One of my current goals is getting more involved in precision shooting and competition. So it shouldn’t be a surprise that precision has been on my mind a lot. As I’ve been diving into more and more research, I’ve come across a few interesting ways of thinking about things that I want to share with you.
Fair warning, this post doesn’t involve as much math as, say, my article on Mils vs Minutes of Angles, but there is some statistical theory.
Defining Accuracy and Precision
Let’s take this back to high school chemistry back in the day. When we talk about precision, we’re specifically talking about repeatable results. If you’re talking about a precision scale, for example, you would expect it to read the exact same value every time you put the same object on it.
Imagine placing a weight on the scale five times. Each time you placed the weight you see the following readings: 22.4 oz, 26.2 oz, 20.7 oz, 23.8 oz, and 24.0 oz.
You would say this scale is imprecise and wouldn’t trust it for serious work. That’s not to say you couldn’t use it for quick and dirty measurements to get a rough idea, but it’s far from a precision instrument.
A precise scale would show nearly the exact same result each and every time.
But that says nothing of accuracy.
If precision means you get the same result each and every time, then accuracy means that the result is actually correct.
To go back to the scale example, let’s say you placed a weight on the scale and it read 23.5 oz for all five readings. That means it’s precise. But if the weight was actually 18 oz, then the scale is inaccurate.
Luckily, you can fix accuracy fixed through calibration and adjustment.
So how does this relate to shooting? The short version is that you should think of your hardware, meaning the rifle and ammunition, as contributors to precision. How the hardware interacts with you, the shooter, contributes to accuracy.
But that distinction is for another day.
On Precision Rifles
Many gun owners get caught in a tailspin chasing “ultimate accuracy” with their rifles. So let’s break that down. A perfectly precise rifle places each and every bullet through the same hole every time. Of course, that is in a perfect world with no other variables.
The key is that last word: variables.
Producing a precision rifle is all about reducing or removing the variables that cause inconsistency. As Russ Miller put it in Episode 8 of the podcast, “Accuracy is the product of uniformity.” He was specifically referring to the behaviors and actions of the shooter, so it was a correct use of “accuracy,” but the same applies to the hardware as well.
We’re talking the machining of the barrel and muzzle, the profile and harmonics of the steel, torque values of components, action to stock fitment, and other mechanical elements. Ammunition must be consistent also, with the same amount of pressure from shot to shot
Since achieving a perfectly precise rifle is practically impossible, the goal for precision rifles is reducing the statistical deviation from shot to shot. One of the best ways to illustrate this that I’ve ever seen came from John Simpson, who I interviewed back in Episode 2.
John uses this example to explain hit probabilities and how they can happen before the shot is ever fired. It’s an elegant visualization, and I’m probably going to mess it up a bit (please forgive me, John!). It also serves as a wonderful example of why 3-Shot Groups are terrible for evaluating a rifle’s capability.
Imagine a grid four rows tall and four columns wide. You have a pair of six-sided dice. One represents elevation and the other is windage.
Each side of the die correlates to a rows or columns. A one or six goes to the outside edges, a 2 or 3 goes to row/column 2, and a 4 or 5 goes to row/column 3. Roll the dice as a pair and mark the intersection of elevation and windage.
For this experiment, I rolled my dice 50 times and logged each combination of elevation and windage. These first two charts represent the results of a 3-shot group and a 5-shot Group.The number in each square is the amount of times the square was “hit.”
For three shots, we have a nice neat little triangle. If you were using this for your zero, then you’d probably assume that the rifle’s “zero” was in the middle, where the 4/5 elevation row and 4/5 windage column intersect.
But look what happened after two more “shots” at the second grid. Now we have something that looks more like a vertical string. A shooting coach might say that you’re breathing is off, but let’s imagine that the rifle is benched and sandbagged. After five shots, you might think the actual “zero” is two squares lower where row 2/3 and column 4/5 intersect.
Let’s keep shooting.
This next grid shows what happens when we reach 10 shots. Now we start seeing repeated hits in a couple of squares, which provides us with more data.
Our estimate of where the “zero” might be seems a lot more likely now. But it took more shots to produce a more complete picture.
Of course, this isn’t a real target- so don’t think of it that way.
This grid doesn’t have any correlation to the actual size of the groups, only that there was some deviation. Put another way, think of each shot as an independent statistical event between the mechanical factors I mentioned earlier. The space where the most shots repeat and others cluster around is the mechanical “zero” of the rifle.
For completeness, here’s the grid showing all 50 “shots” from my dice.
