaussian in X and Y, there's a good reason for this)

2. The standard deviation in X is the same as Y (for convenience).

For these constraints, the average center to center distance of the farthest apart shots of a 10 shot group will average to about 3.8 times the standard deviation of X and Y. Some groups will be bigger, some will be smaller, but if you shoot enough, you'll get a reasonable measure of the standard deviation. In reality, the group size is just an easy way to measure standard deviation. It's not as accurate as measuring every shot, but it's a whole lot easier. The more groups you shoot and average together, the more accurate your estimate will be.

Why 10 shot groups? I don't know, convenience perhaps? You can use 5 shot groups, 50 shot groups, what ever makes you happy. A 5 shot group will always be smaller than a 10 shot group, and a 50 shot group will always be larger, so the 3.8 factor will be different (sorry I don't have those numbers off the top of my head).

My real performance measure is not how big an arbitrary group is, it's really the standard deviation of the shots. I'm just using group size as an estimator because it's easy, and I'm lazy. If your groups are non-Gaussian (like a tight group with a flier because you jerked the trigger) then the group size is not as good an indicator.

So, what I really want to see when I test a gun is that the average size of all the 10 shot groups is below a certain size. This is the metric I like to track.

And to answer Steve S and the original poster's question, I like the groups from the gun off a sand bag to be 1/3 of my offhand groups or less. If you can barely keep ten shots in a the black of a B2 target at 50 feet, then less than an inch should do. If you shoot a little better, you can do the math to get your own requirements. My number for a .22 is 2" at 50 yards. If my gun puts 10 shots inside 2" (on average) then I'm done.

Steve
(like you guys need another one).

[alb]

Steve/tenex,

Actually, group size is a random variable in its own right (it seems to follow a gamma distribution). This is partly what was causing me difficulty in understanding your analysis (it still is).

You can use extreme spread to measure group size, but this is a less efficient measure than minimum covering circle (see the Grubbs paper that I cited in a previous post). It's easy to measure minimum covering circle -- I use a vuegraph transparency with concentric circles printed on it. The relationship between group size (measured using the minimum covering circle method) and number of shots per group is given by a probability density function derived by H. E. Daniels in 1952.

In order to increase the reliability of your group size method, the most efficient technique is to shoot groups with a large number of shots and then drop the worst shots. The military does this when they apply "circular error probability," i.e., a circle that covers 50 percent of your shots. Using CEP, you can get a very reliable measure of the accuracy of your shooting system with only 20 shots -- mean group size is 0.985 times the standard deviation of individual shots in one dimension, and the variance in group size is 0.0313 times the standard deviation of individual shots in one dimension. The reason that this works so well is that you are dropping all of the low probability shots.

Unfortunately, you really can't use CEP with small arms, because you typically get one large ragged hole with a few outliers. So you shoot a larger number of shots per group and drop fewer shots. I recently did accruracy testing on some target loads that I had worked up for my .45. At 50 yards, I was able to shoot 40 shot groups and drop the worst 10 shots. With this approach, my mean group size (for the best 30 shots group) is 1.53 times the standard deviation of individual shots in one dimension, and the variance in group size is 0.025 times the standard deviation of individual shots in one dimension. When I tested at 25 yards, I had to drop fewer shots.

I have software that I developed that allows me to run simulations of any number of shots per group and number of shots dropped per group. Unfortunately, I haven't been able to work out an analytical solution of a probability density function for the relationship between group size, number of shots per group and number of shots dropped per group. Several Ph.D. statistitians have assured me that there may not be an analytical solution.

As to Steve Swartz's point, I like to compare the difference between shooting my Pardini and shooting my Ruger to the difference between driving an Italian sports car and driving a tractor. I started shooting Bullseye last May. At the time that I bought the Pardini, my high score with the Ruger was 768/900. Ten days after I bouht the Pardini, I shot an 832/900. Meanwhile, the person who sold me the Pardini had been shooting in the 850's with it, and when he switched to the S&W Model 41, his scores dropped into the 750's, where they've remained for the past year.

There probably isn't any significant difference in the accuracy of those 3 guns, but the difference in the quality and the aesthetics of their triggers makes a world of diffence to the shooter.

Regards,

Al B.

[SteveT]

Clearly both of you are much (much MUCH) more knowledgeable than I on this subject

What I want to know is (1) what ammo should I use in my gun and (2) How far off call does a shot have to be before I do something about it.

