# Difference between revisions of "Closed Form Precision"

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=== Example 3 === | === Example 3 === | ||

− | ''How meaningful is a 3-shot precision guarantee?'' The first answer is that ''any performance standard that is expressed in terms of group sizes is probably meaningless''. After all, even the most precise gun can shoot an occasionally wide 3-shot group, and a shot-out barrel will with some probability still put three rounds through the same hole. The inherent precision of a gun is revealed over a large number of shots, and it doesn't matter how those shots are sampled: neither the gun nor the target has a memory. If we shoot 100 rounds through a rifle, we could pick any three at random as a 3-shot group. Therefore, a precision guarantee has to hold over | + | ''How meaningful is a 3-shot precision guarantee?'' The first answer is that ''any performance standard that is expressed in terms of group sizes is probably meaningless''. After all, even the most precise gun can shoot an occasionally wide 3-shot group, and a shot-out barrel will with some probability still put three rounds through the same hole. The inherent precision of a gun is revealed over a large number of shots, and it doesn't matter how those shots are sampled: neither the gun nor the target has a memory. If we shoot 100 rounds through a rifle, we could pick any three at random as a 3-shot group. Therefore, a precision guarantee has to hold over a large number of shots and is more aptly expressed in terms of probabilities. E.g., "99% of shots with our rifle will fall within a ½MOA radius." Based on the [[Predicting_Precision#The_Precision_Parameter|Rayleigh CDF]] this 99% claim is the same as saying 3''σ'' = ½MOA, or that the rifle's precision ''σ'' = 1/6MOA. |

There are two qualifications to the preceding: | There are two qualifications to the preceding: | ||

# We do know that the sample size of a 3-shot group is smaller than the sample size of larger groups, simply because [[Measuring_Precision#Bessel_correction_factor|the sample center is closer to the sample shots than the true center]]. This is shown in the correction terms above. For example, the average 3-shot group has a mean radius <math>1/\sqrt{c_B(3)} \approx 80\%</math> the size of the mean radius measured from twenty or more shots. | # We do know that the sample size of a 3-shot group is smaller than the sample size of larger groups, simply because [[Measuring_Precision#Bessel_correction_factor|the sample center is closer to the sample shots than the true center]]. This is shown in the correction terms above. For example, the average 3-shot group has a mean radius <math>1/\sqrt{c_B(3)} \approx 80\%</math> the size of the mean radius measured from twenty or more shots. | ||

# It is not exactly true that guns have no memory: The effects of heating and fouling can create second-order changes in both the precision and point of impact. So it might be necessary to qualify a precision guarantee with limits on barrel temperature and/or prescriptions on how many shots be fired between bore cleaning. And in the limit barrels erode and have a finite lifespan, so especially with higher-power calibers practical limits do exist on how long precision can stay constant. | # It is not exactly true that guns have no memory: The effects of heating and fouling can create second-order changes in both the precision and point of impact. So it might be necessary to qualify a precision guarantee with limits on barrel temperature and/or prescriptions on how many shots be fired between bore cleaning. And in the limit barrels erode and have a finite lifespan, so especially with higher-power calibers practical limits do exist on how long precision can stay constant. |

## Revision as of 01:33, 28 November 2013

## Contents

# The Precision Parameter

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 \(\frac{x}{\sigma^2}e^{-x^2/2\sigma^2}\). 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)

Radial Standard Deviation \(RSD = \sigma \sqrt{2}\). The expected sample RSD of a group of size *n* is

- \(RSD_n = \frac{\sigma}{c_{G}(2n - 1)} \sqrt{2}\)

## 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

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 1

*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

### Example 2

Over many tests I have found my rifle produces 5-shot groups with an average extreme spread of 1MOA. *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 1.24MOA.

### Example 3

*How meaningful is a 3-shot precision guarantee?* The first answer is that *any performance standard that is expressed in terms of group sizes is probably meaningless*. After all, even the most precise gun can shoot an occasionally wide 3-shot group, and a shot-out barrel will with some probability still put three rounds through the same hole. The inherent precision of a gun is revealed over a large number of shots, and it doesn't matter how those shots are sampled: neither the gun nor the target has a memory. If we shoot 100 rounds through a rifle, we could pick any three at random as a 3-shot group. Therefore, a precision guarantee has to hold over a large number of shots and is more aptly expressed in terms of probabilities. E.g., "99% of shots with our rifle will fall within a ½MOA radius." Based on the Rayleigh CDF this 99% claim is the same as saying 3*σ* = ½MOA, or that the rifle's precision *σ* = 1/6MOA.

There are two qualifications to the preceding:

- We do know that the sample size of a 3-shot group is smaller than the sample size of larger groups, simply because the sample center is closer to the sample shots than the true center. This is shown in the correction terms above. For example, the average 3-shot group has a mean radius \(1/\sqrt{c_B(3)} \approx 80\%\) the size of the mean radius measured from twenty or more shots.
- It is not exactly true that guns have no memory: The effects of heating and fouling can create second-order changes in both the precision and point of impact. So it might be necessary to qualify a precision guarantee with limits on barrel temperature and/or prescriptions on how many shots be fired between bore cleaning. And in the limit barrels erode and have a finite lifespan, so especially with higher-power calibers practical limits do exist on how long precision can stay constant.