I am trying to characterize the angular deviation of a set of 3D Cartesian vectors V = {v_i} from a fixed z-axis. V is constructed by discretely sampling a complex physical system so it suffers from noise, sparse sampling, etc. If we work in spherical coordinates, I define the azimuthal angle as "phi" and the altitude or polar angle from the z-axis as "theta" (the "physics" convention described here).
I am most interested the angle theta between the elements of V and the z-axis, so I have constructed an area-normalized histogram P_approx(theta) with a 1 degree bin-width across a theta range from 0 to 180 degrees which serves as an approximation of the true probability distribution P(theta). P_approx(theta) is peaked between 0 and 180 and falls toward zero at theta = 0 and theta = 180. A theta-only histogram is desirable since the system should show azimuthal symmetry and summing over all values of phi improves the statistics of the resulting histogram.
I am reluctant to use P_approx(theta) to characterize the angular behavior in my system since orientations near theta = 90 are favored relative to orientations near theta = 0 and theta = 180 (more surface area of the unit sphere when integrating along phi). For example, if the vector evenly samples the upper hemisphere of a unit sphere (0 < theta < 90, 0 < phi < 360), the P(theta) will still be peaked. This is misleading.
Does anyone know of a more physically-insightful method to characterize the angular preferences of the dataset V?
