Mean Radius
Mean Radius
The Mean Radius is the average distance over all shots to the groups center.
Experimental Summary
yada yada
Given 

Assumptions 

Data transformation  Measure positions \((h_i, v_i)\) for each shot, \(i\). 
Experimental Measure  \(MR = \overline{r_n}\) method
Preliminary Cartesian Calculations
Shot impact positions converted from Cartesian Coordinates
\(\overline{r_n}\)  the average radius of n shots \(\overline{r_n} = \sum_{i=1}^n r_i / n\) \(MR = f(\Re)\) method 
Outlier Tests 
Given
Assumptions
Data transformation
Experimental Measure
Outlier Tests
Theoretical \(r_i\) 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. 
Parameters Needed
yada yada
Variance and Its distribution
yada yada
yada yada
CDF
Mode, Median, Mean
Outlier Tests
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 
Parameters Needed
yada yada
Variance and Its distribution
yada yada
yada yada
CDF
yada yada
Mode, Median, Mean
yada yada
Outlier Tests
yada yada
Theoretical \(r_{\Re}\) Distribution
Distribution for MR where MR calculated from \(\Re\)
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
yada yada
Variance and Its distribution
yada yada
yada yada
CDF
Mode, Median, Mean
Outlier Tests
Studentized Mean Radius
need table for this...
Outlier Tests
See Also
Projectile Dispersion Classifications  Discussion of other models for shot dispersion