The Effect of Muzzle Velocity Variation on Group Size

[denton]

We were having a fun discussion on another thread, https://www.thehighroad.org/index.php?threads/throwing-powder-statistically.868396/unread, which drifted away from the original topic. Mea culpa. At the administrator's suggestion, this is a new thread, addressing the questions we drifted off to.

Some Necessary Basic Ideas

Please bear with me. There are some basics that we need, so as not to be confounded in our discussion.

1. All inferential statistics are estimates. Only rarely will they be perfect. But they are often good enough to be useful.

2. Normality is overrated. Most statistics work quite well even if the data are not normally distributed. Besides, no real world collection of data was ever truly normally distributed.

3. Only rarely do you need 30 samples. That sticks in many people's minds because it is part of teaching Z tests. But nobody uses Z tests anymore, outside the classroom. We've had T tests since about 1920, and they do fine with smaller samples.

4. Standard deviation is a common estimate of dispersion. The bigger your SD is, the wider your distribution is. If you work in SDs rather than ESs, a lot more useful math is possible. SDs do not add linearly, and the way they do add leads to some surprising results. And yes, you can convert back and forth between ES and SD, but that's a story for another day. Anyway, for accurate shooting we want consistency in our loading practices, and smaller SDs are better.

5. When many small sources of error are involved, they have a strong tendency to partially cancel each other.

Applying the Stats to Group Size

Assume that you have a rifle that you know on average prints 1/2" groups at 100 yards. Also assume that you have some way to perfectly control the rifle, and that you're shooting under absolutely ideal conditions, with no wind. Finally assume that you're shooting perfect ammunition, exactly perfectly the same MV. So that's ideal conditions. The interesting question is, if we allow imperfection in the ammunition MV, how much does it distort groups? The answer is helpful in figuring out where to spend our energy.

Under ideal conditions, the rifle will print roughly round groups 5" in diameter at 1000 yards. So that's our reference point for comparison.

95% of shots will fall within plus and minus 2 standard deviations. So for 5" groups, that's plus and minus 2.5", or a vertical and horizontal SD of about 1.25"

Assume that we now switch to less than perfect ammunition, and that the effect of MV variation is to introduce a vertical error with a standard deviation of 1" (by our 95% approximation, that's plus and minus 2", total 4"). We can combine that with our 1.25" SD for ideal conditions by taking the square root of the sum of the squares:

square root (1.25^2 + 1^2) = 1.6"

So again applying our 2 SD = 95% rule, and assuming my foggy old brain has done all of this correctly, we'll now be printing groups 5" wide and 4 X 1.6" = 6.4" high (plus and minus 2 SDs). Our groups will be ellipses, 5" wide and 6.4" high. You'd have to shoot a lot of groups to detect that change.

So controlling MV SD does matter, but not as much as most people intuitively think it does. The more precise the rifle, and the longer the range, the more it matters. But with typical rifles, and shorter ranges, it matters a lot less.

 

[taliv]

this could be a short conversation, as i mostly agree with that.

PRB did an article on it

https://precisionrifleblog.com/2015/04/18/how-much-does-sd-matter/



they were basically arguing that a 5 FPS SD was only a tiny bit better than a 15 FPS SD and that a 10 FPS to 3 FPS improvement only gained you 1% more hits. and if you're shooting at a 20" circle at 1000, with their wind variation, that might be true. but given the x ring on an F class target is only 5", your hit percentage would change pretty dramatically if that simulation was run with a 5" target instead. i mean, just look at the 15 fps SD vertical spread and imagine the target is 1/4 as tall. (you'll have to imagine since I don't have a $200 copy of the software to rerun the simulation)

These days there are a lot of rifles that are capable of shooting competitive F class, even as savage proved, out of the box. So it's pretty reasonable I think to up our expectations a bit. and given how competitive shooting is these days, i'd need all the help i can get.

 

[Offfhand]

SD, Standard Deviation, is often confused with extreme spread. They are NOT the same.

 

[denton]

SD, Standard Deviation, is often confused with extreme spread. They are NOT the same.

