Circular error probable

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In the military science of ballistics, circular error probable (CEP) or circular error probability is a simple measure of a weapon system's precision. It is defined as the radius of a circle into which a warhead, missile, bomb, or projectile will land at least 50% of the time. The CEP of nuclear warheads on U.S. ICBMs was dramatically improved with the advent of the Global Positioning System (GPS).

Some examples of CEP include:

A Trident II warhead has a CEP of 90 meters (with GPS guidance); thus, there is a 50% probability that each warhead will impact within 90 meters (assuming GPS is used).
For Minuteman III warheads, the CEP is 275 meters for the three 170 kt W62 warheads contained in General Electric (GE) Mk 12 reentry vehicles, and 220 meters for the three 335 kt W78 warheads contained in GE Mk 12A reentry vehicles.
In its most accurate mode, a Joint Direct Attack Munition provides a CEP of 13 meters or less when GPS data is available.
Russian newest quasi-ballistic Iskander missile has a CEP of 10 meters or less, when all guidance modes are used, making it one of the most accurate ballistic missiles ever.

The impact of munitions near the target tends to be bivariate normally distributed around the aim point, with most reasonably close, progressively fewer and fewer further away, and very few at long distance. One component of the bivariate normal will represent range errors and the other azimuth errors. Unless the munition is arriving exactly vertically downwards, the standard deviation of range errors is usually larger than the standard deviation of azimuth errors, so the resulting confidence region is elliptical.

Generally, the munition will not be exactly on target, that is, the mean vector will not be (0,0). This is referred to as bias. The mean error squared (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. The square root of the MSE is the circular error probable, commonly abbreviated to CEP. Geometrically, it corresponds to radius of a circle within which 50 % of rounds will land.

The concept of CEP is based on a normally distributed accuracy distribution. 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 less than 0.2%. This is not indicitive of precision-guided munitions, with which the number of 'close misses' is higher. While 50% is a very common definition for CEP, the circle dimension can be defined for percentages. Approximate formulas are available to convert the distributions along the two axes into the equivalent circle radius for the specified percentage.