Circular uniform distribution

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Directional statistics (also called circular statistics) can be a challenge to understand, especially if you are new to the subject. However, if you want to pursue a career in science or engineering, it is essential to have a good understanding of the circular uniform distribution. In this blog post, we will walk you through everything you need to know about circular uniform distribution including its definition, properties and applications.

Definition of Circular Uniform Distribution

A circular uniform distribution is a continuous probability distribution over the unit circle, where the density of the probability is the same (uniform) for all angles.

unit circle
The unit circle.

Uniformity means that all values around the circle are equally likely [1]. To understand it better, think of a compass with 360-degree angles. In a circular uniform distribution, any angle within the compass has an equal chance of being selected.

Properties of Circular Uniform Distribution

The circular uniform distribution has several properties, such as rotational invariance, which means that the probability of the angle between any two vectors is the same regardless of the starting point of the angle.

The circular uniform distribution on the unit circle is closed under summation (modulus 2π); in other words, the sum of independent circular uniformly distributed random variables is also circular uniformly distributed [2]. Modulus 2π refers to the remainder after dividing by 2π.

The circular uniform distribution is obtained by mapping points from [0, 1] to [0, 2π]; periodicity is implicitly assumed [3]. Periodicity is the property of a function that repeats itself after a certain interval. In other words, it returns to the same value after a certain period of time or distance.

Probability densities {\displaystyle P_{N}({\bar {R}})} for small values of N. Densities for N>3″> are normalized to the maximum density, those for <img decoding= and 2 are scaled to aid visibility [4].

The probability density function (PDF) is given by [2]

f(a) = 1/360,

where 𝑎 is an angle in radians and the probability between any two points 𝑎1 and 𝑎2 is

Pr(𝑎1 < 𝑎2 | 𝑎1 ≤ 𝑎2, 𝑎2 ≤ 𝑎1 + 2𝜋) = (𝑎2 − 𝑎1) / (2π).

The Rayleigh test of uniformity is one way to test for circular uniformity; the test returns a p-value between 0 and 1, which is the probability of seeing data similar to or more clustered than the observed sample if circular uniformity is truly behind the observed process [5]. Other tests for circular uniformity include Watson’s test, Kuyper’s test, and Rao’s spacing test.

Similarity to other distributions

The von Mises (circular normal) distribution equals the circular uniform distribution when κ = 0. This indicates that there is little difference between the two models [6]. Stephens [7] provides a table of approximate sample sizes required to identify whether data follow a von Mises or circular uniform distribution.

PDF of the von Mises distribution; when k = 2, it becomes the circular uniform distribution
PDF of the von Mises distribution; when κ = 2, it becomes the circular uniform distribution.

The circular uniform distribution is similar to the normal distribution in the sense that it has a mean and variance. However, one of the significant differences is the range of the distribution. While the normal distribution ranges from negative infinity to infinity, the circular uniform distribution only ranges from 0 to 2π. Also, unlike the normal distribution, the circular uniform distribution is not symmetric.

Applications of Circular Uniform Distribution

The circular uniform distribution has numerous applications in science and engineering, such as directional statistics, robotics, physics, and meteorology. For instance, it is used to model wave propagation, wind direction, and orientation estimation in robotics. It is also widely used in astronomy to determine the orientation of galaxies.


[1] Chapter 230: Circular Data Analysis. NCSS Statistical Software.

[2] Frenandez-Duran et al. (2022). Sums of Independent Circular Random Variables and Maximum Likelihood Circular Uniformity Tests Based on Nonnegative Trigonometric Sums Distributions.

[3] Karmaker, S. (2016). On Some Circular Distributions Induced by Inverse
Stereographic Projection. Graduate Thesis, Concordia University.

[4] Lucaswilkins, CC BY-SA 4.0, via Wikimedia Commons

[5] Landler et. al. (2020). Model selection versus traditional hypothesis testing in circular statistics: a simulation study. The Company of Biologists Ltd | Biology Open (2020) 9, bio049866. doi:10.1242/bio.049866

[6] Bentley, J. (2006). Modeling circular data using a mixture of von mises and uniform distributions. Retrieved April 16, 2023 from:

[7] Stephens, M. A. (1969). Tests for randomness of directions against two circular alternatives. J. Amer. Statist. Assoc., 64:280–289

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