In probability theory and directional statistics, a circular uniform distribution is a probability distribution on the unit circle whose density is uniform for all angles.

The probability density function (pdf) of the circular uniform distribution, e.g. with ${\displaystyle \theta \in [0,2\pi )}$, is:

We consider the circular variable ${\displaystyle z=e^{i\theta }}$ with ${\displaystyle z=1}$ at base angle ${\displaystyle \theta =0}$. In these terms, the circular moments of the circular uniform distribution are all zero, except for ${\displaystyle m_{0}}$:

where ${\displaystyle \delta _{n}}$ is the Kronecker delta symbol.

Here the mean angle is undefined, and the length of the mean resultant is zero.

The sample mean of a set of N measurements ${\displaystyle z_{n}=e^{i\theta _{n}}}$ drawn from a circular uniform distribution is defined as:

where the average sine and cosine are:

and the average resultant length is: