The Mean Radius is the average distance over all shots to the groups center.

# Experimental Summary

Given
• set of n shots {$$(h_1, v_1), (h_2, v_2), ..., (h_n, v_n)$$}
for which all of the (h,v) positions are known
Assumptions
• The dispersion of shot $$i$$ follows a Rayleigh Distribution so that with the conversion from Cartesian coordinates to Polar coordinates, $$\theta$$ will be entirely random and independent of radius.
• $$h_i \sim \mathcal{N}(\bar{h},\sigma_h^2), v_i \sim \mathcal{N}(\bar{v},\sigma_v^2)$$
• 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 Mean Radius.
• No Fliers
Data transformation Measure positions $$(h_i, v_i)$$ for each shot, $$i$$.
Experimental Measure Preliminary Cartesian Calculations
• $$\bar{h} = \frac{1}{n-1} \sum_{i=1}^n h_i^2$$
• $$\bar{v} = \frac{1}{n-1} \sum_{i=1}^n v_i^2$$

Shot impact positions converted from Cartesian Coordinates

• $$r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}$$
Thus the Cartesian Coordinate ($$\bar{h}, \bar{v}$$) is translated to Polar Coordinate $$(r = 0, \theta = 0)$$
• $$\theta$$ Polar Coordinate is unneeded for mean radius calculation and ignored (assumed to be pure noise).

$$\overline{r_n}$$ - the average radius of n shots

$$\overline{r_n} = \sum_{i=1}^n r_i / n$$

Outlier Tests

# Theoretical $$r(1)$$ Distribution

Distribution for a single shot as a function of r.

 Parameters Needed $$\Re$$ - Rayleigh shape parameter fit to experimental shot distribution $$PDF_{r(1)}(r; \Re)$$ $$\frac {r}{\Re^2} \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}$$ $$CDF_{r(1)}(r; \Re)$$ $$1 - \exp\Big \{-\frac {r^2}{2\Re^2} \Big\}$$ Mode of $$PDF_{r(1)$$ $$\Re$$ Median of $$PDF_{r(1)}$$ $$\Re\sqrt{\ln{4}}$$ Mean of $$PDF_{r(1)}$$ $$\Re\sqrt{\frac{\pi}{2}}$$ Variance of $$PDF_{r(1)}$$ $$\frac{(4-\pi)}{2}\Re^2$$ Variance Distribution (h,v) for all points? Yes Symmetric about Mean? No, skewed to larger values. More symmetric as number of shots increases.

# Theoretical $$\overline{r(n)}$$ Distribution

Given:

• $$n$$ shots were taken on a target
• The average mean radius, $$\overline{r(n)}$$, was calculated
• The Rayleigh shape parameter $$\Re$$ for an individual shot is known.

then using $$r$$ as a variable, the properties of the distribution of the average mean radius for $$n$$ shots is investigated in this section.

 Parameters Needed $$n$$ - n of shots in sample $$\Re$$ - Rayleigh shape parameter from individual shot distribution $$PDF(\bar{r_n}; n, \Re)$$ $$\frac{\Gamma(n,2\Re^2)}{n}$$ where $$\Gamma(n,2\Re^2)$$ is the Gamma Distribution $$CDF(r; n, \Re)$$ Mode of PDF) $$\bar{r_n}$$ Median of PDF no closed form, but $$\approx 1.177\bar{r_n}$$ Mean of PDF $$\sqrt{2} \Gamma({\frac{3}{2}})\bar{r_n} = \frac{\sqrt{2}}{2}\sqrt{\pi}\bar{r_n} \approx 1.2533\bar{r_n}$$ Variance Variance Distribution (h,v) for all points? Yes Symmetric about Measure? No, skewed to larger values. More symmetric as number of shots increases. NSPG Invariant Yes Robust No