# Circular error probable

In the military science of ballistics, Circular Error Probability or circular error probable (CEP) is a simple measure of a weapon system's precision. It is defined as the radius of a circle into which a missile, bomb, or projectile will land at least half the time.

For example, an ICBM warhead with a CEP of 100 meters will impact within 100 meters of the target point in at least 50 percent of all attempts.

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 indeed 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, and the resulting confidence regions is elliptical. Generally, the munition will not be exactly on target, i.e. 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 percent of rounds will land.

It should be noted that the concept of CEP is only strictly meaningful if misses are roughly normally distributed. This is generally not true for precision-guided munitions.

Another important thing to remember about CEP is that if 50 percent of rounds land inside the circle, then 50 percent land outside it! Generally, if CEP is n meters 50 percent of rounds land within n meters of the target, 43 percent between n and twice that distance and 7 percent between two and three times that distance. If misses were exactly normally distributed as in this theory, then the proportion of rounds that land farther than three times the CEP from the target is less than 0.2%. With precision-guided munitions, the number of 'close misses' is higher.

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