You are correct.

 

[Varminterror]

Assume that we now switch to less than perfect ammunition, and that the effect of MV variation is to introduce a vertical error with a standard deviation of 1"

Why did you choose to assume a deviation result rather than (as did PRB in their WEZ simulation) simply assuming ES as the manipulated variable? 1” vertical dispersion at 1,000yrds seems arbitrary (5fps-ish), whereas you could have made a more meaningful example by comparing two ES standards, say, 20fps ES vs. 50, as assuredly disparate standards.

This example also neglects the additional wind influence as a result of variable ToF among a larger ES. Smaller than the vertical influence, of course, but undeniable (appears to be slightly more than 10% relative increase for my 6 creed load). A 5” round group increasing in ES will gain in width as well as height.

Demonstrating the difference between an <20ES group vs. a >50ES group would be as simple, and more relevant to how most folks are considering their reloading or ammo selection process in terms of velocity consistency. Don’t even have to run a Monte Carlo. Run a ballistic calculator with your high velocity and then run again with your low velocity, noting the elevation and windage corrections for each.

(Side note for readers: sensitivity testing your calculator in this way is good practice. A shooter should be familiar with establishing their wind bracket to understand their potential to connect at range. A shooter should be tracking themselves in practice to understand their typical accuracy in wind calling, such in critical shots, they know their wind bracket - as in some instances, that information might mean passing on a shot, or spending more time to read the conditions more carefully).

Here’s a real-world example demonstrating a comparison of high ES vs. low ES. I’ve had much better lots of Hornady Black Grendel ammo, but this lot used to fire the target on the right measured over 20 shots to be 78fps ES. On the left, I was developing data for a batch of 6 creed handloads failed on my first waterline attempt, the 3 shots at the bottom, clicked up 0.2mils, and shot the group. That batch of ammo yielded 24fps ES. Both fired at 875 yards, both rifles reliably printing 1/2-3/4moa at 100yrds in my hands. So at range, one group scattered in the wind due to my poor estimation, while the other scattered in the wind AND moreso up and down the target. Statistically, the groups could have been larger, as there could have been a maximum vertical spread ~9 of dispersion in the 6 creed group, rather than the ~6” observed, and the Grendel group could have exhibited ~24” instead of the observed 12”, but as the Monte Carlo simulation by PRB illustrates above - it’s highly unlikely to observe the combination of multivariable maximums together (lowest mechanical hold of a group & lowest speed falling together AND highest mechanical hold & highest speed falling together). Would I do any more load work to shrink my ES below 24? Nope. Would I ever take a load with a 78ES to a match? Nope again. The targets don’t lie - 1) I need more work improving my wind calls, and 2) I shouldn’t compete with that lot of Hornady Black ammo.

 

[denton]

Good questions, and possibly more involved than what we can do here.

For starters, what is your method for combining ESs? If your rifle is shooting 1 MOA, and you add something new that also contributes 1 MOA, how do you combine the two ESs?

Yes, the choice of adding 1" vertical SD is arbitrary. It's a number that's in the ballpark, and is a reasonable example. It's easy enough to substitute another number, to see what happens. The answer is, if there is one source of variation that is much larger than the others, it will almost totally determine the resulting variation.

 

[Weflyfast]

Accuracy and precision for long range shooting.
Bryan Litz
....Bryan really gets deep into ALL the error factors that cause deviations in hit percentage.....
As little as +/-30fps can have little affect at 100 yrds- poke it out to 1000 yrds and you miss by 18 inches....
It's great reading if you are a rifle loonie!

 

[jmorris]

Run a ballistic calculator with your high velocity and then run again with your low velocity, noting the elevation and windage corrections for each.

So many don't see that its that simple.

 

[Varminterror]

For starters, what is your method for combining ESs? If your rifle is shooting 1 MOA, and you add something new that also contributes 1 MOA, how do you combine the two ESs?

