Difference between revisions of "Projectile Dispersion Classifications"
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Before considering the mathematical models that will be used for the actual statistical analysis, let's consider the assumptions of various dispersion models and hence the intrinsic functions of how shots are dispersed. The [http://en.wikipedia.org/wiki/Normal_distribution Normal distribution] is the broadly assumed probability model used for a single random variable and it is characterized by its mean <math>(\bar{x})</math> and standard deviation <math>(\sigma)</math>. Since we are interested in shot dispersion on a twodimensional target we will assume that the two dimensional analog of the Normal distribution, the [http://en.wikipedia.org/wiki/Bivariate_normal_distribution Bivariate Normal Distribution]. This distribution describes, at least approximately, the dispersion of a gunshots about their center point, (<math>\bar{h}</math> and <math>\bar{v}</math>). The bivariate normal distribution also has separate parameters for the standard deviation in each dimension, <math>\sigma_h</math> and <math>\sigma_v</math>, as well as a correlation parameter ''ρ''. The full bivariate normal distribution is thus:<br >  Before considering the mathematical models that will be used for the actual statistical analysis, let's consider the assumptions of various dispersion models and hence the intrinsic functions of how shots are dispersed. The [http://en.wikipedia.org/wiki/Normal_distribution Normal distribution] is the broadly assumed probability model used for a single random variable and it is characterized by its mean <math>(\bar{x})</math> and standard deviation <math>(\sigma)</math>. Since we are interested in shot dispersion on a twodimensional target we will assume that the two dimensional analog of the Normal distribution, the [http://en.wikipedia.org/wiki/Bivariate_normal_distribution Bivariate Normal Distribution]. This distribution describes, at least approximately, the dispersion of a gunshots about their center point, (<math>\bar{h}</math> and <math>\bar{v}</math>). The bivariate normal distribution also has separate parameters for the standard deviation in each dimension, <math>\sigma_h</math> and <math>\sigma_v</math>, as well as a correlation parameter ''ρ''. The full bivariate normal distribution is thus:<br >  
<math>  <math> 
Revision as of 14:16, 25 May 2015
Contents
Modeling Dispersion
Before considering the mathematical models that will be used for the actual statistical analysis, let's consider the assumptions of various dispersion models and hence the intrinsic functions of how shots are dispersed. The Normal distribution is the broadly assumed probability model used for a single random variable and it is characterized by its mean \((\bar{x})\) and standard deviation \((\sigma)\). Since we are interested in shot dispersion on a twodimensional target we will assume that the two dimensional analog of the Normal distribution, the Bivariate Normal Distribution. This distribution describes, at least approximately, the dispersion of a gunshots about their center point, (\(\bar{h}\) and \(\bar{v}\)). The bivariate normal distribution also has separate parameters for the standard deviation in each dimension, \(\sigma_h\) and \(\sigma_v\), as well as a correlation parameter ρ. The full bivariate normal distribution is thus:
\(
f(h,v) =
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1\rho^2}}
\exp\left(
\frac{1}{2(1\rho^2)}\left[
\frac{(h\bar{h})^2}{\sigma_h^2} +
\frac{(v\bar{v})^2}{\sigma_v^2} 
\frac{2\rho(h\bar{h})(v\bar{v})}{\sigma_h \sigma_v}
\right]
\right)
\)
For the overall equation note that the following restrictions apply:
\(1 ≤ \rho ≤ 1\)
\( \sigma_h>0 \) and \( \sigma_v>0 \)
Since we are primarily interested in the dispersion component, the overall assumption is that weapon is properly sighted so that the center of impact is the same as the point of aim. In practice this can be achieved with a simple translation of the horizontal and vertical coordinates from absolute values to values relative to the average point of impact. Therefore the terms controlled by sighting in the gun (\(\bar{h}\) and \(\bar{v}\)) drop out in the simplification of the dispersion equation.
Simplifications into cases
Looking at the overall equation two different mutually exclusive simplifications can be made:
 Either \(\sigma_h = \sigma_v\) (equal variances) or \(\sigma_h \neq \sigma_v\) (unequal variances).
 Obviously if we could measure both \(\sigma_h\) and \(\sigma_v\) with a very high precision (e.g 6 significant figures), then the two quantities would never really be equal. But in many cases the assumption is reasonable. In reality since shooters typically collect only a small amount of data, statistical tests will fail to detect a difference unless the difference is great. In such cases the shot pattern would be noticeably elliptical not round.
