Accuracy

To start, a list of Jargon, Concepts, Variables, and Acronyms:

1. BAC
2. MOA and sub MOA
3. Rayleigh Distribution
4. σ and Mean Radius
5. Accuracy Indicators (MR, P1x, ES, etc.)
6. Computing P1x from CEPYY

Many rifle and component manufacturers guarantee MOA or even Sub MOA performance from their products. Claims of MOA and sub MOA weapon performance is prolific within the shooting community. Using the Ballistic Accuracy Classification system (BAC™) I will show how the different weapon classes within BAC relate to plausible claims of MOA and sub MOA accuracy.

To begin, we neglect environmental effects, especially wind, precipitation, air temperature, and the rest, and choose to focus on effects of rifle, shooter, and ammunition. Essentially, we’re looking at effects that dominate shooting rifles from the bench at 50 to 100 yards. These effects extend to longer ranges and different shooting positions, but these scenarios must consider environmental effects and shooter skill which are neglected in this discussion.

Weapon accuracy is described by the Rayleigh distribution, which is just a two-dimensional version of the common Normal or Gaussian distribution.  It even uses the same parameter (sigma) to characterize the radial dispersion of shots on a target. BAC defines several classes of weapon accuracy based on the single Rayleigh distribution statistical parameter, σ (sigma), which is the Rayleigh analog of the standard deviation for Gaussian distributions. Each of the classes corresponds to a σ that is 1/10 of the designation of the class. In this way, Class 2 has a σ of 0.2, Class 4 has a σ of 0.4, and so on.

For a given weapon system (by which I mean weapon, ammunition, and shooting scenario), once σ is known, and thus the weapon class has been defined, all sorts of interesting information can be readily computed for the weapon.  For example, the Mean Radius (MR) of a class is approximately 1.25 times the σ parameter of that class. The MR is probably the most useful and easily understandable value to the shooter that can be computed from σ because it represents the EXPECTED, or most probable, distance of a given shot from the true point of aim.

I introduce the term Accuracy Indicator to refer to some measure of a subset of a sample of a shot group, real or simulated, that conveys useful and/or interesting information about the accuracy of the weapon, ammunition, shooter, and combinations thereof, that were involved in shooting the group. Such measures as 3-shot and 5-shot extreme spread, mean radius, vertical standard deviation, horizontal standard deviation, group center, and Point of Aim error are all accuracy indicators.

One of my favorite accuracy indicators is the probability that a single shot will be equal to or closer than some distance from the true point of aim (POA) of the shooter.  I will call this indicator the P1x (probability 1 shot less than some x in MOA from true point of aim). The P1x accuracy indicator is practical and interesting for a lot of reasons, including that it is simultaneously its own mean and extreme spread. The P1x is also an inverse of the Circular Error Probable (CEP – i.e. the radius from the true POA for which a given shot is YY% probable, e.g. YY = 90 implies CEP90). While the true point of aim is never known perfectly to the shooter, the statistics reference the P1x indicator from the true POA, and the shooter would like their actual POA to equal the true POA.

CEPYY is computed from σ as

Which is easily inverted to get the P1x result for a given σ, or

Where x is the distance from true POA in question. This can be solved for σ to answer another reasonable question: what σ and x will give me a P1x? For example, suppose we want to know what σ is needed to get a P1-0.5, which is the probability of a single shot landing within 0.5 MOA of the POA, with a probability of 80%. In this case solve for σ to get

With the result that

Which is Class 3.

The reason I like this accuracy indicator so much is that it answers the practical question of whether the success of a proposed shot is probable, and exactly how probable. For example, I think most hunters will be happy if their hunting rifle had a P1-0.5 of at least 50%, meaning that the probability that a single shot will land within 0.5 MOA of the true POA is no worse than 50%. In this scenario, the hunter could reasonably expect to make 300 yard shots as the probability of the shot being within 1.5” of the intended target is 1 out of 2. Further, with a 50% P1-0.5, the P1-1.0 turns out to be much higher, 93.7% in fact, so that in our hunting rifle scenario, the hunter could expect a very good chance of their shot being within 3” of their point of aim at 300 yards (plus a small factor to account for discrepancy between true and intended POA). Such a weapon has a σ of 0.425 which puts it at the high end (lower is better) of Class 4 (Class 4 has σ centered at 0.4 and extends from 0.35 to 0.45). The following figure shows the P1x for x between 0 and 2 MOA for a σ 0.425 weapon.

