In response to damoncali's posting, the referenced article has the theory approximately correct. My approach is somewhat different, in that what I did was to develop a computer simulation by generating a series of random "shots" using a standard distribution, all "shots" (actually X,Y coordinates) were restricted inside a unit circle, i.e., a circle having a radius of one unit. This involved the generation of thousands of "shots". Then the distance of every shot from every other shot was calculated, creating an enormous array of hundreds of thousands of shot pairs, ranging in length from zero to 2 units. Then groups of various sizes are created. For example, a five-shot group consists of 10 shot pairs. So 10 random shot pairs are captured, and the largest pair length becomes the ES. THis process can be repeated to infinity if desired (I didn't go that far). The same thing can be used for 10-shot, 15-shot, 20-shot, or however many shots are desired for a particular group. Remember that the larger the number of shts per group, the more shot pairs are involved. For example, a 20 shot group contains 190 shot pairs. Then you can create multiples of the shot groups, for example, you can create five 10-shot groups, and average the ESs of each of the five groups. Again, this can be replicated as many times as you wish. This process is called a "Monte Carlo" simulation. Any number of statistical analysis operations can then be performed on the results. Remember that every group cannot exceed an ES of 2 units (due to the unit circle restriction), so that becomes the "Circle of Maximum Dispersion" (CMD). This allows the calculation of the CMD multipliers, and many other parameters. I won't go any further into the exact methodology, but once you get into it, it's not that complicated if you know how to work with Microsoft Excel. The important result for the present is just how many groups of each size are necessary to get a very consistent estimate of the average ES for every size group and how many groups are needed to get a very low standard deviation (SD). In my case, I assumed that a SD of no more than 5% of the mean ES would be satisfactory. The results:
Shots per group Number of groups to be averaged for 5% SD
2 about 250
3 about 75
4 about 40
5 20
10 5
15 3
20 2
As you can see, the most efficient group sizes (in terms of how many shots are required to get statistically reliable results) are five 10-shot groups (50 shots), three 15-shot groups (45 shots), or two 20-shot groups (40 shots). Actually, a single 20 shot group is very close to the 5% SD boundary. My preference is the use of five 10-shot groups, as smaller group sizes are very inefficient. I have personally tested the results on the range extensively using a Winchester Model 52 match rifle at 50 yards with match-grade ammunition, and was provided a large amount of U. S. Navy lot acceptance data for the 5.56mm MK262 (Black Hills) round fired at 300 yards from test barrels to analyze. The Navy has adopted the five groups of 10 shots each as their grouping measurement standard, but has taken it further by firing five 10-shot groups from each of two test barrels. If grouping results for either barrel is below their acceptance standard (4" ES), the lot is rejected. So far, no lots have been rejected. (FYI, the ES of the 10-shot groups run about 2.5" or less, rarely above 3", i.e., true "1 MOA, all day"). My analysis of the Navy acceptance data verifies the 5% standard deviation, at least in the average of all lots I examined. Several lots were only slightly higher.
If one wishes to perform a meaningful comparison of precision during load workup, I submit there is no better way to do it than firing at least five 10-shot groups and determining the average ES. It will also provide considerably more information such as good estimates of CMD and CEP, which is, I believe, superior to attempting to measure mean radius. An example:
Five 10-shot groups of a given load are fired at 100 yards. The ESs of each group are averaged, the average ES being 0.75". For 10-shot groups, the CMD multiplier is 1.30, and the CMD is 0.98" (0.75 X 1.30). This means that your rifle and load combination under constant firing conditions is capable of placing EVERY shot into an approximate 0.98 circle at 100 yards (or if you prefer, 0.93 MOA) under identical conditions. If you put much faith in "Mean Radius," you can then use Circular Error Probable (CEP), which is analogous. CEP is the diameter of a circle within which there is a 99% confidence that at least 50% of the shots will strike. My analysis indicates that CEP is 60.4% of CMD, but 60% is OK. So 0.6 X 0.98 = 0.59" (or 0.56 MOA). As statistics does not deal with precise numbers, but probabilities and deviations, it's safer to say that your rifle and load is capable of delivering a CMD of 0.93 and a CEP of 0.56 MOA within +/- 10%. That's pretty good for comparing load performance. Note that all the extraneous factors such as weather conditions, temperature, and wind are not considered in this analysis. If you want more confidence, simply fire more than 5 groups to establish the mean ES.
