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This site explains and demonstrates statistics for analyzing the precision of guns.

In particular:

Synopsis

When testing a gun to estimate its precision the most useful data are the (x, y) coordinates of each impact on sample targets. These allow for closed-form estimates and confidence intervals on the standard deviation of dispersion along each axis, and from the standard deviation we can deduce any standard precision measure for the gun.

If we assume that the inherent dispersion along each axis is roughly identical then we can use the average of the standard deviations, a single parameter σ, to characterize precision. This only requires the radius $$r_i = \sqrt{(x_i - \bar{x})^2 + (y_i - \bar{y})^2}$$ of each impact on a sample target.

The certainty with which we can assess precision increases with the number of shots. Since shooting large samples on a single target risks developing ragged holes where data points are lost, there are three recommended approaches to efficiently build data sets:

1. Use Danielson's 2-shot method: Fire two shots per target and use calipers to measure their distance from each other. This provides two samples per target with radius r = spread / 2.
2. Fire one shot per target. Manually overlay them or use software like OnTarget Target Data System to automatically aggregate them into a single sample group.
3. Use a logging electronic target. (Not yet widely available.)