Circle of Maximum Dispersion

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.

Looking into the problems of measuring the dispersion of a gun/ammunition combination is something that should be done more often. As such, your post is most interesting.
As you write, explaining the details of your method woul require a considerable longer text. Why not write up a longer document and make it available as pdf file on the web?

Having said that, I must also say that I am rather sceptical about the necessity of a new method. In my view, the standard deviation is the way to express group size. Having computers available today, it is no longer difficult to compute. And according to my computer experiments the standard deviation is the most resistant measure to a variable number of shots in the group(s).

Personally, I think anything less than 10 shots has too much chance in it. The NRA uses 5 consecutive 5 shot groups (25 shots) and I think that is a minimum for a basic test. Military tests in the U.S. traditionally use 90 shots: originally in ten shot groups, now in 30 shot groups (starting with M855). And I am still amazed how much variation you can find between 30 shots groups.

The usual shooting journal tests of five shots, or factory targets of three shots do not tell us anything.

P.S. If you want to work a really unploughed (compared to group dispersion) field, look into the change of mean point of impact between shot groups due to the random character of impacts. Or in other words: can I trust my zero?

Edit for clarity: 90 shots in a) 9 groups of 10 shots each or b) in 3 groups of 30 shots each.

The fundamental reasoning for use of the CMD approach is it gives essentially the same result no matter what the group size you prefer, so long as an adequate number of groups are fired, The standard deviation of the group mean will always be lower the more shots are in the group. Note that the SD of the mean is the population SD divided by the square root of the number of data points per group. The actual value of the SD is not particularly important so long as it is consistent (I could write a lot on that point). In any event, the SD is built into the CMD multiplier. That is why the CMD factor is lower for larger group sizes.

Interesting discussion, thanks. For another idea, MBA used the term “CEP” to describe Gyrojet accuracy. The Glossary in my Gyrojet book has these two entries:

"CEP; Circular Error Probable. An unusual (when discussing firearms accuracy) method selected by MBA to describe the dispersion, or accuracy, of its rockets. The term is more often used to describe the precision of a missile warhead or re-entry vehicle, not a small arms projectile. CEP is the radius of a circle into which a Gyrojet or other rocket [or bullet] will impact 50 percent of the time. If a 100-round salvo is launched, 50 rockets will impact inside the circle described and 50 percent will hit outside of it. When CEP is used as a measure of accuracy, it is generally assumed that while 50 percent of the group will hit inside the circle, 93 percent will hit inside a circle two times the radius of the first circle, and the remaining 7 percent will hit in a circle three times the radius of the first circle. (see Mil)

Mil. A term sometimes used by MBA to describe a rocket’s accuracy. A mil is the angle forming an arc that is 1/6400 of the circumference of a circle, or about 1.2 inches at 100 feet."

Although Robert Mainhardt was a gun guy, his real expertise was in nuclear missiles and he chose a term he was familiar with to describe his Gyrojets’ “accuracy.”

Thw CEP and CMD are not so different in concept, as the CMD is a circle of a diameter of which 100% of shots will impact.

“P.S. If you want to work a really unploughed (compared to group dispersion) field, look into the change of mean point of impact between shot groups due to the random character of impacts. Or in other words: can I trust my zero?”

Actually, my Excel model could relatively easily be modified to do that. Instead of calculating the ES for a group, it could calculate the coordinates of the centroid of a group for any group size, then determine the mean distance and standard deviation between centroids of multiple groups. I’d have to think about it for awhile to see if I can come up with something that could be useful. I don’t know what that might be right now, i.e., even if you know that, what good is it?. It might possibly suggest if group size selection might minimize the distance between groups.

Regarding your question:
People do their zeroing based on the assumption that the mean point of impact (MPI) can exactly be determined from a shot group. But if you shoot a number of groups, each will have a different MPI.
In a way, the MPI does a random walk on the target. Therefore one can not exactly determine the “real” position of the MPI, only estimate it.

So the question is to find a statistical measure that tells me: how large is the uncertainty in determining the MPI? What is the probable distance between the MPIs of two consecutive shot groups? I assume it will depend on shot dispersion as well as number of shots in a group.
I know of nothing in the literature that considers this effect. But it would be good to be able to answer a question like: “Is this change of MPI between two ammunition types caused by random effects or does it show a real difference?”

I sat down and modified my model yesterday to do just that - essentially to calculate the centroid of each group and then to calculate the average distance and the greatest distance between centroids of multiple groups. It’s not as large as you might expect - typically 9% to 13% of the calculated CMD (as previously discussed) for the average distance, and closer to 16% to 18% for the maximum distance. It’s really not as dependent upon group size as I expected. I’ll write up more details in a few days.

Now I can provide a little more detail about MPI wander calculations. What I did was to modify my model to generate random groups of 3, 4, 5, and 10 random shots. The centroid of each group was calculated. Then distances between each of the multiple centroids were calculated, from which the mean distance between each pair of group centroids, and the greatest distance determined. For example, for 5 groups, there would be 10 distances calculated (1-2, 1-3, 1-4, 1-5, 2-3, 2-4, 2-5, 3-4, 3-5, and 4-5), from which mean and maximum inter-centroid distances would be determined. All conditions and constraints as the earlier 98% confidence level CMD formulas remain the same. Essentially thousands of data points were calculated this way, and averages derived.

