The uniform distribution for directional and cyclic data. The probability density function f is constant for directions (in radians)For a data set θ₁, θ₂,…, θ_n, a test of circular uniformity amounts to a test of the null hypothesis, H₀, that there is no preferred direction, with the alternative being that the data come from some unimodal circular distribution. The Rayleigh test, introduced by Lord Rayleigh in 1880, uses the test statistic R² given byUnder H₀ the approximate probability of observing a value≥R² is given byFor very large values of n, T=2R²/n is an observation from an approximate chi-squared distribution with two degrees of freedom.

As an alternative, the Ajne test (suggested by the Swede Björn Ajne in 1968) uses as its test statistic the number of observations contained within the semicircle for which this number is least.