It’s not so complicated - it’s a simulation. You’re again oversimplifying without contextual relevance which separates your path from reality. If you measure a given group size and ES, as I described above, you can make estimates of the relative contribution. You’re comparing highs and lows around an average - punch in 3000fps and 3020, you’ll have two vertical correction outputs. Then put in 2985 and 3035, again, 2 more vertical outputs. Subtract the difference between the first pair from the difference of the second pair, that’s as predictive as it could ever need to be for projecting the additional potential vertical from the additional ES.

Comparatively, you can trust the 100yrd group sizes have exceedingly small influence from velocity on vertical dispersion, so comparing your 100yrd vertical dispersion to your 1000 is another empirical method for determining your ES influence (to which I eluded in my first, referencing my vertical dispersions were considerably smaller than the projections suggested it would be).

Yes, the choice of adding 1" vertical SD is arbitrary. It's a number that's in the ballpark, and is a reasonable example.

Run 1” at 1,000yrds out in the ballistic calculator. I’m not convinced 1” is a reasonable example - not many folks shoot well enough at 1,000 to evaluate their groups to the 1/10th moa, and none of us own equipment which is specified to reliably measure velocity precisely enough to deliver on 1” at 1,000 either. In a 6.5 Creedmoor, 1” of additional vertical would correspond with ~2.5fps in velocity shift. The MagnetoSpeed V3 claims 99.5-99.9% accuracy, the LabRadar claims within 0.1%, ProChrono Digital claims within 1%, the Caldwell claims 0.25% error (I’d assume all of these mean +/- their percentage, but oddly none are published as such). So given a 3000fps cartridge, the measurement device isn’t sufficiently accurate to rely upon the validity of precision to within 1” at 1,000 - the common models on market may not repeat to even 6” at 1,000yrds, and the best models on the market only within 2”. *Noting also - empirically, we certainly tend to get closer repeatability than .25%, as it happens far too often that guys land well on their anticipated waterline. But I share the published specs as they are.

Considering most of us strive to drive ES below 20-25 in our long range reloads and it’s quite common to see factory ammo with 50-150fps ES, we have established standards for comparison. So talking about a variable shift of ~2.5fps in a game where the difference is more likely 5-10x that in scale isn’t as relevant as it could be.

 

[jmorris]

The MagnetoSpeed V3 claims 99.5-99.9% accuracy, the LabRadar claims within 0.1%, ProChrono Digital claims within 1%, the Caldwell claims 0.25% error (I’d assume all of these mean +/- their percentage, but oddly none are published as such).

Also spot on, I have shot through multiple chronographs at the same time, if you have three in the group, you get 3 different numbers.

They are also sensitive to where inside the "window" the shots are placed.
Nationals they would set 3 ProChrono units like a "W" so the screens over lapped where the bullet went, no two ever read the same number, they just took the highest number from each shot. The thing I found most interesting is there wasn't a "slow" or "fast" one, they were just randomly different.

 

[murf]

Why did you choose to assume a deviation result rather than (as did PRB in their WEZ simulation) simply assuming ES as the manipulated variable? 1” vertical dispersion at 1,000yrds seems arbitrary (5fps-ish), whereas you could have made a more meaningful example by comparing two ES standards, say, 20fps ES vs. 50, as assuredly disparate standards.

This example also neglects the additional wind influence as a result of variable ToF among a larger ES. Smaller than the vertical influence, of course, but undeniable (appears to be slightly more than 10% relative increase for my 6 creed load). A 5” round group increasing in ES will gain in width as well as height.

Demonstrating the difference between an <20ES group vs. a >50ES group would be as simple, and more relevant to how most folks are considering their reloading or ammo selection process in terms of velocity consistency. Don’t even have to run a Monte Carlo. Run a ballistic calculator with your high velocity and then run again with your low velocity, noting the elevation and windage corrections for each.

(Side note for readers: sensitivity testing your calculator in this way is good practice. A shooter should be familiar with establishing their wind bracket to understand their potential to connect at range. A shooter should be tracking themselves in practice to understand their typical accuracy in wind calling, such in critical shots, they know their wind bracket - as in some instances, that information might mean passing on a shot, or spending more time to read the conditions more carefully).

