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Old October 11, 2012, 10:32 PM   #26
JohnKSa
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It is exactly a simple addition of shooter and bullet error on target.
That is correct if we consider only a single shot.

However, it's important to understand that both the rifle and shooter errors are random so they don't always end up pointing in the same direction. The rifle's error for a particular shot might be in the 5 O'Clock direction while the shooter's error for that same shot might be in the 11 O'Clock direction (opposite the rifle's error) from the aiming point. The effect on the target in that case would actually be a shorter distance from the center than either of the errors ALONE would have caused.

In fact, the odds that both errors will end up pointing in exactly the same direction (or in exactly opposite directions) are vanishingly small.

So when you try to calculate the group sizes generated by a 1MOA shooter and a 2MOA rifle you have to consider that both the magnitude of the error (distance from the aiming point) and the angle of the error (direction of the error from the aiming point--say 10 O'Clock or 6 O'Clock) are random.

For the expected group size to be generated by a simple addition of 1MOA and 2MOA, the angular error for the rifle and the shooter would have to line up for each shot which is impossible in any practical sense. To calculate the actual expected group sizes of a 1MOA shooter using a 2MOA rifle, one must use techniques for adding random variables.
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Old October 12, 2012, 06:20 AM   #27
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John, I understand your reasoning.

My point is, the accuracy of a rifle and its ammo is the largest group they shoot. Half its size is the furthest a bullet will strike from where the rifle's aimed. If the rifle's aimed somewhere inside a 1 MOA circle 'cause that's the area it moves around in when held by a shooter, the furthest a shot will land from the edge of that circle is half the size of the rifle and ammo's accuracy.

So, a shooter aiming at a point on the target holds within 1/2 MOA of that point; his area where the sights align is 1 MOA. His desired impact point's in the middle of it. His rifle and ammo's largest groups are 2 MOA. The furthest a bullet will strike from where it's aimed is half that amount; 1 MOA.

Therefore, the resultant group on target should the shooter fire many shots will be 1 MOA of holding plus 2 MOA of accuracy and that's 3 MOA. The shooter's shot will land somewhere between his desired impact point and 1-1/2 MOA away from it. The fewest number of shots will be at the outside edge, but they're still gonna be there. They have to be counted in the measurement of the group he shoots. Most of the shots will be inside about 2 MOA, but not all of them. About one third will be in the 2 and 3 MOA range.

In talking with ballistic folks at Lake City Army Ammo Plant on their 7.62 NATO match ammo tests, I've gained some insight on the realities of accuracy tests. They would shoot a couple hundred shots per test group and they told me the following. It's been their observations for a "rule of thumb" that about 10% of the shots fall in the outer 10% of the group radius, 20% in the next inner 20%, 30% in the next on and finally 40% in the inner 40% of the group's radius. Which correllates well with the mean radius (their standard measurement of accuracy) covering about 70% of the bullet holes; inner 40% and 30% of the shot holes and about 70% of the group's radius. Oft times there would be 5 or 10 holes by themselves at the outer edge of a 200-shot cluster typically about 6 to 11 inches in diameter depending on the quality of the ammo lot tested. Yet those outliers very precicely represented the furthest a bullet would strike from group center or where the rifle was aimed and have to be counted as part of the group. I've seen a couple of their test targets and it's hard, but one can fairly easy identify each bullet hole for measuring its location with reasonable accuracy. They're fired at 600 yards.

Where the rifle's aimed about a desired point for bullet placement by the shooter, any error in that has to be added to the radius of the shot group to determine the maximum miss distance from the desired point of impact.

Last edited by Bart B.; October 12, 2012 at 07:15 AM.
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Old October 12, 2012, 08:34 AM   #28
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Look at it this way,

When I say "out shoot the rifle" I mean if you have a 2 MOA rifle and constantly shoot that rifle at 2 MOA, then its time to move up in equipment.

When I first got my M1A (1977) it was a standard grade. I reached a point in High Power where I could shoot certain scores and could not improve. I could outshoot the rifle.

