# Figure of Merit

Figure of Merit The Figure Of Merit (FOM) is the average of the ranges for the width and height of the group.

# 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 dispersion would be 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 $$\sigma$$, or to determine the COI, to calculate the FOM.
• No Fliers
Data transformation Preliminary Cartesian Calculations (Conceptually)
• Identify two shots that form the horizontal range and measure $$h_{Range}$$
$$h_{Range} = \max\{\,h_1,\ \dots \ h_n\,\} - \min\{\,h_1,\ \dots\ h_n\}$$
• Identify two shots that form the vertical range and measure $$v_{Range}$$
$$v_{Range} = \max\{\,v_1,\ \dots \ v_n\,\} - \min\{\,v_1,\ \dots\ v_n\}$$
Experimental Measure $$FOM = \frac{1}{2}(h_{Range} + v_{Range})$$

# Theoretical Evaluations

## Theoretical $$FOM$$ Distribution

 Parameters Needed $$PDF(r; \sigma)$$ $$CDF(r; \sigma)$$ Mode of PDF Median of PDF Mean of PDF Variance Variance Distribution (h,v) for all points? Yes Symmetric about Mean? No, skewed to larger values. More symmetric as number of shots increases.