Note that there is no theoretical basis whatsoever for the results. It is simply a brute-force method of calculation, which some operations research personnel would call using the “Monte Carlo” approach. Its accuracy depends on the generation of a large number of calculated (not measured) data points. The procedure involves determining an average group ES at some range, multiplying it by the corresponding CMD diameter multiplier, then calculating the products of the CMD diameter and the mean and maximum inter-centroid multipliers.

(a) 3-shot groups. The average inter-centroid distance multiplier for multiple groups was calculated to be 13.7% of the 3-shot CMD diameter as previously shown. The maximum inter-centroid distance was 19% of the 3-shot CMD

(b) 4-shot groups. The corresponding CMD diameter multipliers are 9.4% and 16.7%

© 5-shot groups. The corresponding CMD diameter multipliers are 9.1% and 14.8%

(d) 10-shot groups. The corresponding CMD diameter multipliers are 10.3% and 13.1%


  1. For 4-shot groups, the average ES for 8 groups fired is determined to be 1.0" at 100 yards. Therefore the CMD diameter is 2.10 (CMD multiplier from above) X 1.0" = 2.10". Therefore, the mean distance between centroids of random multiple 4-shot groups aimed at the same target aiming point (AP) would be 2.10 X 0.094 = 0.2". The corresponding maximum distance between 4-shot groups using the same AP would be 2.10 X .167 = 0.35". These are averages of a very large population, not absolutes. Therefore, if you fired, let’s say, 5 groups of 4 shots each at the same point on the same target at 100 yards, then marked each group after firing with different colored magic markers to distinguish among them, you would expect the ES of the random group centroids to be between 0.2" and 0.35" apart, i.e, the"MPI wander". At 200 yards, the group centroid ES mean and maximum ES values (MPI wander) would be expected to be about doubled, e.g., 0.4" to 0.7".

  2. Three-10 shot groups fired at 100 yards have an average ES of 1.5". CED diameter: 1.35 X 1.5" = 2.02". Using the same information as example 1, the expected mean inter-centroid spacing would be 2.02" X .103 = 0.21" and the maximum random centroid spacing would be 2.02" X 0.131 = 0.26". At 200 yards, the expected group centroid ES wander would then be 0.42" to 0.52".

Now, maybe one shooter in 100,000 might see the utility of performing this procedure, but in any event, this is a way it can be done. At least it’s better than a wild guess.

I am away from home over the weekend; it will take a few days before I can give you the figures I arrived at. They will be in standard deviations, relative to the shot dispersion. This also will give me some needed time to go through your description step by step.

Here are basic results from my computer simulation.

A) What did I do:
First I generated normally distributed “shots” having a mean point of impact (centroid) of x=0 and y=0. The standard deviation of the shot dispersion is 1.0 in both directions.

Then I grouped these shots into groups of 3, 4, 5 … 32 shots and had my computer calculate the statistics for each group.

B) for those more accustomed to extreme spread (ES):
Assume I claim my rifle is able to hold the so-called MOA, which roughly corresponds to 1 inch ES from 5 shots at 100 yards. My simulation shows that on the average a 5-shot ES is 3.067 times the standard deviation. So the standard deviation in this example can be estimated from the ES as 1/3.067 or 0.326 in.

Another result is that a 10-shot group ES on the average is 3.814 times the standard deviation. This shows that a 10-shot group on the average is 3.814/3.067 = 1.24 times larger than a 5-shot group with the same underlying dispersion. ES is very sensitive to shot count.

C) Result regarding randomness of mean point of impact (MPI)
Contrary to real shooting, in a simulation the real MPI is known, which is at x=0, y=0 in this case. Each shot group has its own MPI. Looking at the MPIs of all shot groups, the MPIs are randomly dispersed around x=0 and y=0.
5-shot-MPIs are dispersed with a standard deviation of 0.448 around the real MPI.
10-shot-MPIs only have a value of 0.316 which is 30 percent smaller. So a 10-shot group MPI will on the average be closer to the real MPI.

D) Effect on Zeroing
In reality, the “real” MPI is not known. The shooter only has the MPI of his current shot group and is left wondering how far the MPI of the next shot group may be away from it.
The value for 5-shot groups is 0.633 standard deviations; for 10-shot its 0.448.
That means there always is randomness in the MPI. I can never be sure that an apparent movement is exclusively due to, for example, adjusting the scope.

Remember: relative standard deviation “1.0” in the above examples is the actual shot dispersion of the ammunition/rifle/shooter combination. This may be 0.326 inch for assumed “1 MOA” or any other real world value. Taking 0.326 inch shot dispersion multiplied by 0.633 one arrives at a standard deviation of MPI movement of 0.206 inch. Three times this deviation (0.618) tells us where in 99 percent of the cases the next MPI will be: somewhere plus/minus 0.618 in (15.7 mm) vertically and/or horizontally from the current MPI.

This figure includes extreme cases, but I believe it nevertheless should be considererd and investigated.