# Describing Precision

## Contents

# Units

When we talk about shooting precision we are referring to the amount of dispersion we expect to see of each shot about a center point (which shooters try to adjust to match the point of aim). Precision is like a cone of error that projects out from the muzzle of the gun. I.e., double the distance and the dispersion also doubles. We can describe this error by referring to dispersion at a specific distance. For example, it is common to quote precision in inches of extreme spread at 100 yards, or "inches per hundred yards."

It is more common, however, to describe the angle of the cone at its tip, since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and the smaller the angle at its tip.

One of two popular angular units used by shooters is MOA, though there is some ambiguity in this term. MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree. At some point shooters began to expand the acronym as Minute of Angle. They also took its original value, which **at 100 yards is 3600” tan (1/60 degrees) = 1.047”**, and rounded it to 1” at 100 yards. For clarity the latter unit is called “Shooters MOA,” or SMOA.

The other common unit is the “mil,” which simply means thousandth. For example, **at 100 yards a mil is 100 yards / 1000 = 3.6”**. Some more benign confusion also persists around this term, with some assuming “mil” is short for milliradian, which is another angular unit. Fortunately, a milliradian is almost exactly equal to a mil so there’s no harm interchanging mil, “milrad,” and milliradian.

It is worth noting that 1 mil = 3.6 SMOA <m>\approx</m> 3.44 MOA.

# Sample Sizes

One of the important questions addressed here is *what* to measure in order to determine the intrinsic precision of a shooting system, and what sample size is sufficient to achieve any degree of statistical significance.

Following are common measurements used in the industry:

- Extreme Spread of a 3-shot group, usually at 100 yards. This is statistically almost meaningless, especially when there is no reference to how many 3-shot groups were sampled. (An extended practical, and amusing, critique of the 3-shot group is archived here.)
- Extreme Spread of a 5-shot group, sometimes excluding the worst shot. Hardly any better.
- Average, Max, and Min Extreme Spread of five 5-shot groups. (This is used by the NRA's magazines.)
- The US Army Marksmanship Unit at Ft. Benning, GA uses a minimum of 3 consecutive 10-shot groups fired with the rifle in a machine rest when testing service rifles. The military also often uses the more statistically powerful Mean Radius and Radial Standard Deviation measures.

# Measures

Eight different measures have been used to characterize the dispersion of bullet holes in a sample target. Some are easier to calculate than others.

In the following formulas assume that we are looking at a target reflecting *n* shots and that we are able to determine the center coordinates *x* and *y* for each shot.

## Horizontal and Vertical Variance

- <m>\sigma_h^2 = \frac{\sum^{n}(x_i - \bar{x})^2}{n - 1}, \ \qquad \sigma_v^2 = \frac{\sum^{n}(y_i - \bar{y})^2}{n - 1}</m>

(Often these will be given as standard deviations, which is just the square root of variance.)

## Radial Standard Deviation (RSD)

- <m>RSD = \sqrt{\sigma_h^2 + \sigma_v^2} \ = \sqrt{2}\ \sigma</m>

Note that this average sigma is a key parameter in the math for Measuring Precision.

## Mean Radius

- <m>\bar{R} = \sum_{}^n r_i</m> where <m>r_i = \sqrt{(x_i - \bar{x})^2 + (y_i - \bar{y})^2}</m>

## Circular Error Probable (CEP)

CEP(*p*), for <m>p \in [0, 1)</m>, is the radius of the smallest circle that covers proportion *p* of the shot group. When *p* is not indicated it is assumed to be 50%.

## Diagonal

Let <m>\hat{X} = \max |x_i - x_j|, \ \qquad \hat{Y} = \max |y_i - y_j|</m> — i.e., the ranges of *x* and *y* values.

The diagonal <m>D = \sqrt{\hat{X}^2 + \hat{Y}^2}</m> is the length of the diagonal line through the smallest rectangle covering the sample group.

## Figure of Merit

FoM = <m>(\hat{X} + \hat{Y}) / 2</m> is the average extreme width and height of the group.

## Radius of Covering Circle

This is conceptually just CEP(100%). In contrast to CEP it is easier to measure on a sample target, but it has the unfortunate characteristic of being non-analytic, and unlike CEP its expected value increases with the sample size *n*.

## Extreme Spread

The Extreme Spread <m>ES = \max \sqrt{(x_i - x_j)^2 - (y_i - y_j)^2)}</m> is the largest distance between any two points in the group.

# Which Measure is Best?

Measuring Precision will detail the mathematical relationships between many of these measures. It is worth noting that the first four measures do not vary with group size. I.e., taking more shots increases their confidence interval but doesn't change their expected value. The last four are more popular because they are easier to measure, but they are statistically far weaker because they virtually ignore inner data points.