Most rifle shooters, myself included, are inclined to think of accuracy in terms of 3 or 5 shot extreme spread. I will explore the usefulness of this metric as an accuracy indicator and compare with two other accuracy indicators: MR and P1x.

The figure below shows the probability of a 1 MOA or better 5-shot extreme spread (ES) for BAC classes 1 through 7. Note that the range of probability of 1 MOA or better ES is large for some classes (3, and 4) and small for most of the others (1 and 2 at the high accuracy range where the probability is greater than 90%, and 5-7 at the low accuracy range where the probability is less than 20%).

From the above notes and the graph below, it is my opinion that any BAC class that spans more than a 20% range of probability for a given MOA metric can be assumed to be of that ES class. In this case, BAC Class 3 and Class 4 can be reasonably thought of as MOA weapons. In this case, Class 3 is MOA with greater than 50% probability of 5-shot MOA groups while Class 4 is MOA with less than 50% probability of MOA ES. Classes 1 and 2 are what most of us think of as solidly sub MOA, and weapons at the low end of Class 3 are at the high end of sub MOA performance with a solid probability of 0.8 and 0.9 MOA or better 5-shot groups.

A natural question at this point is why not refine the classes so that classes 3 and 4 are broken up into finer regions on the 1 MOA plot? This is a very good question, and one I struggled with as I would like a more refined system as well. There turns out to be a very good answer: It is very difficult to determine the σ of a given weapon system to a finer degree than we are doing with the BAC system.

There are many reasons why this is true. Foremost, the number of shots required to define a BAC class more precisely than 0.1 MOA per class is very high, somewhere between 100 and 1000. Shooting so many shots starts to effect the accuracy of the rifle as the barrel you are shooting at the start of the session is not exactly the same as the on a the end of the session for a lot of reasons. Second, during a given shooting session, shooter fatigue will set in and start to have an increasingly larger effect on the results. There are other reasons but these are the most significant that immediately come to mind.

Class 5 is worth further consideration. It is not quite an MOA class according to the discussion so far but it is close. What happens if we extend the size of the 5 shot group under consideration from 1 MOA to 1.25 MOA?

So, a Class 5 weapon can reasonably be considered a 1.25 MOA weapon when thinking in terms of extreme spread. How does a Class 5 weapon stack up in terms of P1-0.5 and P1-1.0 accuracy indicators? It turns out that for the center of Class 5 with σ = 0.5, P1-0.5 equals 39% and P1-1.0 is 86%. I consider this very good for hunting and informal target shooting.

On the other hand, I don’t like individual 3-shot groups for determining accuracy as they are statistically meaningless and they give a false sense of accuracy. Consider the probability of each class of weapon shooting a 3-shot extreme spread at 1 MOA or better:

In this case, Class 5 is solidly an MOA performer as up to 30% to 50% of 3-shot groups are expected to be MOA or better. However, 3 shot groups for selecting ammunition that works well for a given rifle, or for zeroing a scope is a grave mistake, and for those and other reasons I reject 3-shot groups for any serious rifle evaluation. I know if feels awesome to have three shots in close proximity on paper, but take the extra courage and risk blowing the group for the greater good of knowing something closer to the truth. In fact, for serious accuracy determination, I shoot a minimum of 10 shots per group, and I usually combine data from three such groups into a single 30 shot result. I love 5 shot groups for fun and friendly competition, but for evaluation 10+ shot groups are vastly superior.

In my next post I will consider an important aspect of rifle shooting accuracy: the shooter. How can we estimate the accuracy of the shooter, and if this quantity is known, how can the accuracy of the rifle and ammunition be separated from the capability of the shooter with that weapon? Especially important to consider: how the precision of the shooter is effected by the weapon. Deficiencies in shooting technique are exaggerated by lighter weight weapons and cartridges with greater recoil, and the ratio between the two. Hold these thoughts until next time.

Countless times I am asked and have asked the question that arguably troubles rifle shooters more than anything: “How accurate is my rifle?” Most shooters, myself included, have settled on 3 or 5 shot groups, and the expected extreme spread of that group. We like to buy rifles that promise groups that are 1 inch or 1 MOA. We take our shiny new weapons to the range and put the manufacturer’s guarantee to the test.

The after-action report finds us sometimes pleased and sometimes not. We all know a single bad range trip can be due to lack of sleep, too much coffee, a bad batch of ammunition, ammunition the weapon simply doesn’t like, or one of many other excuses. These excuses might be legitimate, but there’s no way to know. Adjustments are made, bullet seating depth is changed, different ammo is selected. The next range trip might produce a couple groups in the MOA to sub MOA range and we are then pleased that our rifle is indeed a shooter and we pack up happy until next time.

