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= Synopsis =
 
= Synopsis =
  
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When testing a gun to estimate its precision the most useful data are the (''x'', ''y'') coordinates of each impact on sample targets.  These allow for [[Measuring Precision|closed-form estimates and confidence intervals on the standard deviation of dispersion]] along each axis, and [[Predicting Precision|from the standard deviation we can deduce any standard precision measure]] for the gun.
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When testing a gun, shooter, and/or ammunition the most popular measure is [[Extreme Spread]] or "size" of a sample target group.  However, as we will illustrate throughout this site, [[Extreme Spread]] is not only a statistically inefficient measure but also one frequently and easily abused.
  
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If we assume that the inherent dispersion along each axis is roughly identical then we can use the average of the standard deviations, a single parameter ''σ'', to characterize precision.  This only requires the radius <math>r_i = \sqrt{(x_i - \bar{x})^2 + (y_i - \bar{y})^2}</math> of each impact on a sample target.
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We advocate for [[Describing_Precision#Invariant_Measures|size-invariant measures]] like [[Circular Error Probable]] or Mean Radius.  The expected value of these does not change with the number of shots on target.  Instead [[Measuring_Precision#How_large_a_sample_do_we_need.3F|taking more shots serves only to reduce the statistical error in our measurement]].
  
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The [[Measuring_Precision#How_large_a_sample_do_we_need.3F|certainty with which we can assess precision]] increases with the number of shots.  Since shooting large samples on a single target risks developing [[ragged holes]] where data points are lost, there are three recommended approaches to efficiently build data sets.  These are explained in more detail in [[Measuring Tools]]:
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Since shooting large samples on a single target risks developing [[ragged holes]] where data points are lost, there are three recommended approaches to efficiently build data sets for estimating precision.  These are explained in more detail in [[Measuring Tools]]:
 
# Use [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|Danielson's]] 2-shot method: Fire two shots per target and use calipers to measure their distance from each other.  This provides two samples per target with radius ''r'' = spread / 2.  
 
# Use [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|Danielson's]] 2-shot method: Fire two shots per target and use calipers to measure their distance from each other.  This provides two samples per target with radius ''r'' = spread / 2.  
 
# Fire one shot per target.  Manually overlay them or use software like [http://ontargetshooting.com/tds/ OnTarget Target Data System] to automatically aggregate them into a single sample group.
 
# Fire one shot per target.  Manually overlay them or use software like [http://ontargetshooting.com/tds/ OnTarget Target Data System] to automatically aggregate them into a single sample group.
 
# Use a logging electronic target.  (Not yet widely available.)
 
# Use a logging electronic target.  (Not yet widely available.)
  
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Most shooters are used to expressing precision in terms of extreme spread, which is a statistically inefficient and unbounded measure. Therefore, to facilitate the transition to proper and efficient statistics, we suggest that precision be reported in terms of Mean Diameter, which is 2.5''σ'', and which in expectation will cover 96% of shots fired from a gun.
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Furthermore, if we assume that the inherent shot dispersion is free of directional bias then we can use closed-form expressions to calculate and analyze precision.
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[[:Category:Examples]] of the application of these methods and tools include:
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* Determining the likelihood of a hit on a particular target by a zeroed shooting system.
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* Comparing the inherent precision of different shooting systems.
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* Determining which ammunition shoots better in a particular gun.

Revision as of 17:42, 25 February 2014

This site explains and demonstrates statistics for analyzing the precision of guns.

In particular:

Synopsis

When testing a gun, shooter, and/or ammunition the most popular measure is Extreme Spread or "size" of a sample target group. However, as we will illustrate throughout this site, Extreme Spread is not only a statistically inefficient measure but also one frequently and easily abused.

We advocate for size-invariant measures like Circular Error Probable or Mean Radius. The expected value of these does not change with the number of shots on target. Instead taking more shots serves only to reduce the statistical error in our measurement.

Since shooting large samples on a single target risks developing ragged holes where data points are lost, there are three recommended approaches to efficiently build data sets for estimating precision. These are explained in more detail in Measuring Tools:

  1. Use Danielson's 2-shot method: Fire two shots per target and use calipers to measure their distance from each other. This provides two samples per target with radius r = spread / 2.
  2. Fire one shot per target. Manually overlay them or use software like OnTarget Target Data System to automatically aggregate them into a single sample group.
  3. Use a logging electronic target. (Not yet widely available.)

Furthermore, if we assume that the inherent shot dispersion is free of directional bias then we can use closed-form expressions to calculate and analyze precision.

Category:Examples of the application of these methods and tools include:

  • Determining the likelihood of a hit on a particular target by a zeroed shooting system.
  • Comparing the inherent precision of different shooting systems.
  • Determining which ammunition shoots better in a particular gun.
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