Here’s a real-world example demonstrating a comparison of high ES vs. low ES. I’ve had much better lots of Hornady Black Grendel ammo, but this lot used to fire the target on the right measured over 20 shots to be 78fps ES. On the left, I was developing data for a batch of 6 creed handloads failed on my first waterline attempt, the 3 shots at the bottom, clicked up 0.2mils, and shot the group. That batch of ammo yielded 24fps ES. Both fired at 875 yards, both rifles reliably printing 1/2-3/4moa at 100yrds in my hands. So at range, one group scattered in the wind due to my poor estimation, while the other scattered in the wind AND moreso up and down the target. Statistically, the groups could have been larger, as there could have been a maximum vertical spread ~9 of dispersion in the 6 creed group, rather than the ~6” observed, and the Grendel group could have exhibited ~24” instead of the observed 12”, but as the Monte Carlo simulation by PRB illustrates above - it’s highly unlikely to observe the combination of multivariable maximums together (lowest mechanical hold of a group & lowest speed falling together AND highest mechanical hold & highest speed falling together). Would I do any more load work to shrink my ES below 24? Nope. Would I ever take a load with a 78ES to a match? Nope again. The targets don’t lie - 1) I need more work improving my wind calls, and 2) I shouldn’t compete with that lot of Hornady Black ammo.

View attachment 915947

a picture is worth a thousand words.

murf

 

[denton]

It’s not so complicated - it’s a simulation. You’re again oversimplifying without contextual relevance which separates your path from reality. If you measure a given group size and ES, as I described above, you can make estimates of the relative contribution. You’re comparing highs and lows around an average - punch in 3000fps and 3020, you’ll have two vertical correction outputs. Then put in 2985 and 3035, again, 2 more vertical outputs. Subtract the difference between the first pair from the difference of the second pair, that’s as predictive as it could ever need to be for projecting the additional potential vertical from the additional ES.

Comparatively, you can trust the 100yrd group sizes have exceedingly small influence from velocity on vertical dispersion, so comparing your 100yrd vertical dispersion to your 1000 is another empirical method for determining your ES influence (to which I eluded in my first, referencing my vertical dispersions were considerably smaller than the projections suggested it would be).



Run 1” at 1,000yrds out in the ballistic calculator. I’m not convinced 1” is a reasonable example - not many folks shoot well enough at 1,000 to evaluate their groups to the 1/10th moa, and none of us own equipment which is specified to reliably measure velocity precisely enough to deliver on 1” at 1,000 either. In a 6.5 Creedmoor, 1” of additional vertical would correspond with ~2.5fps in velocity shift. The MagnetoSpeed V3 claims 99.5-99.9% accuracy, the LabRadar claims within 0.1%, ProChrono Digital claims within 1%, the Caldwell claims 0.25% error (I’d assume all of these mean +/- their percentage, but oddly none are published as such). So given a 3000fps cartridge, the measurement device isn’t sufficiently accurate to rely upon the validity of precision to within 1” at 1,000 - the common models on market may not repeat to even 6” at 1,000yrds, and the best models on the market only within 2”. *Noting also - empirically, we certainly tend to get closer repeatability than .25%, as it happens far too often that guys land well on their anticipated waterline. But I share the published specs as they are.

Considering most of us strive to drive ES below 20-25 in our long range reloads and it’s quite common to see factory ammo with 50-150fps ES, we have established standards for comparison. So talking about a variable shift of ~2.5fps in a game where the difference is more likely 5-10x that in scale isn’t as relevant as it could be.

The problem with your approach is that ES is a function of both the process variability and the number of items in the sample. The larger the sample, the larger the ES is likely to be. So if a 5 shot string and a 10 shot string have equal ranges, they represent different underlying amounts of variation in the process. Combining ESs is therefore very messy.

The cure is to switch to working in standard deviations. That does not depend (much) on sample size, and the math for combining them is well understood and well proved on all the innumerable physical processes where it has been applied.

I suppose it is possible that group size could be the first exception, but I don't think so.

I think you may have misunderstood my choice of a 1" SD at 1000 yards. Since 95% of shots will fall within plus and minus 2 standard deviations, that's a 4" range, not 1". That's about .4 MOA.

