Measurement is fundamental to virtually all human endeavors of practical importance. However, many measurements require accounting for error, and Circular Error Probable (CEP) is one method of quantifying deviations from a best estimate, commonly applied in the context of weapons systems and GPS. This article will derive CEP and explore its practical applications.

Deriving CEP

CEP is commonly defined as the radius of a circle in which 50% (i.e. the proportion ) of observed data points are expected to fall. Our goal is to determine the radius representing the circle enclosing the proportion of the data.

(Note: A related measure, R95, uses the same statistical formalism, but ensures that 95% of the data lies within the circle.)

To achieve this, lets summon the Rayleigh distribution, which will allow us to compute a good estimate of the radius. You can think of this distribution as a Gaussian distribution in which the random variable, in the case of the Gaussian would range from to , would be compressed to the domain . The formula for this distribution is as follows:

The density function itself resembles the form of the gaussian, in that they both belong to the exponential family of distributions. However, notice that this distribution is weighted by the variable itself, imposing a sort of controlled “skewing” tendency on the graph. Examples of the probability density function are plotted below:

For a given Rayleigh distribution corresponding to the Gaussian variance , we can find the radius which encircles percent of our data by adjusting the formula below:

Therefore corresponds to the cumulative distribution function (CDF) of the Rayleigh distribution. By rearranging this CDF in terms of , we obtain:

In particular, for , we obtain the CEP:

Consequently, the CEP is obtained by scaling the standard deviation by the factor .