I gave my rifle to the All Guard Armors at the Wilson Matches (National Guard Championships) who converted it to a Super Match. Only then could I improve to the point I could shoot master scores and get my Distinguished Rifle badge.

You can add 2 MOA plus 1 MOA and get 3 MOA all you want. There comes a point where it is time to move on, or move up rifle wise if you want to improve.
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Old October 12, 2012, 09:21 AM   #29
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Kraig, I agree with you. When you can hold better than the rifle and ammo shoots, then you need to get better ammo and rifle. Your rifle and its ammo needs to shoot better than you can hold if best accuracy's the goal. The better that "hardware" is the smaller your groups will be, the higher your scores will be and you'll miss your desired point the least amount.

There are exceptions but I'll not address them here. There's been National champions who've picked a less accurate firearm than what was available because they shot it more accurate than what the could do with the more accurate ones.

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Old October 12, 2012, 09:30 PM   #30
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My point is, the accuracy of a rifle and its ammo is the largest group they shoot. Half its size is the furthest a bullet will strike from where the rifle's aimed. If the rifle's aimed somewhere inside a 1 MOA circle 'cause that's the area it moves around in when held by a shooter, the furthest a shot will land from the edge of that circle is half the size of the rifle and ammo's accuracy.

So, a shooter aiming at a point on the target holds within 1/2 MOA of that point; his area where the sights align is 1 MOA. His desired impact point's in the middle of it. His rifle and ammo's largest groups are 2 MOA. The furthest a bullet will strike from where it's aimed is half that amount; 1 MOA.

Therefore, the resultant group on target should the shooter fire many shots will be 1 MOA of holding plus 2 MOA of accuracy and that's 3 MOA. The shooter's shot will land somewhere between his desired impact point and 1-1/2 MOA away from it. The fewest number of shots will be at the outside edge, but they're still gonna be there. They have to be counted in the measurement of the group he shoots. Most of the shots will be inside about 2 MOA, but not all of them. About one third will be in the 2 and 3 MOA range.
Again, the only way that they will add up to 3 MOA is in the extremely improbable case where the vector angle of the errors lines up perfectly on two shots that go in opposite directions on the target and the magnitudes of all the errors are at their maximums.

Each shot has a random error associated with it. That error has both a magnitude and an angle. If you consider that the shooter contributes some error and the rifle contributes some error, now each shot is the vector sum of two random errors. That is the combination of two random magnitudes and two random angles.

To get a group of 3MOA, you'd first need a shot where the random error magnitudes of both shooter and rifle are essentially at their maximums and the random error angles of both the shooter and the rifle are lined up in the same direction. Then you need a SECOND shot where both random magnitudes are again essentially at their maximums and the two random angles are lined up with respect to each other but in the OPPOSITE direction of the two random angles of the other shot where everything lined up.

So you need two shots in your group where all of the following are true.
  • All four random magnitudes are at or very near their maximums.
  • One of the shots must have two random angles line up very closely.
  • A second shot must have two random angles that line up very closely and that are opposite or very nearly opposite the other two random angles from the other shot.
The odds of having that happen in any reasonable number of shots (let alone a group shot with 3, 5, or even 10 shots) are astronomical. Essentially impossible from a practical standpoint.

The bottom line is that a X MOA shooter with a Y MOA rifle will shoot groups that are larger than the larger of the two accuracy figures but smaller than X+Y MOA.
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Old October 13, 2012, 10:58 AM   #31
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Originally Posted by JohnKSa
The bottom line is that a X MOA shooter with a Y MOA rifle will shoot groups that are larger than the larger of the two accuracy figures but smaller than X+Y MOA.
It's called the Root Mean Square (RMS):

Shooter^2 + Rifle^2 = Group^2

A 1 MOA shooter + a 2 MOA rife will produce, on average, a 2.24 MOA group.
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Old October 20, 2012, 01:14 PM   #32
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Absolutely, a 1 MOA shooter + a 2 MOA rife will produce, on average, a 2.24 MOA group.

