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

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.