Question 1 is pretty straightforward. Whether you use the average of several groups' or one large group's extreme spread, just pick the smallest groups and go shoot.

Question 2 is where I prefer the worst case extreme spread. I will give an example since I don't know enough about the mathematics to express it that way.

Lets say I have a gun that will shoot 4" groups. Even if I shoot 100 shot in a group they will all be in a 4" diameter circle or, put another way, the point of impact (POI) will always be within 2" of the point of aim (POA).

I take my first shot in slowfire and it is a good shot and I make a good call. When I look through the scope and the hole is 6" away from my call. I am going to do what the books say not to do, make a sight adjustment based on just one shot. Of course this is all dependant on a very good call in which I have high confidence.

The next shot is another good shot and good call and it is 1" lower than my call. I am not going to make an adjustment until I take a few more shots because it could have been a totally random event that caused it. If I take a couple more shots and they are all lower than my call, then I will make the adjustment.

Steve Turner

[alb]

Lets say I have a gun that will shoot 4" groups. Even if I shoot 100 shot in a group they will all be in a 4" diameter circle or, put another way, the point of impact (POI) will always be within 2" of the point of aim (POA).

Steve,

Your average 100-shot group will be a little over 51 percent larger than your average 10-shot group, assuming that you don't drop outliers.

As to when you should move your sights, we discussed this about a year ago:

viewtopic.php?t=12528&postdays=0&postorder=asc&start=80

Please see my post dated 5/8/07, at 4:50pm, where I talked about confidence intervals. Basically, when you start adjusting your sights in the middle of a match, all you accomplish is to make your gun 'effectively' less accurate, i.e., you increase the gun's dispersion -- you are moving your POA relative to your POI. Of course, your sights may actually be off, but unless they are really off, it takes a large number of shots to make this determination. The bettery your accuracy, the smaller the amount that constitutes "really off."

You also have to consider another source of error which most people tend to overlook. Your ability to 'call' your shot isn't any more perfect than anything else in the universe. So, was that shot 2 inches off, or was your call 2 inches off, or some combination the two?

Regards,

Al B.

[2650 Plus]

Steves comment about trying to learn when the accuracy is in question is right on. A perfectly executed shot that is not on call can cause the shooter to change something that is perfectly correct. And change for the worst can set the shooter back as a measure of uncertainty is introduced into the shooters progress. When a perfect shot is fired the shooter should be able to look at the target through the spotting scope and instantly be looking through the shot hole. Good Shooting Bill Horton

[alb]

Steves comment about trying to learn when the accuracy is in question is right on. A perfectly executed shot that is not on call can cause the shooter to change something that is perfectly correct. And change for the worst can set the shooter back as a measure of uncertainty is introduced into the shooters progress. When a perfect shot is fired the shooter should be able to look at the target through the spotting scope and instantly be looking through the shot hole. Good Shooting Bill Horton

The more you practice, the better you get. And the more you practice calling your shots, the better you get at that too. Still, nothing's perfect. For someone who shoots in the 2650's, messing with your sights probably isn't going to cost you more than a point or two anyway. And besides, how would you ever know?

[Steve Swartz]

- Buy the best gun you can afford (assuming taking out a second on your home)

- Any "match grade" ammo at this point is fine believe it or not. Don't waste your time testing ammo/lots of ammo within brand (the variation between lot numbers inthe same brand is greater than the variation across brands. In either case the variation is small enought to ignore). Instead, invest your time in

- Train your a** off and don't worry about the gun (you already bought the best) or your ammo (it doesn't really matter as long as you are using match-grade "target" ammo)

Oh and understand how your sights work so that you don't fall into the trap of adjusting them during a match after 5 or ten shots . . .

[tenex]

Steve/tenex,

Actually, group size is a random variable in its own right (it seems to follow a gamma distribution). This is partly what was causing me difficulty in understanding your analysis (it still is).
...

Hey Al,
Don't get too worked up over the distribution of group size. You're applying a geometric measure to a median type measurement (CEP) and you're going to create some nonlinear random process that's going to have some bizzaro probability distribution. The real point is to get an estimate of the standard deviation of the group (without working too hard, see lazy reference above). CEP is good measurement technique, but all the shooters measure extreme spread, and it's something everyone understands. Any linear measure however (extreme spread, min group diameter, etc.), will be proportional to standard deviation, so they all work.

I set up a MonteCarlo simulation with a million or so 10 shot groups to calculate average spread of the group just to give me a really simple method to use at the range. I don't really care if my process is that far off, I can just shoot more groups.

