Editing Circular error probable (section)

==Concept==
[[File:Multivariate Gaussian.png|thumb|Circular bivariate normal distribution]]
[[File:Circular error probable - example.png|thumb|20 hits distribution example]]

The original concept of CEP was based on a [[multivariate normal distribution#Bivariate case|circular bivariate normal]] distribution (CBN) with CEP as a parameter of the CBN just as μ and σ are parameters of the [[normal distribution]]. [[Munition]]s with this distribution behavior tend to cluster around the [[mean]] impact point, with most reasonably close, progressively fewer and fewer further away, and very few at long distance. That is, if CEP is ''n'' metres, 50% of shots land within ''n'' metres of the mean impact, 43.7% between ''n'' and ''2n'', and 6.1% between ''2n'' and ''3n'' metres, and the proportion of shots that land farther than three times the CEP from the mean is only 0.2%.

CEP is not a good measure of accuracy when this distribution behavior is not met. 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 [[Unbiased estimator|bias]].

To incorporate accuracy into 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.

Several methods have been introduced to estimate CEP from shot data. Included in these methods are the plug-in approach of Blischke and Halpin (1966), the Bayesian approach of Spall and Maryak (1992), and the maximum likelihood approach of Winkler and Bickert (2012). The Spall and Maryak approach applies when the shot data represent a mixture of different projectile characteristics (e.g., shots from multiple munitions types or from multiple locations directed at one target).