Closed Form Precision

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Measuring Precision showed how a single parameter σ characterizes the precision of a shooting system.

This σ is the parameter for the Rayleigh distribution with probability density function <m>\frac{x}{\sigma^2}e^{-x^2/2\sigma^2}</m>.

Using the characteristics of the Rayleigh distribution we can immediately compute the three most useful precision measures:

Radial Standard Deviation <m>RSD = \sigma \sqrt{2}</m>. The expected sample RSD of a group of size n is

<m>RSD_n = \sigma \sqrt{\frac{2}{c_{G}(n)}} \approx \sigma \sqrt{2 - \frac{1}{2n} - \frac{7}{16n^2} - \frac{19}{64n^3}}</m>

Mean Radius <m>MR = \sigma \sqrt{\frac{\pi}{2}}</m>. The expected sample MR of a group of size n is

<m>MR_n = \sigma \sqrt{\frac{\pi}{2 c_{B}(n)}}\ = \sigma \sqrt{\frac{\pi (n - 1)}{2 n}}</m>

Circular Error Probable <m>CEP = \sigma \sqrt{\ln(4)}</m>