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Measuring Precision

The following text considers mainly shots from a direct fire weapon firing a single projectile on a vertical target within the line of sight, for example rifle or pistol shots. Such weapons as shotguns, intercontinental missiles, and motars would have some similar characteristics, but also have factors that are neglected in the measurements.

Dispersion 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 adjust to match the point of aim). There are two basic categories of units for dispersion, linear distances and as an angle.

Linear distance typically uses a fixed (and specified) distance. For example the inches in diameter of a group of shots at 100 yards.

Angular Size is another common unit and is the angle at the tip of the cone of fire since this is independent of the distance at which a target is shot. The higher the precision, the tighter the cone and hence the smaller the angle at its tip.

Linear Distance

In countries using the metric system the extreme spread of shots (group size) would typically be measured in centimeters (cm), or perhaps millimeters (mm). Countries (i.e. the USA) still using the British Imperial system would typically measure linear distances in inches.

Mil

The other common linear 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 an angular unit. Fortunately, a milliradian — 3600" tan (1/1000 radians) ≈ 3.600001" inches at 100 yards — is almost exactly equal to a mil so there’s little harm in interchanging mil, mrad, milrad, and milliradian.

Angular Size

The overall assumption is that the 2-dimensional precision is like a cone that projects linearly from the muzzle of the gun - i.e., double the distance and the dispersion also doubles. In many instances this model is sufficient. In reality this isn't true for all cases.

For example due to projectile spin and aerodynamics there is some point at which a projectile's flight would degrade faster than the linear distance. So a 1 inch group at 100 yards might become a 10 inch group at 500 yards, and a three foot group at 1000 yards.

Another example is given by cases documented where a projectile "goes to sleep." Essentially the violent exit of the projectile from the muzzle results in an projectile instability which is damped by air resistance. In this case a group might be 0.5 inches at 50 yards, but just 3/4 of an inch at 100 yards. Thus the linear group size at a longer distance is larger, but not geometrically larger. Note however that if you were using an angular measure, then the group size would be smaller at 100 yards than 50 yards.

Minute Of Arc

One of two popular angular units used by shooters is MOA, though there is some ambiguity in this term. From high school geometry a circle is divided into 360 degrees, and each degree is divided into 60 minutes. Thus MOA was initially short for Minute of Arc, or arc minute, which is one sixtieth of one degree.

At 100 yards (3600 inches) one MOA is 3600" tan (1/60 degrees) = 1.047".

Shooter's Minute of Angle

At some point shooters began to expand the acronym as Minute of Angle. They also rounded its correct value to 1” at 100 yards, though for clarity the latter unit is properly called "Shooters MOA," or SMOA.

Conversions between measuring units

See Angular Size for detailed illustrations and conversion formulas.

Dispersion Measures

Precision Measures diagrammed on a 10-shot 100-yard group. Data in Media:SCAR17_150gr_100yd.xls

Nine 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:

1. We are looking at a target reflecting n shots
2. We are able to determine the center coordinates h and v as needed for analysis. For example for extreme spread we just need to be able to measure the distance between the two widest shots, but for the radial standard deviation we need the horizontal and vertical positions of each shot on the target.

Some additional mathematical symbols and variables:
    NSPG - Number of Shots Per Group
    \(\bar{h}\) is defined as \(\bar{h} \equiv \sum_{i=1}^n h_i / n\)
    \(\bar{v}\) is defined as \(\bar{v} \equiv \sum_{i=1}^n v_i / n\)

NSPG Invariant Measures

It is worth noting that following measures in this section do not increase with the number of shots per group (NSPG). That means that more shots tightens the precision of the measurement but doesn't change its expected value.

• Mean Radius
• Circular Error Probable
• Horizontal and Vertical Variances
• Radial Standard Deviation of the Rayleigh Distribution
• Bivariate Gaussian Distribution Parameters

Mean Radius (MR)

The Mean Radius is the average distance over all shots to the groups center.

Assumptions	Rayleigh Distribution for Shots \(\sigma_h = \sigma_v\) \(\rho = 0\) No Flyers
Experimental Measure	\(\bar{r} = \sum_{i=1}^n r_i / n\)     where \(r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}\)
PDF(\(r_i\))
CDF(\(r_i\))
Variables in PDF & CDF	\(r\)
Fitted parameters	\(\mu_h, \mu_v\)
Mode of PDF(\(r_i\))
Median of PDF(\(r_i\))
Mean of PDF(\(r_i\))
PDF(\(\bar{r}\))
CDF(\(\bar{r}\))
(h,v) for all points?	Yes
Symmetric about Measure?	No, but the distribution will become more symmetric as the number of shots in a group increases.
NSPG Invariant	Yes

Formula:
    \(\bar{r} = \sum_{i=1}^n r_i / n\) where \(r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}\)

As we will see in Closed Form Precision, the Mean Radius is typically only 6% larger than the Circular Error Probable. Since this is within the margin of error of most real-world usage the terms MR and CEP may be interchanged in casual usage.

Circular Error Probable (CEP)

\(CEP_p\), for \(p \in [0, 1)\), 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 \(CEP_{0.5}\), which is the median shot radius (50% radius).

