The theory is that the standard deviation of group sizes is a function of the number of shots in the group, and that this estimator of the population standard deviation is "better" than calculated estimates of small sample standard deviations. Certainly, this estimator is easier to calculate.

The standard deviation of the group size distributions is taken as .275 (for 5 shot groups) and .179 (for 10 shot groups) times the mean. Thus, the distribution of five shot groups sizes has a standard deviation of .275 times average group size and the distribution of 10 shot group sizes has a standard deviation of .179 times average group size.

Where do the .275 and .179 come from?

I used the 2002 and 2004 Cast Bullet Association (CBA) National Match results for groups, 45ACP 50 yard pistol groups from "Cast Bullets" by E. H. Harrison and my records of shooting a Savage Model 12BVSS in 223 and a M54 Winchester in 30/30 at 100 yards.

Average group size, standard deviation of group size, and standard deviation of group size over average group size were calculated for many shots.

Note the consistency of the Standard deviation/Avg. numbers in both tables.

Explaining the tables: In 2002, at 100 yards, there were 196 five shot groups shot, with 49 averages of four groups shot Each of 49 shooters shot 4 groups. Each set of four groups was averaged, the standard deviation was calculated, and the statistic "standard deviation/average" was calculated. The average of all 49 averages was 1.117". The average of all 49 standard deviations was .314". The average of all "standard deviation/average" statistics was .275. Then for these 196 groups, on the average, the standard deviation was 27.5% of the average or arithmetic mean, of the group size.

Why is the standard deviation of group size a fraction of average group size?

Well, first off, because it is. The standard deviation of any distribution is a fraction of the mean. What I'm proposing here is that the standard deviation of group size is a CONSTANT fraction of the mean, and that that fraction varies with n. Then for any n = number of shots in the groups, the distributions are scale models of each other, varying with group size. I'm a little uncomfortable with this, because, as an example, the standard deviation of bullet weights seems fairly independent of mean bullet weight. Four hundred grain bullets vary +/- half a grain; sixty-grain bullets vary +/- half a grain. My weighing experiences tell me that the standard deviations of cast bullet weights is independent of bullet weight. However, for close average group sizes, it doesn't matter.

Then there's the fact that the data are remarkably similar. My experience with Economic and gun-related data is that data bounces around furiously. This doesn't. Looking at the rightmost column on each table shows lovely consistency.

And again, there's the fact that somebody else came to the conclusion long before I did.

"Shot Group Statistics", "Matching Ammo To The Rifle" and "Estimating Ammo Quality From Shot Group Diameters" by Jeroen Hogema explained the notion that the standard deviation of group size is a constant fraction of group size, that fraction varying with n. These and more are to be found on the Internet at: http://home-2.worldonline.nl/~jhogema/ballist.htm

I sent Jeroen a copy of this article for his review and comment; He did some simulation to estimate the relationship between the standard deviation and arithmetic mean of shot groups.

His result is that the standard deviation of five shot groups is .2691 times group diameter, and the standard deviation of ten shot groups is .1947 times group diameter. These multipliers are in substantial agreement with the multipliers I derived from real-world data.

Here is a letter from Jeroen Hogema, explaining his method:

From: Jeroen Hogema

To:     Joe Brennan

Date:  9 November 2004

• The following assumptions were used.

• The location of the X-co-ordinates of the shots (Left-Right) vary according to a normal distribution.

• The location of the Y-co-ordinates of the shots (High-Low) vary according to a normal distribution.

• The x- and y-distributions are independent (i.e. no stringing of the shot group).

     The standard deviations of Left-Right and High-Low deviations are equal (i.e. a circular shot group, not an elliptical one). This common sd is referred to as the ‘shot group sd’. Not to be confused with the sd of group diameters as discussed below.

The rifle is accurately zeroed, i.e. the means of the X and Y distributions are zero. Thus, the centre of a shot group will, on average, be in the ‘10’.

For a given value of this common shot group sd, I did the following.

‘Fired’ (simulated) a number of shots (either 5 or 10), from the resulting shot group, measured the resulting shot group diameter (centre-centre) and kept that as the observations-to-be-analyzed; repeated this many times, thus collecting my observations.

Then calculated the resulting mean and SD of the observed shot group diameters.

Repeated this process for a wide range of shot group SD’s.

Results are shown in Fig. 1.

Fig. 1 Relationship between standard deviation and mean of shot group diameters as a function of the number of shots in a group (results from simulations and fitted functions).

     Clearly, there is a linear relationship between the sd and the mean of the shot group diameter. The slopes that I found, compared to what you wrote:

I’d say that we agree.

Total number of shots simulated to generate Fig. 1:  6,600,000

Note: I suspected that small-group-shooters had small standard deviations; that their groups didn't vary much. Regression analysis for average group size vs. standard deviation for the CBA data failed to show any relationship.

Note: There are other solutions than n1 = n2, these can be found by fiddling with the EXCEL spreadsheet/program, but I don't know how to include them in a chart without confusing both me and the reader.

2005 CBA Nationals

2006 CBA Nationals