Extreme Spread

Experimental Summary

Given	set of n shots {\( (h_1, v_1), (h_2, v_2), ..., (h_n, v_n) \)} All of the (h,v) positions do not need to be known so a ragged hole will suffice.
Assumptions	Ideally the shots would follow a Rayleigh Distribution \(\bar{h} \sim \mathcal{N}(\bar{h},\sigma_h^2), \bar{v} \sim \mathcal{N}(\bar{v},\sigma_v^2)\) Horizontal and vertical dispersion are independent. \(\sigma_h = \sigma_v\) (realistically \(\sigma_h \approx \sigma_v\)) \(\rho = 0\) \(PDF_{r_i}(r) = \frac{r}{\sigma^2}e^{-r^2/2\sigma^2}\) Note: It is not necessary to fit \(\sigma\) to calculate the Figure of Merit. No Fliers
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.

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.

Theoretical \(ES\) Distribution

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.

Mode, Median, Mean, Variance, %RSD of Mean

Table columns for "ES Values from Monte Carlo Simulation" Table

number of shots
Mode
Median
Mean
"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. + value means positively skewed, - value means negatively skewed.
Variance
%RSD

Theoretical \(ES Values from Monte Carlo Simulation\) Distribution

number of shots	Mode	Median	Mean	"Normality Error"	Variance	%RSD
2
3
4
5
6
7
9
10

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