And the biggest group will be 1 + 2 = 3. 3 MOA. This happens when all the errors or variables add up directly.

If all the errors for a given bunch of shots cancel each other out, the smallest groups will be zero MOA. Or close enough to not matter.

Ain't math great?
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Old October 20, 2012, 03:35 PM   #33
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Indeed -- as number of groups goes to infinity, the outliers will be 3MOA and 1MOA, but the average "should" be as stated. This thread is a fascinating example of everybody being right, but in different ways, and talking "past" each other.

All of that, of course, assumes that the deviations are truly random (and thus, evenly distributed). They probably aren't, since there are certain phenomena that create these deviations. The result will likely not be random. I would guess that some of the deviations, such as a wobbling hold or inconsistent cheekweld, "tend" to produce linear deviations (rather than "planar", if that distinction communicates) such that you might well approach the "extreme case" of 3MOA more often than root-mean-square indicates.

Nevertheless, 3MOA is a "worst case" given the conditions. Math IS awesome, especially since it includes probability and statistics .

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Old October 20, 2012, 07:13 PM   #34
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And the biggest group will be 1 + 2 = 3. 3 MOA. This happens when all the errors or variables add up directly.
Correct.

Assuming that the 1 and 2 MOA figures are hard limits--no shots ever go larger than those limits for the shooter and rifle respectively. (This isn't really a good assumption. In reality the figures aren't hard limits, they are a very simplified representation of the probability distribution of the shots on the target. That's why the MOA figures for 10 shots groups is larger than that obtained shooting 3 shot groups. The more shots you get, the better the representation of the probability distribution you get and the farther the extreme shots are likely to be from each other. But the math gets complicated and I don't want to mess with it so I'm going to assume these are hard limits. It will serve to illustrate the probabilities that explain why it's problematic to think about it in terms of straight addition of accuracy figures.)

AND

Assuming the shot distribution over the magnitude limits are uniform. (This isn't really a good assumption either but it will suffice for illustration. The shot distribution would probably be better modeled with a normal (Gaussian) distrubition but I don't want to fiddle with the math.)

AND

Assuming that we accept magnitudes that are 95% or greater as being sufficiently large.

Then you'd need a group containing about 400 shots in order to get JUST the magnitudes to line up so you'd have a really good chance of getting a group that was 2.9MOA or greater.

That completely ignores the error angles which would also have to line up just right to get a group that large. So...

Making the same general assumptions as above and assuming that the error angles have to line up within 18 degrees (5%) to make everything work, then you'd end up again needing a group with maybe 400 shots before you'd have a really good chance of having JUST the angles lined up just right.

If we assume that the magnitude and angle errors are independent of each other and independent from shot to shot.

Then, to get the magnitudes AND angles to line up just right all at the same time--based on the above assumptions, we would need a group with something like 160,000 shots in it to have a really good chance of getting everything to line up just right.

That's why it's much better to talk about averages when we combine accuracy figures than to try to define maximums. Averages provide useful information. If you talk about maximum group sizes instead of averages, the resultant figures aren't very representative of what is likely to be encountered in the real world.
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Old October 21, 2012, 02:19 AM   #35
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we would need a group with something like 25,600 Billion shots in it
I want to learn to shoot long range. I really do... but that really is more reloading than I'd had in mind.
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Old October 21, 2012, 08:36 AM   #36
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John, how many call vs. impact plots for 20-shot groups have you made with 1/2 moa gear fired hand held?

And isn't the average of 2 3 4 5 6 7 & 8 the same as 4 4 5 5 5 5 6 & 6? If these are groups fired with two systems, one is more accurate than the other in spite of having the same average.

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Old October 21, 2012, 11:58 AM   #37
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Corrected my post above. Got a 4 where there should have been a 2 and since it was an exponent, it really made a difference.
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If these are groups fired with two systems, one is more accurate than the other in spite of having the same average.
To have any chance of providing a reasonable answer when combining accuracy figures, the figures need to actually be representative accuracy figures.