To me, it looks like our “guess” of a zero being around the intersection of row 2/3 and column 4/5 was about right. Interestingly, we made that determination after our first five shots, but could not have done it after the first three.
So there’s a lesson for you: five shots is far more useful for zeroing and grouping than three shots. Ten shots are even better. More than that, if you could keep a log book of every shot you ever fired through this rifle, it would provide the maximum amount of data on that rifle’s performance.
This exercise isn’t a perfect illustration, but I think you’re getting the point. A more precise rifle is like using a smaller grid, or dice with fewer sides.
In contrast, an imprecise rifle might be like rolling with twenty-sided dice. By removing the possible variables, you improve the overall statistical probabilities and keep things “tighter.”
But this is only a statistical illustration of probabilities. How do we translate that into the real world?
The Wrong Way
Let’s start with what not to do. Go to any message board or gun shop, and someone will regale you with tales of their “Sub-MOA All Day” rifle. To prove it, they will pull a target out of their wallet and show you a nice tight little three-shot cluster.
“See,” they say, “my rifle is a beast. This here is a three-quarter inch group.”
Our friend here is using what we call extreme spread. He measured the distance between the two furthest impacts and then claims this group size represents the totality of the rifle’s capability.
Here’s the thing, this exercise tells you nothing. As you saw from our earlier dice-rolling experiment, three shots don’t really tell you much statistically. Was this three-shot group cherry-picked because it was an outlier? What didn’t you see?
If you were to take our friend’s rifle out to the range, how likely are you to land a hit at 600 yards with it? How do you know?
Circular Error Probable and Mean Radius
Back when I was in the ICBM business, I spent a lot of time learning about how we test the precision of our ground-based nuclear missiles. Over the decades, the US Government has amassed a lot of test data from each and every test launch.
Every once and a while, I was briefed on something called the Circular Error Probable, or CEP. In short, imagine a circle drawn around the point of aim. The CEP is the diameter of the circle in which 50% of the impacts will occur. It’s an average of every test impact and how far away it fell from the center.
The more precise the weapon system, the smaller the CEP becomes. The ICBM engineering program has a goal CEP for each weapon system.
A similar concept, though not identical, is the Mean Radius. I first came across this concept while reading military small arms acquisition requirements. It’s different than what most people use to measure weapon precision, but it’s far more useful.
Defining the Mean Radius
Let’s start by shooting any statistically significant shot group. I’ll use five as a nice round number, but 10 or more is better. To find the mean radius, we must first find the mathematical center of the group. I’m not going into the nitty-gritty here, but internet-famous precision-nerd Molon did a great primer on it years ago.
Once you know the center, then you measure the distance of each impact from that center point. Once you have that, divide by the number of shots and you have the mean radius.
In this example, which is not to scale, I represented the center of the shot group as the green ‘x.’ Each red circle is an impact.
The dotted circle represents the mean radius. Whatever that measurement is, say 1.25 MOA, I can confidently say that 50% of my shots will land within 1.25 MOA of my point of aim.
Now, there’s some nuance here because of the statistical math involved, but you get the idea. The more shots you take and plot, the more confident you can be.
One of the reasons that a lot of new folks don’t like using mean radius is that it usually looks worse on paper than a one-off extreme spread. It looks much nicer to show off that one-off group and say “this is what my rifle can do.”
But the truth is that knowing your rifle’s actual level of statistical precision is far more useful to you in the long run. If you know your mean radius, and you know the statistical center of your groups (i.e. your zero), then you’ve already got much of the battle won.
So What’s the Right Amount of Accuracy
I already said a lot of people go chasing their tails trying to obtain the highest level of precision out there. But they don’t really need it. A rifle capable of 1 MOA is plenty to get the average newbie started. Spend the rest on the ammo.
I hope you found this article useful. I want to thank John Simpson for providing the rolling dice example for statistical deviation, it really did make a huge difference to my understanding and I hope I did it justice.
Knowing you have a precise rifle is one thing, but it’s only a component. Just as important is how you, the shooter, interact with that rifle to place accurate hits on target.
How do you determine the center of a 5 shot group? Or a 3 shot group?
Hey Jason, thank’s for asking. The short answer is that you need to do some measuring and plotting. I could go into it here, but Molon did a great job of it in an old thread on ARFCOM (link here) You’ll have to find the lowest impact on the target and consider that the “baseline.” Then you you measure how far above each shot is from that up to the topmost impact. Average each of those measurements to get the center of the vertical string. Repeat the process again using the leftmost impact as the horizontal center. Where the vertical… Read more »