Circular Error Probable (CEP)

The engineering

CEP is the radius of the circle, centered on the mean point of impact, that contains 50% of the shots. Given the down-range (range) and cross-range (bearing) standard deviations of the impact scatter, this card returns the classic linear-combination CEP used in targeting and accuracy budgets.

The simple 0.5887·(σ_r+σ_b) form assumes a roughly circular error ellipse. When one axis dominates — σ_min/σ_max below about 0.2 — that formula overestimates, so the card switches to the elongated-ellipse weighting. Remember CEP is a median, not a mean: expect a heavy outlier tail, and never confuse CEP with the 1σ radius.

Where this math comes from

CEP grew out of WWII and early Cold War bombing-accuracy analysis, where planners needed a single intuitive accuracy number instead of a two-axis covariance. The bivariate-normal impact model gave the exact 50% radius, but analysts wanted something computable by hand — hence the linear-combination fits.

RAND Corporation formalized the practical approximations in its accuracy memoranda, tabulating CEP as a function of the ratio of the two axis sigmas. The 0.5887 coefficient (= √(2·ln2) applied to the averaged sigma for the circular case) and the elongated-ellipse weights became the standard back-of-envelope tools carried into missile and munition specs.

1. 1947USAAF/USAF operations analystsAdopt circular error probable as the standard bombing accuracy metric.
2. 1952RAND CorporationPublishes CEP approximation notes relating CEP to the range/bearing sigma ratio.
3. 1960Ballistic missile programsCEP becomes the headline accuracy figure in ICBM and cruise-missile specifications.
4. 1980MIL-STD accuracy conventionsStandardize CEP, R90, and R95 reporting for guided weapons.

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