Precision Models

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Models of Dispersion

We present four options for measuring and analyzing precision:

Before selecting one consider the following background:

General Bivariate Normal

The normal, a.k.a. Gaussian, distribution is the broadly accepted model of a random variable like the dispersion of a physical gunshot from its center point. The normal distribution is parameterized by its mean and standard deviation, or $$(\mu, \sigma)$$. As explained in What is Precision? we are only interested in the dispersion component, since the center point of impact is controlled by sighting in the gun (i.e., adjusting its aiming device). Therefore we will assume that a gunner can dial $$\mu \approx 0$$ and leave that parameter out of the question in what follows.

Since we are interested in shot dispersion on a two-dimensional target we will look at a bivariate normal distribution, which has separate parameters for the standard deviation in each dimension, $$\sigma_x, \sigma_y$$, as well as a correlation parameter ρ.

Uncorrelated Bivariate Normal

We don't have any compelling evidence that in general there is, or should be, correlation between the horizontal and vertical dispersion of gunshots. Therefore, for most of our analysis we will assume ρ = 0.

We do know that targets can often exhibit vertical or horizontal stringing, and therefore $$\sigma_x \neq \sigma_y$$. To the extent these parameters are not equal they produce elliptical instead of circular shot groups.

However, we know some of the significant sources of stringing and can potentially factor them out:

1. The primary source of x-specific variance is crosswind. If we measure the wind while shooting we can bound and remove a “wind variance” term from that axis. E.g., "Suppose the orthogonal component of wind is ranging at random from 0-10mph during the shooting. Given lag-time t this will expand the no-wind horizontal dispersion at the target by $$\sigma_w$$."[1] Since variances are additive we could adjust $$\sigma_x$$ via the equation $${\sigma'}_x^2 = \sigma_x^2 - \sigma_w^2$$.
2. The primary source of y-specific variance is muzzle velocity, which we can actually measure with a chronograph (or assert) and then remove from that axis. E.g., "If standard deviation of muzzle velocity is $$\sigma_{mv}$$ then, given the bullet's ballistic model for the given target distance, the vertical spread attributable to that is some $$\sigma_v$$. Here too we can remove this known source of dispersion from our samples via the equation $${\sigma'}_y^2 = \sigma_y^2 - \sigma_v^2$$. This adjustment is shown in several of the examples:

Statistical Analysis of Dispersion

In view of the preceding:

1. The Closed Form Precision model requires that we assume the shot group is, or can be normalized to be, a fairly symmetric bivariate Gaussian process, but allows for the most convenient estimators and analysis. Therefore, whenever $$\sigma_x \approx \sigma_y$$ we prefer this approach.
2. Circular Error Probable disregards any ellipticity in the actual shot process in order to characterize precision using a single parameter. Since most of precision estimation is for the purposes of comparing loads, rifles, and shooters, we need a single number and we don't care if the dispersion is elliptic: tighter is always better.
3. Elliptic Error Probable allows for a full characterization of the General Bivariate Normal model. For some applications – e.g., computing hit probabilities on non-circular targets – we want to preserve statistically significant ellipticity. And for a few – e.g., harmonic dampening, and perhaps shooter technique correction – the orientation of the ellipse produced by non-zero ρ could be helpful if it can be estimated.
4. Extreme Spread and the other Range Statistics, which increase with number of shots per group n, do not have any useful functional forms. The characteristics of these measures have to be derived from Monte Carlo simulation. They are the least efficient statistics but are also the most commonly used because they are so easy to measure in the field and so familiar to shooters.

One practical question that many shooters raise is what to do with outliers, known in the sport as "fliers." We address fliers here.

Tools

See Measuring Tools for convenient ways of measuring and analyzing precision.

References

1. Wind deflection is a function of the ballistic curve and distance, but can be expressed as a simple product of the cross-wind velocity and lag time. For more information on the "lag rule" see Bryan Litz, Applied Ballistics for Long Range Shooting, 2nd Edition (2011) A4; or Robert McCoy, Modern Exterior Ballistics, 2nd Edition (2012) 7.27.