 Either \(\rho = 0\) (uncorrelated) or \(\rho \neq 0\) (correlated).
The pair of mutually exclusive assumptions thus results in four cases for analytical evaluation.
Error Propagation
In general shooting has a number of different error sources. The "basic" assumption is that the error sources are normally distributed and independent. Thus for example the vertical error component would be something like:
\(\sigma_v^2 = \sigma_{v,Weapon}^2 + \sigma_{v,Ammunition}^2 + \sigma_{v,shooter}^2\)
and there would similar equations for \(\sigma_h^2\) and \(\sigma_{RSD}^2\).
Notes:
 The error sources can't be measured independently. Only the total error is observable. This is a very important consideration since the total error thus limits the precision which any of the individual error factors can be determined.
 If any one of the error sources is more than about three times the sum of the other two, then that error source essentially controls the overall error of the measurement.
 These error sources could be decomposed further into "finer" errors. For instance consider the multitude of variables that a handloader controls when loading a cartridge.
 The fact that a very small amount of data is typically collected by a shooter greatly limits the overall precision of the mathematical analysis. In statistics this is known as the "small sample" problem. For example to get "good" estimates for the normal distribution parameters \(\bar{x}\) and \(\sigma\), 30 measurements are desired.
 In general for a given sample size of n measurements the mean, \(\bar{x}\), is determined relatively (i.e as a percentage) much more precisely than the standard deviation, \(\sigma\) (or other dispersion factor).
Stringing
To the extent that either \(\sigma_h \neq \sigma_v\) or \(\rho \neq 0\) then elliptical shot groups will result instead of circular shot groups. If the shot groups are not round then we have three options. We can:
 Isolate the error source experimentally and remove it (for instance weigh gunpowder carefully to remove vertical stringing).
 Use a mathematical model for analysis that allows for stringing.
 Scale the raw data to remove the stringing.
Obviously the experimental reason for stinging may not be obvious and easy to remove. Experimental designs to isolate and quantify the source of the stringing are beyond this basic discussion at this point, but possible.
There isn't any theoretical ballistic requirement that requires correlation between the horizontal and vertical dispersion of gunshots. Therefore, most statistical measures implicitly assume \(\rho = 0\). In general if \(\rho \neq 0\), then there would be an elliptical shaped group with the major axis oriented at some angle to the horizontal or vertical axis.
We do know that targets can often exhibit vertical or horizontal stringing as evidenced by an elliptical shaped group along the vertical or horizontal axis respectively. Obviously in such cases \(\sigma_h \neq \sigma_v\).
 (1) The primary source of horizontal stringing is crosswind.
 If we measure the wind while shooting we can bound and remove a “wind correction” term from that axis. E.g., "Suppose the orthogonal component of wind is ranging at random from 010mph during the shooting. Given lagtime t this will expand the nowind horizontal dispersion at the target by \(\sigma_{wind}\)."^{[1]} Since variances are additive we could adjust \(\sigma_h\) via the equation \({\sigma'}_h^2 = \sigma_h^2  \sigma_{wind}^2\).
 (2) The primary source of vertical stringing is muzzle velocity.
 We can actually measure with a chronograph and then correct for that source of variation. E.g., If standard deviation of muzzle velocity is \(\s_{mv}\) then, given the bullet's ballistic model for the given target distance, the vertical spread attributable to that is some \(\s_v\). Here too we can remove this known source of dispersion from our samples via the equation \(sigma_v^2 = \sigma_v^2  f(\sigma_{mv}^2\). This adjustment is shown in several of the examples:
Four Special Cases for Dispersion
Neglecting flyers, and assuming perfect aim, the overall assumptions about shot dispersion result in four cases for statistical analysis.
In the four cases below the assumptions use approximately equal to \((\approx)\)) rather than strictly equal to (=). This is an acknowledgement that we are dividing the cases into ones that are close enough to be useful, even though they most certainly are not exact. There is absolutely no method by which the true population values for \(\bar{h}, \bar{v},\sigma_h\) and \(\sigma_v\) can be determined. We can only get experimental estimates from calculations based on sample data for the factors \(\mu_h, \mu_v, s_h\) and \(s_v\) and these estimates are at best only good to a scant few significant figures. Thus the difference between approximately equal to and strictly equal to is really under experimental control. In other words, we can theoretically make the measurements as precise as we want by collecting more data, but practically there are limits.
