denton
Member
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.
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.
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