So if you're getting a wide variance across your groups (as with the first system in your example)--especially with a relatively small number of groups then you won't be able to provide a representative accuracy figure and the combination won't be representative either.
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Old October 21, 2012, 03:31 PM   #38
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While the odds of both sources of variables adding to the maximum is one in a huge number, nobody can predict the shot it will happen on. Might be in the last thousand, middle hundred or the first ten. Therefore I'll include it in the accuracy measurement of the shooter and his equipment.

PS
Mean radius of a many shot group is better than the average of several few shot ones.

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Old October 21, 2012, 03:37 PM   #39
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Sure, it won't hurt anything. On the contrary, you'll be pleased to find that the groups shot will be always be significantly smaller than the accuracy measurement you've chosen to espouse would suggest they should be. It'll be like Christmas every time you measure a group.
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Mean radius of a many shot group is better than the average of several few shot ones.
Group size is a convenient measurement and that's why it's used. There are certainly better ways to characterize accuracy.

I believe the military uses Mean Radius rather than extreme spread because they are less interested with how far the farthest two shots are from each other and more interested in how far the shots land, on average, from the aim point.
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Old October 21, 2012, 03:51 PM   #40
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I agree but 2.999999 moa is close enough to 3.000000 moa to be insignificant.
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Old October 21, 2012, 03:57 PM   #41
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I agree but 2.999999 moa is close enough to 3.000000 moa to be insignificant.
For whatever it's worth, that is a true statement.

The problem is that the figure in question (when combining 1MOA and 2MOA) is not 2.999999 MOA, it's about 2.24MOA. 2.24MOA is significantly different from 3 MOA--especially to someone who's concerned with precision.

You'd actually be better off in this case approximating the result by simply using the larger of the two accuracy figures and ignoring the other entirely than you would be by adding them together.
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Old October 21, 2012, 07:32 PM   #42
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The problem is that the figure in question (when combining 1MOA and 2MOA) is not 2.999999 MOA, it's about 2.24MOA. 2.24MOA is significantly different from 3 MOA--especially to someone who's concerned with precision.
Except when you test your free recoiling rifle and ammo with a 30 shot group and it's 2 MOA then dry fire 30 shots from prone and their call plot's 1 MOA, you can well count on the groups you shoot from prone will actually be larger than 3 MOA. Possibly 3.5 MOA.

Go do the statistics on that to explain why that's reality.
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Old October 21, 2012, 08:04 PM   #43
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Go do the statistics on that to explain why that's reality.
The issue has nothing to do with statistics, the problem is that there are two unjustifed assumptions in the hypothetical provided.

First, a call plot from a dryfire session does not tell the complete story as far shooter accuracy is concerned.

Second, the free-recoiling accuracy of a rifle does not fully characterize its performance from all shooting positions.

I responded to a comment about the combination of a 1MOA shooter and a 2MOA rifle. My response was based on the very reasonable assumption that those numbers were actually representative of the live-fire capability of the two systems and that the testing used to generate those representative figures (including the combined figure) is done under conditions that are reasonably similar.

If one has truly representative accuracy figures for the two systems and those two figures are 2MOA and 1MOA, then the resultant combination for the two systems, under similar testing conditions is going to be about 2.24MOA.

Nothing I've said was intended, nor should it be construed, to mean that it's impossible to construct a scenario whereby it's possible to combine a 1MOA shooter and a 2MOA rifle and get groups that are larger than 2.24MOA. That is certainly possible.
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Old October 22, 2012, 08:26 AM   #44
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First, a call plot from a dryfire session does not tell the complete story as far shooter accuracy is concerned.
I agree. It's been known for over a century that the muzzle axis doesn't point to the same place when the bullet leaves as it did when the firing pin fired the round. But it's the best thing going these days outside of laser pointers that some international teams have used.

