Closed Form Precision

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Previous: Measuring Precision

The Symmetric Precision Parameter σ

Measuring Precision showed how a single parameter σ (sigma) 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 , and 99% within .

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

Mean Radius (MR)

Mean Radius \(MR = \sigma \sqrt{\frac{\pi}{2}} \ \approx 1.25 \ \sigma\).

\(1 - e^{-\frac{\pi}{4}} \approx 54\%\) of shots should fall within the mean radius. 96% of shots should fall within the Mean Diameter (MD = 2 MR).

The expected sample MR of a group of size n is

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

(This sample size adjustment doesn't use the Gaussian correction factor because the mean radius is not an estimator for σ, even though in the limit the true value of one is a constant product of the other.)

Circular Error Probable (CEP)

Circular Error Probable \(CEP = \sigma \sqrt{\ln(4)} \ \approx 1.18 \ \sigma\). 50% of shots should fall within the circular error probable.

In theory CEP is the median radius, but especially for small n the sample median is a very bad estimator for true median. Nevertheless, if you want to know the expected sample median radius of a group of size n it turns out the following is a good estimate:

  \(CEP_n = \sigma \frac{\sqrt{\ln(4)}}{c_{G}(n) c_{R}(n)}\)

Summary Sigma Probabilities

Name Radius in Sigmas Shots Covered
1 39%
CEP 1.18 50%
MR 1.25 54%
2 86%
MD 2.5 96%
3 99%

Typical values of σ

A lower bound on σ is probably that displayed by rail guns in 100-yard competition. On average they can place 10 rounds into a quarter-inch group, which as we will see shortly suggests σ = 0.070MOA, or under 0.025mil.

The U.S. Precision Sniper Rifle specification requires a statistically significant number of 10-round groups fall under 1MOA. This means σ = 0.28MOA, or under 0.1mil.

The specification for the M110 semi-automatic sniper rifle (MIL-PRF-32316) as well as the M24 sniper rifle (MIL-R-71126) requires MR below 0.65SMOA, which means σ = 0.5MOA. The latter spec indicates that an M24 barrel is not considered worn out until MR exceeds 1.2MOA, or σ = 1MOA!

XM193 ammunition specifications require 10-round groups to fall under 2MOA. This means σ = 0.6MOA or 0.2mil, and it is a good minimum precision standard for light rifles.

Spread Measures

Median values for size statistics when σ = 1. Bands cover 50% of samples around each median.

The three measures that vary with n (Extreme Spread, Diagonal, and Figure of Merit) are range statistics that lack convenient functional forms. However both the mean and standard deviation of their expected value, as well as quantiles, scale directly with σ, so it is sufficient to calculate those statistics once for σ = 1 and multiply the resulting values by the desired σ. Media:Sigma1RangeStatistics.xls contains quantiles and moments for n up to 100.

Example

What extreme spread should I expect for 5-shot groups from my rifle? The extreme spread median from the table for 5 shots is 3.0 σ. I've determined my rifle has precision σ = ½MOA. If I take five shots at 100 yards we would expect half my groups to be less than 3.0/2 = 1.5MOA \(\approx\) 1.6".

Multiplying the rest of the distribution data for that row by my 0.5MOA we can also say that the extreme spread of my 5-shot groups should exhibit the following distributions:

  • 50% between (1.2, 1.8)MOA
  • 80% between (1.0, 2.1)MOA
  • 95% between (0.8, 2.4)MOA



Next: Examples