Extreme Spread
Experimental Summary
Given |
All of the (h,v) positions do not need to be known so a ragged hole will suffice. |
Assumptions |
|
Data transformation | Identify two holes, \(i, j\) which are the farthest apart. |
Experimental Measure | \(ES = \sqrt{(x_i - x_j)^2 - (y_i - y_j)^2)}\), |
Given
The requirements for this test are very basic. Just a target with \(n\) shots, and some measuring device. Assuming an Extreme spread of under 6 inches then a vernier caliper is used. A measurement is possible to a few thousandths of an inch which is vastly more precision than is usually required. From longer distance a ruler, or perhaps a tape measure.
Assumptions
None are needed to make measurement. However making assumptions about the dispersion will enable theoretical predictions about the measurement.
Data transformation
The data transformation for a human has simple requirements, just the ability to locate the holes which are the furthest apart. If the target has a ragged hole it can be a bit tricky, but the edges of the hole should have enough curvature to make shot location possible.
Experimental Measure
If measuring on the range, then the center of the hole is difficult to locate. Typically a vernier caliper would be used to measure the distance from the outside edges of the holes, then the bullet caliber subtracted to get a c-t-c measurement.
If using a computer then the center location would be a matter programming. For example a mouse might be used simply to point out the holes, or to drop a dot at the center of the hole, or to drag a circle over the hole. The computer would then make the c-t-c measurement.
Outlier Tests
Theoretical \(ES\) Distribution
Assuming that the shots are Rayleigh distributed allows us to make some theoretical estimates. Since the distribution is positively skewed: Mean > Median > Mode.
Parameters Needed | |
\(PDF(r; \sigma)\) | no direct evaluation, must be simulated via Monte Carlo |
\(CDF(r; \sigma)\) | no direct evaluation, must be simulated via Monte Carlo |
Mode of PDF) | depends on \(n\), in general Mode increases as number of shots increases. |
Median of PDF | depends on \(n\), in general Median increases as number of shots increases. |
Mean of PDF | depends on \(n\), in general Median increases as number of shots increases |
Variance | no direct evaluation, must be simulated via Monte Carlo |
Variance Distribution | |
(h,v) for all points? | yes for simulation. |
Symmetric about Mean? | No, skewed to larger values.
More symmetric about mean as the number of shots increases. |
Parameters Needed
CDF
Mode, Median, Mean, Variance, %RSD of Mean
number of shots | the number of shots used per target |
Mode | The peak of the distribution |
Median | The 50th percentile |
Mean | the average value over the whole distrubtion |
"Normality Error" | As sort of a crude indication of normality let's use the value:
"Normality Error" = \( \frac{\frac{CDF(25) - CDF(75)}{2} - Mean}{Mean} {\dot 100}\) So we measure half the distance between the 25th percentile and the 75th percentile to determine where the Mean should be if the distribution was symmetrical, and determine the % error based on the actual value of the mean.
|
Variance | |
%RSD | The relative standard deviation which is a ratio of the standard deviation to the mean as a percentage. |
number of shots | Mode | Median | Mean | "Normality Error" | Variance | %RSD |
---|---|---|---|---|---|---|
2 | 1.772 | 0.932 | ||||
3 | 2.406 | 0.887 | ||||
4 | 2.787 | 0.856 | ||||
5 | 3.066 | 0.828 | ||||
6 | 3.277 | 0.806 | ||||
7 | 3.443 | 0.783 | ||||
9 | 3.710 | 0.754 | ||||
10 | 3.813 | 0.745 |
The tabular values can be used in a number of ways:
- Estimate ES values for different group sizes.
- Given Mean of 5 shots = 3.066 and measured group size is 1.53 inches
- a 3-shot group would be given by measured size times ratios of Means
- \(1.53 \frac{2.406}{3.066} = 1.20\) inches
- a 10-shot group would be given by measured size times ratios of Means
- \(1.53 \frac{3.813}{3.066} = 1.90\) inches
- a 3-shot group would be given by measured size times ratios of Means
Sample Variance and Its distribution
Outlier Tests
See Also
Dispersion Assumptions - A discussion of the different cases for shot dispersion
Other measurements practical for range use are:
- Covering Circle Radius - about same precision as Extreme Spread if Rayleigh distributed
- Diagonal - somewhat better precision than Extreme Spread if Rayleigh distributed
- Figure of Merit - somewhat better precision than Extreme Spread if Rayleigh distributed