Quote:
Second, the free-recoiling accuracy of a rifle does not fully characterize its performance from all shooting positions.
I disagree. Free recoiling rifles held the same way as people do will do that. Gripped at the stock's fore end and its butt plate held against a back support. Firing one so mounted lets the rifle to move rearward as the muzzle axis rises while the bullet goes down the barrel. Such machine rests do exist and have been and are used by top level competitors since the 1960's. Here's the one used by David Tubb:



The rifles accuracy with the ammo it shoots is fixed. It doesn't know how its held nor where it's pointed when shot. It performs the same for each and every shot. Where it's pointed and how it's held for each shot it totally controlled by the shooter. Which is why there's 1/4 to almost 2 MOA change in zero across several shooting positions for the same rifle and ammo combination.

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Old October 22, 2012, 08:47 AM   #45
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Go buy "Magpul: The Art of The Precision Rifle. Watch all the DVDs. It is a very informative video series. It helped me with the proper shooter position and with a slight pull I had when squeezing the trigger. I shoot a .308, and .243 at up to 800 yards, and a 50bmg, and 338 Win Mag past 800... The best thing is practice and being able replicate results over and over.
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Old October 22, 2012, 10:56 AM   #46
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i luv statistics!!!!!!!!
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Old October 22, 2012, 10:34 PM   #47
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It's been known for over a century that the muzzle axis doesn't point to the same place when the bullet leaves as it did when the firing pin fired the round.
And that is precisely the reason that the following statement can't be true.
Quote:
The rifles accuracy with the ammo it shoots is fixed. It doesn't know how its held nor where it's pointed when shot.
Because the muzzle moves slightly in recoil between the instant of firing and the time the shot leaves the barrel, it will shoot to slightly different points of aim depending on how the recoil movement is affected by the way the rifle is held/restrained in recoil.

It shouldn't affect the accuracy on target too much as long as the hold is consistent, because from a given position and with a consistent hold the recoil behavior of the rifle should be pretty consistent too.

However, the fact remains that careful shooting from a well-designed machine rest will produce accuracy results that a human can not duplicate from a formal shooting position, even if that shooter can completely eliminate aiming error, wobble and flinch.

All that is assuming a rifle set up so that things like hand positioning and sling tension don't affect the point of impact to any significant extent. Those things are additional complications that can certainly make a big difference in the accuracy of some rifles when shot from different shooting positions.
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Old October 23, 2012, 07:22 AM   #48
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Regarding rifles fired from good machine rests.....

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It shouldn't affect the accuracy on target too much as long as the hold is consistent, because from a given position and with a consistent hold the recoil behavior of the rifle should be pretty consistent too.
Right you are. In my opinion, recoil behavior will be extremely consistant and won't affect the rifle or ammo's accuracy at all.

A friend shot a dozen or so 10-shot groups testing his Win. 70 based match rifle clamped in an identical machine rest as I posted above at 600 yards. The 308 Win. ammo groups were between 3/4ths inch and 1-1/2 inch. He then shot a 40-shot group with the same ammo that went into 1-7/8ths inch. When he hand-held the rifle slung up in prone shooting the same ammo, his groups were in the 7 to 8 inch range on target at the Nationals. He won all three of the 600 yard matches that year. Not too shabby for full length sized cases either.

The best competitors shooting prone hold their aiming area inside about 3/4 MOA. Then try to break their shots inside 1/2 MOA. Such is the way things are when our pulse beat wiggles the line on a Richter scale of vibrations. Best I've observed was in 1997 watching Corky Tyson set the 600-yard prone record shooting all 20 shots inside about 4 inches using aperture front and rear sights.

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Old October 23, 2012, 11:56 PM   #49
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Best I've observed was in 1997 watching Corky Tyson set the 600-yard prone record shooting all 20 shots inside about 4 inches using aperture front and rear sights.
I wish I had seen that. I really enjoy rifle shooting, but I do precious little of it these days.
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