Horizontal and Vertical Variances

Assumptions:

• \(\sigma_h \neq \sigma_v\)
• \(\rho = 0\)
• No Flyers

    \(\sigma_h^2 = \frac{\sum^{n}(h_i - \bar{h})^2}{n - 1}, \quad \sigma_v^2 = \frac{\sum^{n}(v_i - \bar{v})^2}{n - 1}\)

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

Radial Standard Deviation of the Rayleigh Distribution

From high school mathematics one should remember the two coordinate systems - Cartesian Coordinates and Polar Coordinates. Essentially the Rayleigh distribution converts shots from the Cartesian Coordinate system to the Polar Coordinate system. It is implicit in the coordinate conversion that the origin for the polar coordinate system is at the average point of impact. Thus for the polar coordinates the radial positioon of each shot will be relative to origin, or the average point of impact. Each shot will then have two coordinates, an angle, \(\theta, \) and the radius, r. Given that the shot distribution assumptions hold, then the angle should be entirely random and is of no interest. The two-variable problem has thus been reduced to a one-variable problem of determining the distribution for the shot radius.

Given that the conversion for the radial distance for each shot i from Cartesian Coordinates is:
    \(r_i = \sqrt{(h_i - \bar{h})^2 + (v_i - \bar{v})^2}\)
then the mean radius for the sample of i shots can be calculated in a straight forward manner using:
    \(\bar{r} = \frac{\sum_{i=1}^n r_i}{n}\)
and likewise for the standard deviation of the radius sample:
    \(s_r = \sqrt{\frac{\sum^{n}(r_i - \bar{r})^2}{n - 1}}\)

Now assuming that the shot dispersion follows the Bivariate Gaussian Distribution and that the following simplifying assumptions are true:

• \(\sigma_h = \sigma_v\)
• \(\rho = 0\)
• No Flyers

then the equation for the PDF for an individual shot is given by the Rayleigh distribution function which is:
    \(f(r) = \frac{r}{\sigma_{RSD}^2} e^{-r^2/2\sigma_{RSD}^2)}, \quad r \geq 0,\)
where \(\sigma_{RSD}\) is the single scale parameter of the distribution and is called the Radial Standard Deviation. Solving the distribution function for the mean radius and the standard deviation of the radius shows that they both are a proportional to \(\sigma_{RSD}\).

For the mean radius:
    \(\bar{r} = \sqrt{\pi/2} \sigma_{RSD} \approx 1.2533 \sigma_{RSD}\)
and for the standard deviation of the radius:
    \(\sigma_r = \sqrt{\frac{4 - \pi}{2}} \sigma_{RSD} \approx 0.6551 \sigma_{RSD}\)

There is an additional association which needs to be mentioned. Given the assumption \(\sigma_h = \sigma_v\), then according to the strict derivation of the Rayleigh distribution, \(\sigma_{RSD} = \sigma_h = \sigma_v\). In reality for the sample of shots \(s_h \approx s_v\) which means that \(s_{RSD} = (s_h + s_v)/2\)

This bit of mathematical magic is due to the fact that the error of a shot from the polar origin has been broken into two parts, an angular error and a radial error. The implicit assumption here is that the angular part of the error is entirely random and hence not significant in characterizing the distribution of the radius. Thus that part of the error in a shot's position has been isolated and removed. This mathematical manipulation isn't "free." The essence is that the Rayleigh model places an even greater dependency on the assumptions when making predictions about confidence intervals which use the standard deviation. In plainer language if the assumptions don't hold, then a small error in the estimated \(\sigma_{RSD}\) results in larger errors in the confidence interval predictions.

Bivariate Gaussian Distribution Parameters

Assumptions:

• Full Bivariate Gaussian Distribution for shot dispersion
  • \(\sigma_h \neq \sigma_v\)
  • \(\rho \neq 0\)
• No Flyers

The full bivariate Gaussian distribution is:
    \( f(h,v) = \frac{1}{2 \pi \sigma_h \sigma_v \sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)}\left[ \frac{(h-\bar{h})^2}{\sigma_h^2} + \frac{(v-\bar{v})^2}{\sigma_v^2} - \frac{2\rho(x-\bar{h})(y-\bar{v})}{\sigma_h \sigma_v} \right] \right), \)

In general case, when \(\sigma_h \neq \sigma_v\), then the actual standard deviation

of the radius \(r_i\) is not easy to calculate and is given by the formula:

    \(\frac{\sigma_h^2}{\pi} (\pi - 2 K^2(1 - \frac{\sigma_v^2}{\sigma_h^2})) + \sigma_v^2\) where K is the complete elliptic integral.

NSPG Variant Measures

The following measures in this section do increase with group size. They are more commonly used because they are obtained by direct measurements with either no calculations, or very simple calculations. But they are statistically weaker because they mostly ignore inner data points.

• Group Size (GS)
• Diagonal (D)
• Figure of Merit (FOM)
• Covering Circle Radius (CCR)

In general a ragged hole does not present a problem for these measures. However a ragged hole might be an experimental problem depending, for example, if the projectile syle does not cut relatively clean holes, or if the target tears.

