            Using Quantitative Methods to Analyze Shooter Skill
                           by John E. Leslie III
                    Copyright 1993 by John E. Leslie III
                            All Rights Reserved


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 ingredients, or to analyze the result of a 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 which 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 which
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 which 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 computationaly 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: Frank E. Grubbs, 1965).