The formulas for the distributions are given in terms of the population parameters (i.e. \(\bar{h}, \bar{v}, \sigma_h, \mbox{and } \sigma_v)\) rather than the experimentally determined factors (i.e. \(\mu_h, \mu_v, s_h, \mbox{and } s_v)\) on purpose to emphasize the theoretical nature of the assumptions. Of course the "true" population parameters are unknown, and we estimate them with the experimentally fitted values about which there is some error.
Given:
 \(\sigma_h \approx \sigma_v\)
 \(\rho \approx 0\)
then the mathematical formula for the dispersion distribution would be the Rayleigh distribution:
\(f(r) = \frac{r}{\sigma_{RSD}^2} e^{r^2/(2\sigma_{RSD}^2)}, \quad r \geq 0,\) and \(\sigma_{RSD}\) is the distribution shape factor known as the Radial Standard Deviation.
This is really the best case for shot dispersion. Shot groups would be round. Strictly, for the Rayleigh distribution to apply, then \(\sigma_h = \sigma_v\), in which case \(\sigma_{RSD} = \sigma_h = \sigma_v\). For the "loose" application of the Rayleigh distribution to apply, then \(\sigma_{RSD} \approx (\sigma_h + \sigma_v)/2\)
The following statistical measurements are appropriate:
 Circular Error Probable (CEP)
 Covering Circle Radius (CCR)
 Group Size (GS)
 Figure of Merit (FOM)
 Mean Radius (MR)
 Rayleigh Distribution
Notes:
 In this case that the FOM, and Group Size are different measurements.
 The group size would only depend on the two shots most distant in separation. The FOM would depend on two to four shots. For a large number of shots we'd typically expect four different shots to define the extremes for horizontal and vertical deflection.
 For the measures for the CCR, the GS and the FOM measurements a target would a ragged hole would be acceptable, but for the rest of the measures the {h,v} positions of each shot must be known.
 Experimentally the radial distance for each shot, i, is \(r_i = \sqrt{(h_i  \mu_h)^2 + (v_i  \mu_v)^2}\)
 The conversion to polar coordinates results in each shot having coordinates \((r, \theta)\). (a) The conversion implicitly assumes that the polar coordinates have been translated so that the center is at the Cartesian Coordinate of the true center of the population \((\bar{h}, \bar{v})\). (b) The distribution of \(\theta\) is assumed to be entirely random and hence irrelevant. This assumption is testable. (c) The distribution is thus converted from a twovariable distribution to a onevariable distribution.
Given:
 \(\sigma_h \approx \sigma_v\)
 \(\rho \neq 0\)
 The {h,v} position of each shot must be known.
The following statistical measurement is appropriate:
 Elliptic Error Probable
Given:
 \(\sigma_h \neq \sigma_v\)
 \(\rho \approx 0\)
 The {h,v} position of each shot must be known.
then the mathematical formula for the dispersion distribution would be:
\(
f(h,v) =
\frac{1}{2 \pi \sigma_h \sigma_v}
\exp\left(
\frac{1}{2}\left[
\frac{h^2}{\sigma_h^2} +
\frac{v^2}{\sigma_v^2}
\right]
\right)
\)
The following statistical measurements are appropriate:
 Diagonal
 Individual Horizontal and Vertical variances
In this case the horizontal and vertical standard deviations could be determined independently from the horizontal and vertical measurements respectively.
Given:
 \(\sigma_h \neq \sigma_v\)
 \(\rho \neq 0\)
 The {h,v} position of each shot must be known.
then the mathematical formula for the dispersion distribution would be:
\(
f(h,v) =
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1\rho^2}}
\exp\left(
\frac{1}{2(1\rho^2)}\left[
\frac{h^2}{\sigma_h^2} +
\frac{v^2}{\sigma_v^2} 
\frac{2\rho h v}{\sigma_h \sigma_v}
\right]
\right)
\)
This mathematical formula will be called the General Bivariate Gaussian distribution. This is really the most complex case for shot dispersion. Shot groups would be elliptical or eggshaped.
