Closed Form Precision

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

Rayleigh distribution of shots given σ

This σ is the parameter for the Rayleigh distribution with probability density function \(\frac{x}{\sigma^2}e^{-x^2/2\sigma^2}\). The associated Cumulative Distribution Function gives us the probability that a shot falls within a given radius of the center: \[Pr(r \leq \alpha) = 1 - e^{-\alpha^2 / 2 \sigma}\] Therefore, we expect 39% of shots to fall within a circle of radius σ, 86% within 2σ, and 99% within 3σ.

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

Radial Standard Deviation \(RSD = \sigma \sqrt{2}\). The expected sample RSD of a group of size n is \[RSD_n = \sigma \sqrt{\frac{2}{c_{G}(n)}} \approx \sigma \sqrt{2 - \frac{1}{2n} - \frac{7}{16n^2} - \frac{19}{64n^3}}\]

Mean Radius \(MR = \sigma \sqrt{\frac{\pi}{2}}\). The expected sample MR of a group of size n is \[MR_n = \sigma \sqrt{\frac{\pi}{2 c_{B}(n)}}\ = \sigma \sqrt{\frac{\pi (n - 1)}{2 n}}\]

Circular Error Probable \(CEP = \sigma \sqrt{\ln(4)}\). The expected sample CEP of a group of size n is \[CEP_n = \sigma \frac{\sqrt{\ln(4)}}{c_{G}(n) c_{R}(n)}\]

The three measures that vary with n are range statistics without convenient functional forms. However both the mean and standard deviation of their expected value scales with σ, so it is sufficient to calculate them once for σ = 1 and multiply the resulting values by the actual σ. Medoa:Sigma1ShotStatistics.ods contains those values for n up to 100.