Finding the Accuracy Envelope

I very much enjoyed the interesting article by Rick Moritz in The Black Powder Cartridge News No. 120 (Winter 2022). However, I hope I can provide some help with the answer to one of his questions, “How many shots does it take to identify a MOA rifle?”

About 20 years ago, I was asking the same question and enlisted the aid of a former co-worker, Jeff Grimm, to find an answer using Monte Carlo simulations. Most of the files from that work have since been lost, but your article prompted me to recreate some of what we had done earlier, and readers may be surprised at what we found.

The short answer to Rick’s question is that if you want to know with certainty whether you have a minute-of-angle (MOA, 1.05 inches at 100 yards) rifle, you will need to do a lot of shooting – more than any sane person would want, I think. Jeff and I found that the average size of even 200-shot groups (center-to-center of the outside shots) is still smaller than the true accuracy envelope. It comes close, and even 100-shot groups are pretty close, but who wants to wear out barrels doing that? Fortunately, a rifle’s accuracy can be estimated (not measured) with far fewer shows, as I’ll explain.

First, I need to explain that we did not simulate groups shot in mirage or windy weather. Or, with the forearm sling stud neatly backed up to the front edge of a rest. Or, from a rifle held precariously against the shoulder socket. Any simulation that tried to account for fliers and their idiosyncratic causes would be of little help toward a general understanding of group statistics. Instead, we assumed that shots would be “in control” in the sense used by Joe Brennen (ref. 1). This means simply that the causes of wild shots or stringing have been corrected or avoided. Controlled groups are round, or would be if more shots are fired, and completely bounded by a circle that I’ll call the accuracy envelope. Groups might be tightened, and the accuracy envelope narrowed, by tweaking the load or other refinements, but the erratic bugaboos of rifle accuracy have been tamed.

To simulate controlled groups, we allowed shots to miss center in random directions and by random distances from zero up to the radius of the accuracy envelope. Some typical three, five, and 10-shot groups based on this algorithm are illustrated in the nearby panel, where the accuracy envelope is the outer circle, the true center of accuracy is at the intersection of dashed lines, and the geometric group center is shown by the “+” symbol. Samples of a few shots generally will not produce what anyone would call round groups, and these are not. But, as the illustration of a 40-shot group begins to suggest, the accuracy envelope would be completely filled with holes if a great many shots were fired, and of course, none of the hole centers would land outside.

Note that the density of holes tends to diminish from the center outward. This is a mathematical consequence of the simulation algorithm, and the math is very close to measurements taken by Andrew Knez (ref. 2) from actual targets shot with 22 rimfire match ammunition. Qualitatively, our virtual groups appear realistic based on the many photographs and facsimiles of targets with 10 to 40 shots shown in Ned Roberts’ fascinating book (ref. 3). If this gives you confidence that the simulations approximate reality, as it does me, then read along as I work toward some conclusions. Incidentally, I should explain here that groups were plotted on X-Y axes with unitless values ranging from -0.5 to +0.5, with a 1.0 diameter accuracy envelope. Inches, centimeters, or MOA come into play only when estimating the accuracy envelope after measuring real targets.

All told, I ran simulations of three through 10-shot groups with 200 replicates of each. Jeff ran it out to 200-shot groups and did 10,000 replicates of each. I “fired” over 10,000 shots, and he did more than 19 million! But he ran a Fortran program on a university mainframe, and I fiddled with an Excel spreadsheet on my laptop. Even so, the new results are virtually identical to the old where I have those earlier data for comparison (specifically, columns two and three of the table). Three things stand out.

The first is that although group center rarely lands on true center, it is usually not far off. The center of three-shot groups was, on average, only .15-MOA away from true center if scaled to a 1-MOA accuracy envelope. That seems encouraging, but if you are sighting in a 1-MOA rifle, the center of a single three-shot group might be anywhere from 0- to 1-MOA away from true center, and we cannot guess direction without supporting information from other groups. This is why sight adjustments can sometimes be frustrating.

The second point is that controlled groups sometimes contain what look like accidental fliers but are not. The accuracy envelope for new loads is a mystery until you have fired enough groups to get a sense of scale. Targets like the bottom middle one in the illustration can give a person false visions of tack-driving accuracy – accuracy that is seemingly just around a corner and never quite found. I know the temptation, and it causes a waste of powder. If there is no reason to believe that an outlying shot is accidental or caused by a gust of wind, then that shot may be the most informative hole in the entire target.

The third and most important finding, alluded to earlier, is that groups are almost never as large as the accuracy envelope. Actually, none of my simulated groups were that large, although a few came close. On the other hand, many groups were far smaller. To express this in familiar terms by scaling the accuracy envelope to 3 inches: among my 200 simulations of three-shot groups, 20 percent were smaller than 1 inch, but only 2 percent were larger than 2.5 inches. By contrast, no 10-shot groups were smaller than 1 inch, but 37 percent were larger than 2.5 inches.

The graph shows how group size approaches the accuracy envelope as the number of shots increases, and the table puts numbers to the curve. For example, the accuracy envelope is twice the size of an average three-shot group but only 29 percent larger than an average 10-shot group. But, again, these are average relationships, and an estimate of accuracy based on one group will probably be wrong. However, the information in the table can be used to calculate 90 percent confidence brackets around estimated accuracy. For example, the accuracy envelope estimated from a single, 1-inch group of three shots is 2.06 inches. However, we can “confidently” say (if 90 percent certainty gives you confidence) only that the true accuracy envelope is somewhere between 1.26 and 5.81 inches. (If that 1-inch group was shot at 100 yards, it was almost certainly not from a “MOA rifle.”) A single 10-shot group of 1-inch shrinks the confidence interval for estimated accuracy to between 1.06 and 1.65 inches.

The simulations provide persuasive support for basing accuracy testing on a 10-shot standard, although the true accuracy envelope is probably larger and maybe 65 percent larger. Ten-shot groups seem to be Rick’s standard, and that is the one I have always used for my Sharps rifles. A few, small three-shot groups do not prove much.

References:

1. Brennen, Joe. 2005. Group sizes and statistics. Single Shot Rifle Journal 59(1): 28-29, 56-57.

2. Knez, Andrew, Jr. 2012. Some thoughts on testing .22 caliber rimfire ammo. Precision Shooting 59(11): 8-11.

3. Roberts, Ned H. 1952. The Muzzle-Loading Cap Lock Rifle. The Stackpole Company, Harrisburg, Pennsylvania.