Group Size (GS)

The Group Size is is the largest center-to-center distance between any two points, i and j, in the group. It is often called the Extreme Spread in the statistical literature.

Assumptions:

• Rayleigh Distribution for shot dispersion
  • \(\sigma_h = \sigma_v\)
  • \(\rho = 0\)
• No Flyers

Formula:
    \(ES = \max \sqrt{(h_i - h_j)^2 - (v_i - v_j)^2)}\)

Note: Be careful with with the phrase extreme spread. Shooters will often refer to the range of values from a chronograph as the extereme spread. Context should allow an easy determinatioon of the correct meaning of the phrase.

Diagonal (D)

The Diagonal is the length of the diagonal line through the smallest rectangle covering the sample group. Note that it is implicit that the rectangle is oriented along the horizontal and vertical axes. The diagonal may be determined by two to four points depending on the pattern of shots within the group.

Assumptions:

• \(\sigma_h \neq \sigma_v\)
• \(\rho = 0\)
• No Flyers

Formula:
\(ES = \sqrt{(h_high - h_low)^2 - (v_high - v_low)^2)}\) where \((h_high - h_low)\) and \((v_high - v_low)\) are the observed horizontal and vertcal ranges respecively.

Figure of Merit (FOM)

The Figure of Merit is the average extreme width and height of the group. The FOM may be determined by two to four points depending on the pattern within the group.

Assumptions:

• Rayleigh Distribution for shot dispersion
  • \(\sigma_h = \sigma_v\)
  • \(\rho = 0\)
• No Flyers

Formula:
    \(ES = ((h_{high} - h_{low}) + (v_{high} - v_{low})) / 2\)

The FOM would be reasonable when \(\sigma_h \approx \sigma_v\). However if it is known that \(\sigma_h \ne \sigma_v\), then using the measurement makes no sense. It would be better to use the Diagonal measurement.

Covering Circle Radius (CCR)

The Covering Circle Radius is the radius of the smallest circle containing all shot centers. This will pass through at least the two extreme shots (in which case CCR = (Group Size) / 2 ) or at most it will pass through three outside shots.

Assumptions:

• Rayleigh Distribution for shot dispersion
  • \(\sigma_h = \sigma_v\)
  • \(\rho = 0\)
• No Flyers

Which Measure is Best?

Precision Models discusses in more detail the assumptions about shot dispersion. The disconcerting truth is that there is no universally best measurement. All measurements are dependent on assumptions about the "true" distribution for the dispersion of individual shots, and about the presence of true "outliers" in the data. In practice the effect of neither of these factors is known.

The lack of an absolute truth may be mitigated with an expectation of picking reasonable assumptions and a mathematical model that is good enough. In essence start with a simple assumptions and model, and if the data indicates that the assumptions or model are inadequate, then increase the complexity of the model. Here complexity of the model essentially means the number of parameters which are determined experimentally. So the Rayleigh model has three experimental parameters (average horizontal position, average vertical position and the standard deviation of the radius), but the full bivariate Gaussian distribution has five ((average horizontal position, average vertical position, the horizontal standard deviation, the vertical standard deviation and ρ). The drawback here is that since the full bivariate Gaussian distribution has more parameters to fit experimentally, it would require more data to obtain a good experimental fit.

Shooters use the term flyer to denote the statistical term outlier. An outlier denotes an expected "good shot" with an abnormally large dispersion. So a shot that is much father than average from the center of the group would be a flyer. On the other hand, let's assume that the shooter realizes that his rifle was canted as the rifle discharges. The shooter would call that a "bad shot" before determining the shot position and would ignore that shot when making his measurements regardless of where the projectile landed.

It is convenient to consider the Rayleigh distribution function (or the full bivariate Gaussian as appropriate) as the gold standard given the situation that the underlying assumptions about shot dispersion and the lack of outliers holds. In this situation the Rayleigh model is 100% efficient since it makes as much use of the statistical data as is theoretically possible. In statistics the standard deviation of a measurement divided by the measurement expresses the error as a dimensionless per centage. The effiency of various measures can be thus compared by using the ratios of the variances, the relative standard deviations squared.

However one must be careful to not be too swayed by theory as opposed to experimental reality. In reality the conformance to theory is only due to a lack of enough experimental data to infer that the theory is incorrect. Also for the Rayleigh model neither the position of the center, nor the average radius, nor the standard deviation of the radius are [robust estimators].

Examples

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 by shooters or in the firearm industry:

• Extreme Spread of one 3-shot group, usually at 100 yards.

This is statistically poor, especially when there is no reference to how many 3-shot groups were sampled.

([extended practical, and amusing, critique of the 3-shot group is archived here].)

• Extreme Spread of one 5-shot group, sometimes excluding the worst shot. Hardly any better.
• Average, Max, and Min Extreme Spread of five 5-shot groups.

(This is the protocol used by the NRA's magazines and is actually rather efficient.)

• 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. Armed forces also often explicitly uses the more statistically powerful Mean Radius and Circular Error Probable measures.

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Next: Precision Models