Measuring Precision
The following text considers mainly shots from a direct fire weapon firing a single projectile on a vertical target within the line of sight, for example rifle or pistol shots. Such weapons as shotguns, intercontinental missiles, and motars would have some similar characteristics, but also have factors that are neglected in the measurements.
Dispersion Units
When we talk about shooting precision we are referring to the amount of dispersion we expect to see of each shot about a center point (which shooters adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle.
Linear distance typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards.
Angular Size is another common unit and is the angle at the tip of the cone of fire since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.
Linear Distance
In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches.
Mil
The other common linear unit is the mil, which simply means thousandth. For example, at 100 yards a mil is 100 yards / 1000 = 3.6". Some more benign confusion also persists around this term, with some assuming "mil" is short for milliradian, which is an angular unit. Fortunately, a milliradian — 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards — is almost exactly equal to a mil so there’s little harm in interchanging mil, mrad, milrad, and milliradian.
Angular Size
The overall assumption is that the 2dimensional precision is like a cone that projects linearly from the muzzle of the gun  i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases.
For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and a three foot group at 1000 yards.
Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the projectile from the muzzle results in an projectile instability which is damped by air resistance. In this case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a longer distance is larger, but not geometrically larger. Note however that if you were using an angular measure, then the group size would be smaller at 100 yards than 50 yards.
Minute Of Arc
One of two popular angular units used by shooters is MOA, though there is some ambiguity in this term. From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree.
At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047".
Shooter's Minute of Angle
At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to 1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or SMOA.
Conversions between measuring units
See Angular Size for detailed illustrations and conversion formulas.
Dispersion Measures
Nine different measures have been used to characterize the dispersion of bullet holes in a sample target.
Some are easier to calculate than others.
In the following formulas assume that:
 We are looking at a target reflecting n shots
 We are able to determine the center coordinates h and v as needed for analysis. For example for
extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target.
Some additional mathematical symbols and variables:
NSPG  Number of Shots Per Group
\(\mu_h\) is defined as \(\mu_h \equiv \sum_{i=1}^n h_i / n\)
\(\mu_v\) is defined as \(\mu_v \equiv \sum_{i=1}^n v_i / n\)
NSPG Invariant Measures
It is worth noting that following measures in this section do not increase with the number of shots per group (NSPG). That means that more shots tightens the precision of the measurement but doesn't change its expected value.
 Mean Radius
 Circular Error Probable
 Horizontal and Vertical Variances
 Radial Standard Deviation of the Rayleigh Distribution
 Bivariate Gaussian Distribution Parameters
Mean Radius (MR)
The Mean Radius is the average distance over all shots to the groups center.
Assumptions 

Experimental Measure  \(\bar{r} = \sum_{i=1}^n r_i / n\) where \(r_i = \sqrt{(h_i  \mu_h)^2 + (v_i  \mu_v)^2}\) 
Mode of PDF  
Median of PDF  
Mean of PDF  
CDF  
Variables in PDF & CDF  \(r\) 
Fitted parameters  \(\mu_h, \mu_v\) 
(h,v) for all points?  Yes 
Symmetric about Measure?  No, but the distribution will become more symmetric as the number of shots in a group increases. 
NSPG Invariant  Yes 
Formula:
\(\bar{r} = \sum_{i=1}^n r_i / n\) where \(r_i = \sqrt{(h_i  \mu_h)^2 + (v_i  \mu_v)^2}\)
As we will see in Closed Form Precision, the Mean Radius is typically only 6% larger than the Circular Error Probable. Since this is within the margin of error of most realworld usage the terms MR and CEP may be interchanged in casual usage.
Circular Error Probable (CEP)
\(CEP_p\), for \(p \in [0, 1)\), is the radius of the smallest circle that covers proportion p of the shot group. When p is not indicated it is assumed to be \(CEP_{0.5}\), which is the median shot radius (50% radius).
Horizontal and Vertical Variances
Assumptions:
 \(\sigma_h \neq \sigma_v\)
 \(\rho = 0\)
 No Flyers
\(\sigma_h^2 = \frac{\sum^{n}(h_i  \bar{h})^2}{n  1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i  \bar{v})^2}{n  1}\)
Often these will be given as standard deviations, which is just the square root of variance.