 

[Varminterror]

You’re assigning in your own posts that “range” - ES - is 4x SD. I’ve ignored that particular fallacy until now that you’ve promoted it as defense for a more suitable method. “Either you is or you ain’t,” as they say. Ignoring also, of course, the 1/2moa rifle at 100 remains mechanically to deliver 1/2moa at 1,000 as well.

Again - you’re remaining far from reality if contending a 4” additional range - even a 4” Range is only a 10fps ES, in a scale comparing ~20-25 FPS ES for high shot count groups of phenomenal ammo to ~75-150fps ES for poor ammunition. And again, one of the most popular chrony’s on the market has an accuracy +/-30fps for a 3000fps cartridge.

I’m just not certain why, unlike the others of us who have ran these statistical simulations, you wouldn’t simply use a Monte Carlo simulator and an actual ballistic calculation for the statistically distributed outcomes of our variables? The PRB article outlines the Litz WEZ model relatively well with a relatively simple set of outputs - one depicted above - depending upon the selected manipulated variable. Why make a statistical projection for ballistics without using any ballistic math or any real-world empirical link for relevant scale at all?

 

[denton]

You’re assigning in your own posts that “range” - ES - is 4x SD. I’ve ignored that particular fallacy until now that you’ve promoted it as defense for a more suitable method. “Either you is or you ain’t,” as they say. Ignoring also, of course, the 1/2moa rifle at 100 remains mechanically to deliver 1/2moa at 1,000 as well.

Again - you’re remaining far from reality if contending a 4” additional range - even a 4” Range is only a 10fps ES, in a scale comparing ~20-25 FPS ES for high shot count groups of phenomenal ammo to ~75-150fps ES for poor ammunition. And again, one of the most popular chrony’s on the market has an accuracy +/-30fps for a 3000fps cartridge.

I’m just not certain why, unlike the others of us who have ran these statistical simulations, you wouldn’t simply use a Monte Carlo simulator and an actual ballistic calculation for the statistically distributed outcomes of our variables? The PRB article outlines the Litz WEZ model relatively well with a relatively simple set of outputs - one depicted above - depending upon the selected manipulated variable. Why make a statistical projection for ballistics without using any ballistic math or any real-world empirical link for relevant scale at all?

So many questions.... Well, you are right to challenge things that don't make sense to you. I don't mind.

Why Convert to SD?

There is no easy way to combine variation from several sources using ES. So the trick is to transform the data into a domain where we can handle the math, and then, if you like, transform back to ES.

Converting Range to SD

The proper way to do it is to use the d2 constants, but those require that you know the number of samples, as I noted earlier. If you don't know the number of samples, dividing range by 4 is a quick and dirty estimate. With small samples, your estimate of population SD is very imprecise, probably far greater than the error introduced by using the quick and dirty method. So quick and dirty is OK.

Monte Carlo

There is a saying that when the going gets tough, the tough go to Monte Carlo. It's exactly the method I used for developing conversion factors for 3, 5, and 7 shot groups. It works.

But Monte Carlo requires that for each source of variation, you stipulate a distribution, a mean, and a standard deviation.

Unless the esteemed Mr. Litz has come up with a new approach to Monte Carlo, he has done all those conversions for you in his software, just as I did here, but more elegantly packaged.

Why Not Monte Carlo?

This Covid19 thing has me a bit bored. But I'm not bored enough to spend an afternoon creating a simulation. The required math for my solution only takes a couple of minutes. So in this case, it's much more efficient. Done correctly, they both produce the same answer.

Why 1" Vertical SD?

A quick trip to RSI Shooting Software yielded that a 3000 FPS 90 grain 6mm projectile drops about an extra 3" at 1000 yards with the loss of 20 FPS MV. So I thought, why not use a 4" change, since that yields an SD of 1, which is quick and simple.

I certainly did not mean for you to be stuck forever with the numbers I chose. If you think something else is more representative, the math is all laid out for you and you can plug in whatever numbers you're curious about.