Difference between revisions of "Closed Form Precision"

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(Spread Measures)
(Spread Measures)
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The three measures that vary with ''n'' ([[Describing_Precision#Extreme_Spread|Extreme Spread]], [[Describing_Precision#Diagonal|Diagonal]], and [[Describing_Precision#Figure_of_Merit|Figure of Merit]]) are range statistics that lack 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 desired ''σ''.  [[Media:Sigma1ShotStatistics.ods]] contains those values for ''n'' up to 100.
 
The three measures that vary with ''n'' ([[Describing_Precision#Extreme_Spread|Extreme Spread]], [[Describing_Precision#Diagonal|Diagonal]], and [[Describing_Precision#Figure_of_Merit|Figure of Merit]]) are range statistics that lack 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 desired ''σ''.  [[Media:Sigma1ShotStatistics.ods]] contains those values for ''n'' up to 100.
  
=== Example ===
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=== Example 1 ===
 
The extreme spread value [[Media:Sigma1ShotStatistics.ods|from the table]] for 5 shots is 3.06''σ''.  I've determined my rifle has precision ''σ'' = ½MOA.  If I take five shots at 100 yards I would expect an extreme spread of 3.06/2 = 1.53MOA <math>\approx</math> 1.6".  And since the standard deviation of that value is 0.83/2 = 0.415''σ'' I expect that [https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule 68% of the time] my five shot groups will have an extreme spread in the range (1.12, 1.94) MOA &mdash; i.e., between 1.2" and 2.0".
 
The extreme spread value [[Media:Sigma1ShotStatistics.ods|from the table]] for 5 shots is 3.06''σ''.  I've determined my rifle has precision ''σ'' = ½MOA.  If I take five shots at 100 yards I would expect an extreme spread of 3.06/2 = 1.53MOA <math>\approx</math> 1.6".  And since the standard deviation of that value is 0.83/2 = 0.415''σ'' I expect that [https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule 68% of the time] my five shot groups will have an extreme spread in the range (1.12, 1.94) MOA &mdash; i.e., between 1.2" and 2.0".
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=== Example 2 ===
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Over many tests I have found my rifle produces 5-shot groups with an average extreme spread of 1".  ''What extreme spread should I expect if I instead start shooting 10-shot groups?''  [[Media:Sigma1ShotStatistics.ods|The table]] shows that the ratio of expected extreme spreads on 10-shot groups is 1.24 times the value on 5-shot groups.  So my a priori expectation would be for 10-shot groups to average 1.24".

Revision as of 14:12, 22 November 2013

The Precision Parameter

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 , and 99% within .

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

Radial Standard Deviation (RSD)

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)

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)

Circular Error Probable \(CEP = \sigma \sqrt{\ln(4)}\). The expected sample CEP of a group of size n is

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

Spread Measures

Expected and standard deviation of values for size statistics when σ = 1

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 scales with σ, so it is sufficient to calculate them once for σ = 1 and multiply the resulting values by the desired σ. Media:Sigma1ShotStatistics.ods contains those values for n up to 100.

Example 1

The extreme spread value from the table for 5 shots is 3.06σ. I've determined my rifle has precision σ = ½MOA. If I take five shots at 100 yards I would expect an extreme spread of 3.06/2 = 1.53MOA \(\approx\) 1.6". And since the standard deviation of that value is 0.83/2 = 0.415σ I expect that 68% of the time my five shot groups will have an extreme spread in the range (1.12, 1.94) MOA — i.e., between 1.2" and 2.0".

Example 2

Over many tests I have found my rifle produces 5-shot groups with an average extreme spread of 1". What extreme spread should I expect if I instead start shooting 10-shot groups? The table shows that the ratio of expected extreme spreads on 10-shot groups is 1.24 times the value on 5-shot groups. So my a priori expectation would be for 10-shot groups to average 1.24".