Radial Standard Deviation of the Rayleigh Distribution
From high school mathematics one should remember the two coordinate systems  Cartesian Coordinates and Polar Coordinates. Essentially the Rayleigh distribution converts shots from the Cartesian Coordinate system to the Polar Coordinate system. It is implicit in the coordinate conversion that the origin for the polar coordinate system is at the average point of impact. Thus for the polar coordinates the radial positioon of each shot will be relative to origin, or the average point of impact. Each shot will then have two coordinates, an angle, \(\theta, \) and the radius, r. Given that the shot distribution assumptions hold, then the angle should be entirely random and is of no interest. The twovariable problem has thus been reduced to a onevariable problem of determining the distribution for the shot radius.
Given that the conversion for the radial distance for each shot i from Cartesian Coordinates is:
\(r_i = \sqrt{(h_i  \bar{h})^2 + (v_i  \bar{v})^2}\)
then the mean radius for the sample of i shots can be calculated in a straight forward manner using:
\(\bar{r} = \frac{\sum_{i=1}^n r_i}{n}\)
and likewise for the standard deviation of the radius sample:
\(s_r = \sqrt{\frac{\sum^{n}(r_i  \bar{r})^2}{n  1}}\)
Now assuming that the shot dispersion follows the Bivariate Gaussian Distribution and that the following simplifying assumptions are true:
 \(\sigma_h = \sigma_v\)
 \(\rho = 0\)
 No Flyers
then the equation for the PDF for an individual shot is given by the Rayleigh distribution function which is:
\(f(r) = \frac{r}{\sigma_{RSD}^2} e^{r^2/2\sigma_{RSD}^2)}, \quad r \geq 0,\)
where \(\sigma_{RSD}\) is the single scale parameter of the distribution and is called the Radial Standard Deviation. Solving the distribution function for the mean radius and the standard deviation of the radius shows that they both are a proportional to \(\sigma_{RSD}\).
For the mean radius:
\(\bar{r} = \sqrt{\pi/2} \sigma_{RSD} \approx 1.2533 \sigma_{RSD}\)
and for the standard deviation of the radius:
\(\sigma_r = \sqrt{\frac{4  \pi}{2}} \sigma_{RSD} \approx 0.6551 \sigma_{RSD}\)
There is an additional association which needs to be mentioned. Given the assumption \(\sigma_h = \sigma_v\), then according to the strict derivation of the Rayleigh distribution, \(\sigma_{RSD} = \sigma_h = \sigma_v\). In reality for the sample of shots \(s_h \approx s_v\) which means that \(s_{RSD} = (s_h + s_v)/2\)
This bit of mathematical magic is due to the fact that the error of a shot from the polar origin has been broken into two parts, an angular error and a radial error. The implicit assumption here is that the angular part of the error is entirely random and hence not significant in characterizing the distribution of the radius. Thus that part of the error in a shot's position has been isolated and removed. This mathematical manipulation isn't "free." The essence is that the Rayleigh model places an even greater dependency on the assumptions when making predictions about confidence intervals which use the standard deviation. In plainer language if the assumptions don't hold, then a small error in the estimated \(\sigma_{RSD}\) results in larger errors in the confidence interval predictions.
Bivariate Gaussian Distribution Parameters
Assumptions:
 Full Bivariate Gaussian Distribution for shot dispersion
 \(\sigma_h \neq \sigma_v\)
 \(\rho \neq 0\)
 No Flyers
The full bivariate Gaussian distribution is:
\(
f(h,v) =
\frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1\rho^2}}
\exp\left(
\frac{1}{2(1\rho^2)}\left[
\frac{(h\bar{h})^2}{\sigma_h^2} +
\frac{(v\bar{v})^2}{\sigma_v^2} 
\frac{2\rho(x\bar{h})(y\bar{v})}{\sigma_h \sigma_v}
\right]
\right),
\)
In general case, when \(\sigma_h \neq \sigma_v\), then the actual standard deviation
of the radius \(r_i\) is not easy to calculate and is given by the formula:
\(\frac{\sigma_h^2}{\pi} (\pi  2 K^2(1  \frac{\sigma_v^2}{\sigma_h^2})) + \sigma_v^2\) where K is the complete elliptic integral.
NSPG Variant Measures
The following measures in this section do increase with group size. They are more commonly used because they are obtained by direct measurements with either no calculations, or very simple calculations. But they are statistically weaker because they mostly ignore inner data points.
