Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Expected value ▷ Circular sample mean

Theorem: Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a sample from a circular random variable $X$. Then, the sample mean of $x$ is given by

\[\label{eq:meancirc-samp} \bar{x} = \arctan \left( \frac{\sum_{i=1}^n \sin x_i}{\sum_{i=1}^n \cos x_i} \right)\]

where $\sin$ and $\cos$ are the sine and cosine function, respectively, and $\arctan$ is the arctangent function.

Proof: For observations of a circular random variable satisfying

\[\label{eq:rvar-circ-samp-x} 0 \leq x_i < 2 \pi, \; i = 1,\ldots,n \; ,\]

calculating the sample mean as the arithmetic mean is not appropriate as this does not preserve the periodic nature of the random variable.

Instead and in accordance with the definition of the circular expected value, observation values are transformed to points on the unit circle ($r = 1$):

\[\label{eq:rvar-circ-samp-y} y_i = \left[ \begin{matrix} \cos x_i \\ \sin x_i \end{matrix} \right], \; i = 1,\ldots,n \; .\]

Then, we calculate the sample mean of transformed data points

\[\label{eq:y-mean} \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i = \left[ \begin{matrix} \frac{1}{n} \sum_{i=1}^n \cos x_i \\ \frac{1}{n} \sum_{i=1}^n \sin x_i \end{matrix} \right]\]

and equating this sample mean with its polar-coordinate representation

\[\label{eq:y-mean-pc} \left[ \begin{matrix} \frac{1}{n} \sum_{i=1}^n \cos x_i \\ \frac{1}{n} \sum_{i=1}^n \sin x_i \end{matrix} \right] = \left[ \begin{matrix} \bar{r} \cos \bar{x} \\ \bar{r} \sin \bar{x} \end{matrix} \right] \; ,\]

we obtain the solution for $\bar{x}$ by solving the equation system:

\[\label{eq:x-mean} \begin{split} \frac{\bar{r} \sin \bar{x}}{\bar{r} \cos \bar{x}} &= \frac{\frac{1}{n} \sum_{i=1}^n \sin x_i}{\frac{1}{n} \sum_{i=1}^n \cos x_i} \\ \frac{\sin \bar{x}}{\cos \bar{x}} &= \frac{\sum_{i=1}^n \sin x_i}{\sum_{i=1}^n \cos x_i} \\ \tan \bar{x} &= \frac{\sum_{i=1}^n \sin x_i}{\sum_{i=1}^n \cos x_i} \\ \bar{x} &= \arctan \left( \frac{\sum_{i=1}^n \sin x_i}{\sum_{i=1}^n \cos x_i} \right) \; . \end{split}\]
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Metadata: ID: P533 | shortcut: meancirc-samp | author: JoramSoch | date: 2026-04-20, 16:38.