How is this average possible?

Question

I am studying atomic physics from Physics of Atoms and Molecules (B. H. Bransden) and in some part he says:

\begin{equation} \left\langle\cos^2 \theta\right\rangle=\dfrac{1}{3} \end{equation}

But from everything I know this should be equals to $\dfrac{1}{2}$.

Visual proof (page 168 of the book)

Answer 1

Depends on what $\langle \rangle$ and $\theta$ mean. If $\theta$ is colatitude on a sphere, where the area element is $\sin(\theta)\; d\theta d\phi/(4\pi)$, then they are right.