Projectile Dispersion Classifications

Dispersion Assumptions

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 two-dimensional target we will assume that the Bivariate Normal Distribution, the two dimensional analog of the Normal distribution, applies. This distribution describes, at least approximately, the dispersion of a gunshots about their true center point, (\(\mu_h\) and \(\mu_v\)). 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:
    \( f(h,v; \mu_h, \mu_v, \sigma_h, \sigma_v, \rho) = \frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)}\left[ \frac{(h-\mu_h)^2}{\sigma_h^2} + \frac{(v-\mu_v)^2}{\sigma_v^2} - \frac{2\rho(h-\mu_h)(v-\mu_v)}{\sigma_h \sigma_v} \right] \right) \)

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.

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 \(\mu_h, \mu_v, s_h\) and \(s_v\) can be determined. We can only get experimental estimates from calculations based on sample data for the factors \(\bar{h}, \bar{v},\sigma_h\) and \(\sigma_v\) 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 some 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. \(\mu_h, \mu_v, \sigma_h, \mbox{and } \sigma_v\)) rather than the experimentally determined factors (i.e. \(\bar{h}, \bar{v}, s_h, \mbox{and } s_v\)) on purpose to emphasize the theoretical nature of the assumptions. Of course the "true" population parameters are unknown, and we estimate them with the corresponding experimentally fitted values about which there is some error.

Case 1, Equal variances and uncorrelated (Rayleigh Distribution)

Given:

1. \(\sigma_h \approx \sigma_v\)
   
2. \(\rho \approx 0\)
   

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:

1. In this case that the FOM, and Group Size are different measurements.
2. 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.
3. 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.
4. Experimentally the radial distance for each shot, i, is \(r_i = \sqrt{(h_i - \mu_h)^2 + (v_i - \mu_v)^2}\)
5. 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 two-variable distribution to a one-variable distribution.

Case 2, Equal variances and correlated

Given:

1. \(\sigma_h \approx \sigma_v\)
   
2. \(\rho \neq 0\)
   
3. The {h,v} position of each shot must be known.

The following statistical measurement is appropriate:

• Elliptic Error Probable

Case 3, Unequal variances and uncorrelated

Given:

1. \(\sigma_h \neq \sigma_v\)
   
2. \(\rho \approx 0\)
   
3. 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.

Case 4, Unequal variances and correlated (Hoyt Distribution)

Given:

1. \(\sigma_h \neq \sigma_v\)
   
2. \(\rho \neq 0\)
   
3. 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 egg-shaped.

Related topics

See also the following topics which are closely related:

• Error Propagation - A basic discussion of how errors propagate when making measurements.
• Stringing - Definition of stringing and how it can be handled

References

---

Next: Precision Models