Circular error probable
In the military science of ballistics, circular error probable (CEP) (also circular error probability or circle of equal probability) is a measure of a weapon system's precision. It is defined as the radius of a circle, centered on the mean, whose boundary is expected to include the landing points of 50% of the rounds. That is, if a given bomb design has a CEP of 100 metres (330 ft), when 100 are targeted at the same point, 50 will fall within a 100 m circle around their average impact point. (The distance between the target point and the average impact point is referred to as bias.)
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 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 meters, 50% of rounds land within n meters of the mean impact, 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 mean is approximately 0.32%.
CEP is not a good measure of accuracy when this distribution behavior is 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 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).
Conversion between CEP, RMS, 2DRMS, and R95
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. Approximate formulae are available to convert the distributions along the two axes into the equivalent circle radius for the specified percentage.
|Root mean square (RMS)||63 to 68|
|Circular error probability (CEP)||50|
|Twice the distance root mean square (2DRMS)||95 to 98|
|95% radius (R95)||95|
- Nelson, William (1988). "Use of Circular Error Probability in Target Detection" (PDF). Bedford, MA: The MITRE Corporation; United States Air Force.
Ehrlich, Robert (1985). Waging Nuclear Peace: The Technology and Politics of Nuclear Weapons. Albany, NY: State University of New York Press. p. 63.
- Circular Error Probable (CEP), Air Force Operational Test and Evaluation Center Technical Paper 6, Ver 2, July 1987, p. 1
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- Frank van Diggelen, "GNSS Accuracy – Lies, Damn Lies and Statistics", GPS World, Vol 18 No. 1, January 2007. Sequel to previous article with similar title  
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- Blischke, W. R. and Halpin, A. H. (1966). "Asymptotic properties of some estimators of quantiles of circular error." Journal of the American Statistical Association, vol. 61 (315), pp. 618–632. URL http://www.jstor.org/stable/2282775
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- Grubbs, F. E. (1964). Statistical measures of accuracy for riflemen and missile engineers. Ann Arbor, ML: Edwards Brothers. 
- Spall, J. C. and Maryak, J. L. (1992). "A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data." Journal of the American Statistical Association, vol. 87 (419), pp. 676–681. URL http://www.jstor.org/stable/2290205
- Daniel Wollschläger (2014), "Analyzing shape, accuracy, and precision of shooting results with shotGroups".  Reference manual for shotGroups, an R package 
- Winkler, V. and Bickert, B. (2012). "Estimation of the circular error probability for a Doppler-Beam-Sharpening-Radar-Mode," in EUSAR. 9th European Conference on Synthetic Aperture Radar, pp. 368-371, 23-26 April 2012. URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6217081&isnumber=6215928