While not directly associated with ammunition, some may be interested in a little hobby computer project I did yesterday. It may not be a totally original idea, but it was interesting to consider the ramifications. Please contact me if you want further discussion, as there’s no way I can explain everything in a few sentences.
The Circle of Maximum Dispersion
As a result of some discussions among friends regarding grouping capability, I decided to examine the topic more closely. There are many methods of quantifying grouping capability (often erroneously called “accuracy,” which is a separate topic), probably the best known of which is “extreme spread,” or ES. This measurement involves only firing a group of shots at a target at some known distance, then measuring the distance between the centers of the two most widely spaced bullet holes of the group. Due to the variability and randomness of the grouping, this simple ES measurement provides only a crude indication of the grouping potential of a given system which comprises the ammunition, the gun, the shooter, and the firing environment, all of which are consistent and uniform.
In order to perform a more scientific analysis of grouping capability, it seemed to me that a better approach would be to develop a concept which I call the Circle of Maximum Dispersion, hereafter CMD. CMD is a circular area of a diameter which at any distance will have a very high probability of containing ALL shots, no matter how many, fired by the system at that distance. This dispenses with arguments involving how many groups or how many shots per group provides the best indication of a system’s grouping capability.
The mathematical determination of the diameter of the CMD is not a difficult exercise, wherein average ES data obtained by firing of a relatively small number of shots (30-50) in small groups (2 to 10 shots each) can be translated into the diameter of the CMD at the same distance. What I did was to develop a mathematical model in Microsoft Excel which simulates randomly firing of thousands of rounds in different group sizes into a constrained unit circle, i.e. a circle with a radius of one unit. That data can then be massaged by standard statistical techniques to arrive at a circle of a diameter which will contain all bullet holes. The Excel model is based upon the premise that any group of shots is composed of a series of two shot subgroups. or shot pairs. For example, a two-shot group has one shot pair, a three-shot group has three shot pairs (1-2, 1-3, and 2-3), a four-shot group has six (1-2, 1-3, 1-4, 2-3, 2-4, and 3-4), a five-shot group has ten, a ten-shot group has 45, and so on. Another way to look at this is that a single 10-shot group is the equivalent of firing 45 two-shot groups, indicating the far greater data richness of large groups. Excel can easily generate what is essentially an infinite number of constrained random shots characterized by the coordinates of each, calculating the length of each shot pair, combines them into groups, then analyzes them, shot pair by shot pair, using statistical methods. The final result is a CMD Multiplier. This is a factor by which the average ES obtained from a number of groups fired can be multiplied to determine the CMD diameter, with 98% statistical confidence.
To discuss details of the exact methods used would require far too much space, but the simplified results of the Excel analysis are as follows:
Group size: 2 shots; Number of groups to be fired; 25; CMD Multiplier: 3.44 (98% confidence)
Group size: 3 shots; Number of groups to be fired: 10; CMD Multiplier: 2.58
Group size: 4 shots; Number of groups to be fired: 8; CMD Multiplier: 2.12
Group size: 5 shots; Number of groups to be fired: 5; CMD Multiplier: 2.04
Group size: 10 shots; Number of groups to be fired: 3; CMD Multiplier: 1.51
Group size: 15 shots; Number of groups to be fired: 2; CMD Multiplier: 1.26
Group size: 20 shots; Number of groups to be fired: 2; CMD Multiplier: 1.25
Example: The average ES of five 5-shot groups fired at 100 yards is 0.5”. The CMD, with 98% statistical confidence, will not exceed a maximum diameter of 0.5 X 2.04 = 1.02” at 100 yards. At other distances, the CMD would increase or decrease in proportion to the distance, i.e, at 200 yards, CMD would be 2 X 1.02” = 2.04”.
Essentially, what is happening is that the CMD multiplier is the ratio of the average group extreme spread of a given number of shots to the maximum dispersion at any given range involving an infinite number of shots, i.e., the CMD. Therefore, in order to get a reasonable estimate of the average ES, numerous groups must be fired.
For several practical considerations, my belief is that using 5 and 10 shot group sizes is best, but any group size will work. Using a larger number of groups than recommended will improve the confidence level, as well as the number of shots per group, but at the cost of more time and ammunition required. Using fewer groups is not recommended, as the standard deviation of the average group size will increase, invalidating the calculated CMD multiplier and lowering the confidence level. One criterion of the mathematical model was to choose a number of groups which would result in a standard deviation of 10% or less of the average CMD diameter.
There are a number of other interesting ramifications of the CMD approach I will not fully explore. For example, the simulation predicts that with a 99+% confidence, 50% of all shots fired will strike within 60% of the CMD diameter. Using the previous example, for a CMD of 2.04", AT LEAST 50% of hits will be within a circle which is 60% of CMD, or 0.6 X 2.04" = 1.22" at 200 yards. I consider this a far superior measure of “accuracy” than the often-used mean radius.