# Elliptic Error Probable

**Previous:** Precision Models

# Preserving Asymmetry

There are two significant sources of dispersion asymmetry in ballistics:

- Crosswind variance will cause excess dispersion only in the horizontal axis.
- Muzzle velocity variance will cause excess dispersion that appears only in the vertical axis, and which can become quite significant at longer shooting distances.

Given (*x, y*) coordinate data of impacts on a target, we can preserve asymmetry in the precision models and their application by treating each axis separately. As described in Closed Form Precision, for variance in one dimension:

- \(\hat{\sigma}_x^2 = s_x^2 = \frac{1}{n-1}\sum{(x_i - \bar{x})^2}\)

The confidence interval for variance in one dimension uses \(\chi^2\) values with (*n* - 1) degrees of freedom:

- \(\hat{\sigma}_x^2 \in \left[\, \frac{(n-1)s_x^2}{\chi_{\frac{\alpha}{2},n-1}^2}, \ \frac{(n-1)s_x^2}{\chi_{1-\frac{\alpha}{2},n-1}^2} \,\right]\)

As before, the estimates can be brought into unbiased standard deviation terms by multiplying the square root of estimated variance by the Gaussian correction term\[\hat{\sigma}_x=c_G(n) \sqrt{s_x^2}\]

# Covering Ellipse

The radii (*a, b*) of an ellipse that will cover proportion *p* of shots are given by the Rayleigh quantile function:

- \(a=\sigma_x \sqrt{-2 \ln(1-p)}\)
- \(b=\sigma_y \sqrt{-2 \ln(1-p)}\)

For example, here is a simulation of shots with \(\sigma_x=3, \sigma_y=1\) overlaid with the 50% covering ellipse:

# Other target shapes

As detailed in Section 2.3 of ARL-TR-6494^{[1]}, numerical integration can calculate the hit probabilities for arbitrary target shapes and sizes.

- ↑ Strohm, Luke (2013)
*An Introduction to the Sources of Delivery Error for Direct-Fire Ballistic Projectiles*