Circular error probable
In the military science
Military science
Military science is the process of translating national defence policy to produce military capability by employing military scientists, including theorists, researchers, experimental scientists, applied scientists, designers, engineers, test technicians, and military personnel responsible for...

 of ballistics
Ballistics is the science of mechanics that deals with the flight, behavior, and effects of projectiles, especially bullets, gravity bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.A ballistic body is a body which is...

, circular error probable (CEP) (also circular error probability or circle of equal probability) is an intuitive measure of a weapon system's precision
Accuracy and precision
In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...

. It is defined as the radius of a circle, centered about the mean, whose boundary is expected to include 50% of the population within it.

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 aim 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 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 distribution behavior is often not met. Precision-guided munition
Precision-guided munition
A precision-guided munition is a guided munition intended to precisely hit a specific target, and to minimize damage to things other than the target....

s generally have more "close misses" and so are not normally distributed. Munitions may also have larger standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

 of range errors than the standard deviation of azimuth (deflection) errors, resulting in an elliptical confidence region
Confidence region
In statistics, a confidence region is a multi-dimensional generalization of a confidence interval. It is a set of points in an n-dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem, although other shapes can occur.The confidence region is...

. 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 apply 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
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

 of the range error plus the variance of the azimuth error plus the covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

 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
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

 of a circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 within which 50% of rounds will land.

Conversion between CEP, RMS, 2DRMS, and R95

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.
Accuracy Measure Probability (%)
Root mean square
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...

63 to 68
Circular error probability (CEP) 50
Twice the distance root mean square (2DRMS) 95 to 98
95% radius (R95) 95

From/to CEP RMS R95 2DRMS
CEP - 1.2 2.1 2.4
RMS 0.83 - 1.7 2.0
R95 0.48 0.59 - 1.2
2DRMS 0.42 0.5 0.83 -
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