SUMMARY

Neither total score nor group size, two of the most popular measures of shooting results, adequately measure shooting "skill", (consistency of shot
placement).    Total score reflects sight zero along with skill, while group size is overly sensitive to outliers.    Better statistics to use would examine consistency but not be dominated by the group's outliers.

TOTAL SCORE

The most basic measure of shooting prowess is total score.    Unfortunately, total score is a function of two factors:   how close the average shot is to the center scoring ring, (group placement), and how close each of the shots are to each of the others, (group dispersion).

Adjustable sights make group placement, by and large, a mechanical factor.    By measuring the distance between the center of a group of a large number of
shots and the center of the bullseye, sight corrections can be calculated and adjustments made so that the center of the shot group coincides with the center of the target black.    This is generally not what is meant by skill.    Skill in target shooting means combining all of the various ingredients, such as good position, sight alignment, breath control, proper trigger squeeze, accurate ammunition, good wind doping, etc., to get the tightest, most consistent group possible.    
To accurately measure the effects of the combination of all of these change in any single ingredient, you must examine how closely the shots strike on the
target.    In short, you must look exclusively at group dispersion.

GROUP SIZE

Group size is the distance between the two most extreme shots of a group, usually expressed in either inches or minutes of angle (MOA).    While this measure
does indeed break the mechanical factor out from the skill factor by looking only at group dispersion, it describes the worst case, ignoring the probability of that worst case reoccurring.    A group of twenty shots that are all within one MOA of the group center, except for one outlier which was thrown six MOA out to the left, has the same group size as a group of twenty shots evenly distributed within a seven MOA circle.    Clearly, the first group is better than the second, (the shot thrown out seems to have a low probability of reoccurring), but how do you compare groups when the difference is not so obvious ?    One popular way is to ignore the group's outliers.    

While there are some mathematical ways to determine when to discard an outlier, the most common procedure for adjusting for outliers is the "if only . . . "
method.    How many times have we heard, (or said), "if it wasn't for this outlier, this would be a half inch group!"    The "if only . . . " method is uncomplicated, but also unscientific, and can allow us to see what we want to see rather than what is actually there.    More objective ways to handle outliers are necessary to accurately compare group dispersions.

AVERAGE RADIUS

In his article "Target Evaluation ~ Computer Style", (Shooting Sports USA, March 1993, p.6), Lannie Dietle listed several statistics that he uses to measure
group dispersion, including average radius.    This is the sum of the distance of each shot from the group center divided by the total number of shots.    In other words, it is simply the average distance of the shots from the group center.    While outliers are included in this calculation, their impact is diluted by the averaging process.

RADIAL STANDARD DEVIATION

Another measure of group dispersion is called radial standard deviation (RSD).    The RSD describes the "typical" deviation of a shot away from the center of the group.    The tighter the group, the smaller this typical deviation will be.    This measure is similar to the more common standard deviation statistic except that, rather than describing the traditional one-dimensional, or univariate, distribution of values around an average, it describes a two-dimensional, or bivariate, distribution of values around, (or that radiate out from), a center point.

AVERAGE RADIUS VS. RSD

While both the average radius and the RSD describe group dispersion and compensate for outliers, they differ in one very important way.    Consider that the ten shot string which you just fired is only a sample group of the large number of shots which you have already fired or will eventually fire using the same combination of position, equipment, ammunition, etc.    This sample may be different from the larger group or "universe", just as ten consecutive coin flips may not result in exactly half heads and half tails, even though we expect that a large number of coin flips would.    The average radius and the RSD describe the current sample group distribution.    However, you can use the RSD to estimate the standard deviation of the universe of all shots that already have been or eventually will be fired under these identical circumstances.

ESTIMATED UNIVERSE STANDARD DEVIATION

By examining the estimated universe population standard deviation, you can compensate for any random error due to the small size of the sample group. Just as three "heads" in a row would not convince you that you had a two-headed coin, perhaps the smaller dispersion of a ten shot sample group indicates larger universe dispersion, (poorer long-term consistency), than a slightly larger dispersion of a twenty shot sample group.    By using this statistic, sample groups of different numbers of shots can be compared.

BENEFITS

Decoupling the mechanical factor of shooting results from the skill factor can help provide shooters with more accurate answers to many questions.    A target shooter considering altering his position would, of course, be very interested in the effect this change would have on the size of his shot group,
regardless of the effect the change had on the firearm's "zero."    He could measure the effect of the change very simply by comparing the estimated universe
dispersion calculated before the change to the same statistic calculated after the change.    If the estimate went down, (the forecasted group is tighter), he should adopt the new position;   otherwise he should keep his old position.

Hand loaders should also find these statistics extremely useful.    Rather than examining the group size of a new load, since we have seen the problems with that statistic, the estimated universe dispersion of the current load could be compared to those of other loads.    In this manner, the best load for a particular firearm could be reliably determined.    This method would also apply to comparing different factory loads.    Another example of the uses of the estimated universe dispersion would be to determine a shooter's progress.    Graphing the universe's standard deviation's progress over time would present a visual
representation of a shooter's improvement.    A team coach could even use these statistics to compare shooters and select her best performers for team matches.    While none of these questions were impossible to answer before, they required carefully controlled conditions and some scheme to rationalize away any outliers in order to analyze group dispersion.    Using the statistics discussed here, these and other questions can be answered in the normal course of shooting.

USING COMPUTERS

Part of the reason that total score and group size are such popular measures of shooting skill is that they are easy to calculate.    Total score involves addition or, perhaps if you want an average, some division.    Group size can be found with a ruler or a drawing compass.    The average radius and radial standard deviation both involve numerous measurements and calculations, including finding squares and square roots of many numbers.    This is a problem best solved using a computer.    In fact, there are many PC programs that can help you with these calculations.    Like Mr. Dietle, I have written a program to do this.    It is called ScorStat, and runs on IBM and compatible computers.    There are several other programs available, as well.   While these programs differ in their approaches to the problem, they strive to achieve many of the same goals.    I would expect to see additional computer programs become available as more people become interested in this approach.

CONCLUSION

Statistics such as average radius and radial standard deviation are superior measures of shooting consistency to the more common total score and group size.
They ignore sight zero, are not dominated by outliers and the RSD can be adjusted for the random sampling error associated with small sample sizes.    These measures, however, are computationally intense and are more easily solved by using computer software developed for the purpose.

FURTHER READING

For those who desire further reading on the subject, I suggest . . . "Statistical Measures of Accuracy for Riflemen and Missile Engineers" . . . by Frank E. Grubbs, Ph.D. (Harve De Grace, MD: by the author, 4109 Webster Road, 1991