I have a set of unit-length 3D vectors (represented as cartesian 3-tuples) with 180$^\circ$ equivalence (rotated through any plane passing through the origin i.e. $v=-v$), all situated at the origin (really their position is irrelevant). I'm interested in calculating the degree of anisotropy exhibited in the orientations of the vectors. The measurement should be 0 when the vectors orientations are uniformly distributed across the half unit-sphere, and say, 1 when all the vectors are parallel (identical).
I'm not sure exactly what statistic this is — at first I naively thought it would be the the directional variance of the set, however this clearly isn't right. The case where the vectors are split evenly into 3 orthogonal orientations should have much greater anisotropy than the uniform distribution, and even greater anisotropy if the groups were more closely oriented (not orthogonal).
I'm considering iterating through the vectors and either placing each one in a bin with other vectors of sufficiently close orientation, or creating a new bin if no previously considered vector has sufficiently similar orientation. However, this involves choosing a threshold, which I would like to avoid, and I feel like there must be a more appropriate metric.
Could anyone point me in the right direction (no pun intended)?