“If you shoot a large number of groups from the same ammunition lot (let the computer do it, because no private person has the resources), the ES sizes you find are spread over a very wide range.”

I’ve done extensive computer grouping simulations. You can actually get very good consistency in extreme spreads by averaging the ESs measured from multiple groups if you use enough groups. The numbers of groups depends upon the number of shots per group. For example, the Navy’s practice of using 5 groups of 10 shots each will produce a mean ES with a standard deviation of approximately 5%. For two such 5 X 10 groups (which is actually what the Navy uses), the SD is somewhat lower than 5%. Regarding mean radius (MR), it is very closely related to ES if large numbers of shots are fired, and the relationship approaches MR = .25 X ES. For individual groups of small numbers of shots, the relationship between ES and MR can indeed be all over the place, the ratio typically being between 0.3 to 0.4 (or more). And I have substantiated these findings by actual test firing.

I have found that even single groups of 30 shots or more will produce very consistent values of ES and MR, also with very low SDs, typically 10% or less. And this information can very readily be translated into another measure, which I call the Mean Circle of Maximum Dispersion (MSMR), which can be determined fairly precisely with as few as 30 shots. The MSMR is the diameter of a circle which encompasses, theoretically, an infinite number of shots fired under identical conditions. Additionally, it has the benefit of allowing good estimates of the diameter which will contain various percentages of shots. An example is that for a large number of shots, 50% of them will be contained within a circle described by the MR. And as far as I am concerned, that’s about the only usefulness of the MR.

I have also discovered that the use of 3-shot groups is essentially worthless in understanding anything about grouping, as the average distribution of ESs of 3-shot groups is typically about 9:1 (least to most). Therefore, if you like using 3-shot groups, you’d better shoot lots of them and average the results. It’s actually statistically better to fire large numbers of 2-shot groups, e.g., 15 2-shot groups is will give more consistent results (lower SD) than 10 3-shot groups. I consider firing at least 5 groups of six shots (or perhaps six groups of 5 shots) to be about the minimum required to produce reliable grouping data. The NRA uses 5 groups of 5 shots for their accuracy evaluations, and that is also fairly efficient. If you want to depend on MR s a measurement, you cannot depend upon just measurements from one group, as it is far too variable between individual groups. You need to add together every distance between the shot and the group centroid for every group fired - not simply averaging the MR of each group.