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In the military science of ballistics, circular error probable (CEP) (also circular error probability or circle of equal probability^{[1]}) is a measure of a weapon system's precision. It is defined as the radius of a circle, centered about the mean, whose boundary is expected to include the landing points of 50% of the rounds.^{[2]}^{[3]}
The original concept of CEP was based on a circular bivariate normal distribution (CBN) with CEP as a parameter of the CBN just as μ and σ are parameters of the normal distribution. Munitions with this distribution behavior tend to cluster around the aim point, with most reasonably close, progressively fewer and fewer further away, and very few at long distance. That is, if CEP is n meters, 50% of rounds land within n meters of the target, 43% between n and 2n, and 7% between 2n and 3n meters, and the proportion of rounds that land farther than three times the CEP from the target is approximately 0.32%.
This distribution behavior is often not met. Precision-guided munitions generally have more "close misses" and so are not normally distributed. Munitions may also have larger standard deviation of range errors than the standard deviation of azimuth (deflection) errors, resulting in an elliptical confidence region. Munition samples may not be exactly on target, that is, the mean vector will not be (0,0). This is referred to as bias.
To apply the CEP concept in these conditions, CEP can be defined as the square root of the mean square error (MSE). The MSE will be the sum of the variance of the range error plus the variance of the azimuth error plus the covariance of the range error with the azimuth error plus the square of the bias. Thus the MSE results from pooling all these sources of error, geometrically corresponding to radius of a circle within which 50% of rounds will land.
While 50% is a very common definition for CEP, the circle dimension can be defined for percentages. Percentiles can be determined by recognizing that the squared distance defined by two uncorrelated orthogonal Gaussian random variables (one for each axis) is chi-square distributed.^{[4]} Approximate formulae are available to convert the distributions along the two axes into the equivalent circle radius for the specified percentage.^{[5]}^{[4]}
Ehrlich, Robert (1985). Waging Nuclear Peace: The Technology and Politics of Nuclear Weapons. Albany, NY: State University of New York Press. p. 63.
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