Extreme Spread

Experimental Summary

Given	set of n shots {\( (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) \)} All of the (h,v) positions do not need to be known so a ragged hole will suffice.
Assumptions	Ideally the shot would follow a Rayleigh Distribution \(\bar{h} \sim \mathcal{N}(\bar{h},\sigma_h^2), \bar{v} \sim \mathcal{N}(\bar{v},\sigma_v^2)\) Horizontal and vertical dispersion are independent. \(\sigma_h = \sigma_v\) (realistically \(\sigma_h \approx \sigma_v\)) \(\rho = 0\) \(PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}\) Note: It is not necessary to fit \(\sigma\) to calculate the Figure of Merit. No Fliers
Data transformation	Identify two holes, \(i, j\) which are the farthest apart.
Experimental Measure	\(ES = \sqrt{(x_i - x_j)^2 - (y_i - y_j)^2)}\),

Given

The requirements for this test are very basic. Just a target with \(n\) shots.

Assumptions

None are needed to make measurement. However making assumptions about the dispersion will all theoretical calculations about the measurement.

Data transformation

The data transformation for a human has simple requirements, just the ability to locate the holes which are the furthest apart. If the target has a ragged hole it can be a bit tricky, but the edges of the hole should have enough curvature to make shot location possible.

Experimental Measure

If measuring on the range, then the center of the hole is difficult to locate. Typically a vernier caliper would be used to measure the distance from the outside edges of the holes, then the bullet caliber subtracted to get a c-t-c measurement.

If using a computer then the center location would be a matter programming. For example a mouse might be used simply to point out the holes, or to drop a dot at the center of the hole, or to drag a circle over the hole.

Theoretical \(ES\) Distribution

Assuming that the shots are Rayleigh distributed allows us to make some theoretical estimates.

Theoretical \(ES\) Distribution

Parameters Needed
\(PDF(r; \sigma)\)	no direct evaluation, must be simulated via Monte Carlo
\(CDF(r; \sigma)\)	no direct evaluation, must be simulated via Monte Carlo
Mode of PDF)	depends on \(n\), in general
Median of PDF
Mean of PDF
Variance	no direct evaluation, must be simulated via Monte Carlo
Variance Distribution
(h,v) for all points?	Yes
Symmetric about Mean?	No, skewed to larger values. More symmetric as number of shots increases.

See Also

Dispersion Assumptions - A discussion of the different cases for shot dispersion

Other measurements practical for range use are:

• Covering Circle Radius - about same precision as Extreme Spread if Rayleigh distributed
• Diagonal - somewhat better precision than Extreme Spread if Rayleigh distributed
• Figure of Merit - somewhat better precision than Extreme Spread if Rayleigh distributed