Mean Radius

Mean Radius

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

Experimental Summary

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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.

Theoretical \(r(1)\) Distribution

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.

Parameters Needed

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Variance and Its distribution

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PDF

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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.

Theoretical \(\bar{r_n}\) Distribution

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

Parameters Needed

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Variance and Its distribution

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PDF

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CDF

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Mode, Median, Mean

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Outlier Tests

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Studentized Mean Radius

need table for this...

See Also

Projectile Dispersion Classifications - Discussion of other models for shot dispersion