 Group Size (GS)
 Diagonal (D)
 Figure of Merit (FOM)
 Covering Circle Radius (CCR)
In general a ragged hole does not present a problem for these measures. However a ragged hole might be an experimental problem depending, for example, if the projectile style does not cut relatively clean holes, or if the target tears.
Group Size (GS)
The Group Size is is the largest centertocenter distance between any two points, i and j, in the group. It is often called the Extreme Spread in the statistical literature.
Assumptions:
 Rayleigh Distribution for shot dispersion
 \(\sigma_h = \sigma_v\)
 \(\rho = 0\)
 No Flyers
Formula:
\(ES = \max \sqrt{(h_i  h_j)^2  (v_i  v_j)^2)}\)
Note: Be careful with with the phrase extreme spread. Shooters will often refer to the range of values from a chronograph as the extreme spread. Context should allow an easy determination of the correct meaning of the phrase.
Diagonal (D)
The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be determined by two to four points depending on the pattern of shots within the group.
Assumptions:
 \(\sigma_h \neq \sigma_v\)
 \(\rho = 0\)
 No Flyers
Formula:
\(ES = \sqrt{(h<sub>high</sub>  h<sub>low</sub>)^2  (v<sub>high</sub>  v<sub>low</sub>)^2)}\)
where \((h<sub>high</sub>  h<sub>low</sub>)\) and \((v<sub>high</sub>  v<sub>low</sub>)\) are the observed horizontal and vertcal ranges respecively.
Figure of Merit (FOM)
The Figure of Merit is the average extreme width and height of the group. The FOM may be determined by two to four points depending on the pattern within the group.
Assumptions:
 Rayleigh Distribution for shot dispersion
 \(\sigma_h = \sigma_v\)
 \(\rho = 0\)
 No Flyers
Formula:
\(ES = ((h_{high}  h_{low}) + (v_{high}  v_{low})) / 2\)
The FOM would be reasonable when \(\sigma_h \approx \sigma_v\). However if it is known that \(\sigma_h \ne \sigma_v\), then using the measurement makes no sense. It would be better to use the Diagonal measurement.
Covering Circle Radius (CCR)
The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will pass through at least the two extreme shots (in which case CCR = (Group Size) / 2 ) or at most it will pass through three outside shots.
Assumptions:
 Rayleigh Distribution for shot dispersion
 \(\sigma_h = \sigma_v\)
 \(\rho = 0\)
 No Flyers
Which Measure is Best?
Precision Models discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no universally best measurement. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "outliers" in the data. In practice the effect of neither of these factors is known.
The lack of an absolute truth may be mitigated with an expectation of picking reasonable assumptions and a mathematical model that is good enough. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate Gaussian distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate Gaussian distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit.
Shooters use the term flyer to denote the statistical term outlier. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed.
It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate) as the gold standard given the situation that the underlying assumptions about shot dispersion and the lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless per centage. The effiency of various measures can be thus compared by using the ratios of the variances, the relative standard deviations squared.
However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough experimental data to infer that the theory is incorrect. Also for the Rayleigh model neither the position of the center, nor the average radius, nor the standard deviation of the radius are [robust estimators].
Examples
One of the important questions addressed here is what to measure in order to determine the intrinsic precision of a shooting system, and what sample size is sufficient to achieve any degree of statistical significance.
Following are common measurements used by shooters or in the firearm industry:
 Extreme Spread of one 3shot group, usually at 100 yards.
 This is statistically poor, especially when there is no reference to how many 3shot groups were sampled.
 ([extended practical, and amusing, critique of the 3shot group is archived here].)
 Extreme Spread of one 5shot group, sometimes excluding the worst shot. Hardly any better.
 Average, Max, and Min Extreme Spread of five 5shot groups.
 The US Army Marksmanship Unit at Ft. Benning, GA uses a minimum of 3 consecutive 10shot groups fired with the rifle in a machine rest when testing service rifles. Armed forces also often explicitly uses the more statistically powerful Mean Radius and Circular Error Probable measures.
References
 ↑ Wind deflection is a function of the ballistic curve and distance, but can be expressed as a simple product of the crosswind velocity and lag time. For more information on the "lag rule" see Bryan Litz, Applied Ballistics for Long Range Shooting, 2^{nd} Edition (2011) A4; or Robert McCoy, Modern Exterior Ballistics, 2^{nd} Edition (2012) 7.27.
Next: Precision Models