To answer SteveT's question, I shot 3D archery for a few years, and spent way too much of my life chasing sights. What I started doing for air pistol, is to shoot a practice match and digitize all my targets so I could have a record of where all 60 shots went. Then I could calculate the sight position that would have given me the best score. It was usually very close to where the sights were already. I also seemed to always have a target with a group off center, so if I fiddled with the sights I would have made it worse, not better. That's just my experience, your mileage may vary but I'm pretty reluctant to move the sights once I gotten sighted in.

Steve.

[alb]

I set up a MonteCarlo simulation with a million or so 10 shot groups to calculate average spread of the group just to give me a really simple method to use at the range. I don't really care if my process is that far off, I can just shoot more groups.

Steve,

Thank God for Monte Carlo simulations! Personally, though, I don't like to waste a lot of time doing accuracy testing, so I use a method that produces a very reliable result with minimal effort. The one that I described works well for me. Dropping outliers gives a much more reliable result than shooting an equivalent number of smaller groups.

The military uses CEP because it works so well, and when you're spending several thousand dollars per artillary shell, you want to minimize your expense. You just have to remember to shoot the same number of shots and drop the same number of shots for each target that you want to compare. I do find the math part of it interesting though -- far more interesting than the accuracy testing itself.

It was usually very close to where the sights were already. I also seemed to always have a target with a group off center, so if I fiddled with the sights I would have made it worse, not better. That's just my experience, your mileage may vary but I'm pretty reluctant to move the sights once I gotten sighted in.

Actually, my mileage doesn't vary -- not even a little bit! The engineering contingent on this forum (including me) have been saying for some time now that messing with your sights makes matters worse. The thread that I pointed Seve Turner to discusses this point ad nauseum. But you simply can't convince people that don't understand randomness of this.

There have been more threads than I can count were someone will write, "I shot 5 shots in a tight group that was high right, and then another 5 shots that were low left. What's going on?", followed by all sorts of replies diagnosing possible problems with changing grip over time, changing sight alignment over time, fatigue, cosmic rays, etc. How about, the shooter isn't pointing the gun at the exact same spot on the target each time he breaks the shot (no one can), and the small clusters of shots that sometimes appear are the result of random clumping. When you shoot enough shots at the same target, the clumps disappear.

I see this phenomenon all of the time, because I'm too lazy to change my targets all that often when I'm practicing. If you only shoot 10 shots per target, however, your perceived reality may be different.

I can see that group size is a function of the variance of the shots in the group. I know that variances sum algebraically. And I know that the group size is a function of the standard deviation of the shots, since it has the same dimensions. Maybe after a good night's sleep, it will make more sense.

Regards,

Al B.

[tenex]

...
I take my first shot in slowfire and it is a good shot and I make a good call. When I look through the scope and the hole is 6" away from my call. I am going to do what the books say not to do, make a sight adjustment based on just one shot. Of course this is all dependant on a very good call in which I have high confidence.

The next shot is another good shot and good call and it is 1" lower than my call. I am not going to make an adjustment until I take a few more shots because it could have been a totally random event that caused it. If I take a couple more shots and they are all lower than my call, then I will make the adjustment.

Steve Turner

Hi Steve,
I made up an Excel spread sheet with a graph of 10 shots + the mean. In this case, the zero of the gun is at 0, so you can see what the 10 shot average is. Just keep hitting F9 and you'll get another group. You can see how for some groups the mean is right in the middle, but for some, you'd swear you needed to make a sight correction.

This is great fun to play with, I'll do anything to avoid making dinner or cutting the grass. I think it's an eye opener, remember that the gun is properly sighted in regardless of where the center of the group is. I use stuff like this to visualize what I'm really doing when I start moving my sights, I hope you can run it.

As I said earlier, when I used to shoot compound bow I was an obsessive compulsive sight cranker, now I'm on a 12 step program and my wife doesn't let me move my sights unless I can provide enough data to support it.

Maybe if I'm good this week I'll get to take 2 clicks....

Sincerely,
Steve.

[Steve Swartz]

Al:

Have you ever thought of changing your name to Steve?

I'm just sayin' . . .

[alb]

Al:

Have you ever thought of changing your name to Steve?

I'm just sayin' . . .

Steve,

Don't we have enough Steve's already?

Al B.

P.S: After a good night's sleep, Steve's method for adding error contributions makes perfect sense to me now.