Then we get really sophisticated, especially if we are hand loading our ammunition. 50 rounds, in a blue plastic box, the first 5 sets of 5 rounds varying powder charge by a few tenths of a grain, the second varying bullet seating depth. We shoot our 10 groups and look for trends, and we see the groups open up or close down in a manner that appears to vary proportional to the change in the parameter of interest. And who doesn’t love a good looking target after a trip to the range.

I’d seen some interesting measurements of group statistics that got me thinking. Extreme spread was never completely satisfying from a practical point of view, and achieving a small extreme spread had become more like winning a football game than providing meaningful feedback about the outcome of a shooting session. Here’s where things could have gone one of a couple different directions. If I were to keep on keeping on, business as usual, keep chasing MOA and sub-MOA 5-shot groups with extreme spread as the metric, I start to run the risk of fooling myself. Fliers are the shooters equivalent of Mulligans in golf, and we use “called fliers” as a means to tighten up the group size. I find called fliers very unsatisfying and if the group didn’t stay together, it didn’t stay together and that was that. Still, not enough useful information.

Not surprisingly, all these shooting sessions, all this data, and it’s clear there’s more meaningful information. The problem of measuring shooting accuracy clearly has a statistical nature, and concepts like mean and standard deviation, already long in use when thinking about muzzle velocity, must be applicable in some way. The internet knows everything and Bing searches quickly lead to Ballistipedia. Go there and you will find the application of some serious statistical expertise to the problem of thinking about and measuring rifle accuracy. The problem turns out to be a lot more complicated and nuanced than I thought.

Once I got through my five stages of grief, having given up the old and beloved way of thinking about accuracy, serious study of the proper way to analyze the measurements of little holes in paper began. It took me a couple months of part time study, but I finally gained a good understanding of the statistics of measuring rifle accuracy. The problem is complicated and I think the subject is not one that is suitable to most shooters who just want to know the practical accuracy of their weapons so they can be aware of what shots are reasonable for them to take, and what is the probability of a given outcome.

Getting back to the beginning of this post, as a company that sells rifle barrels and components, we want to be able to tell our customers what level of accuracy they can expect from our products. Now we know there is no meaningful answer to the question “Will my rifle shoot MOA groups?” The hope is for a sea-change in the way rifle accuracy enthusiasts think about and discuss their topic. The methods presented at Ballistipedia are correct but not generally accessible, and the practical implications of the results of detailed statistical analysis of shooting data are hard to discern. A system is needed that boils down the results of the statistical analysis of a sample of shooting data into different classes of performance. Once a weapon and ammunition combination has been classified, the shooter can easily assess the likelihood of a particular outcome of a hypothetical scenario. The shooter can know how likely they are to shoot a 5-shot group with MOA or better extreme spread. They can also know the probability of any given shot being within 1/2″ from the true point of aim. They can also know how many shots are needed to get a good estimate of the true point of aim of their weapon so that they can achieve a good zero for their scope.

To this end, Bison Armory has worked with Ballistipedia.com to create the Ballistic Accuracy Classification system.

We are still working to classify our barrels, upper assemblies, and rifles, but to start we guarantee that our products are at least Class 5 as defined in the BAC system. In practical terms, this means a shooter can expect at least 39% of their shots to fall within 1/2″ of the true point of aim, 9% of 5-shot groups to measure 1″ or better extreme spread, and 35% of 3-shot groups will achieve this measure. Our data indicates that our products are probably typically Class 4 but we’ve not gathered enough data yet to make a guarantee i this regard. For Class 4, a shooter can expect 54% of their shots to fall within 1/2″ of the true point of aim, and 26% of 5 shot groups will have a mean radius of 1″ or smaller, and 57% of 3-shot groups will be this small.

Our match grade, heavy barreled, and Fulcrum products should approach or achieve Class 3 status with the right ammunition. Of course, the true classification of a given barrel in practice will depend on the quality of the build, the ammunition used and its suitability to the weapon, the setup and performance of the shooter, the trigger, the total weight of the weapon, the quality of the optics, the tightness of the fit between upper and lower receiver, and other factors.

Once you know the class for your weapon, you can know with confidence important metrics like the probability of hitting a given target at a given range with a single shot.
No other rifle manufacturer we know of goes to these lengths to give a meaningful accuracy guarantee for their products. It takes time, it takes commitment to the truth above marketing, and it takes dedication to rifle shooting as a discipline. Don’t be fooled by phony accuracy guarantees. Demand the truth, embrace the truth, and then every shot will count.