Upper panel: evolution of the Rayleigh-Watson (blue dashed line, and right-hand scale; the correction to – Eq. (4) – has been made for a confidence level α of 1%) and Bingham ℬ (red dot-dashed line, and right-hand scale) statistics as a function of the number of systems. The horizontal blue dashed curve corresponds to a first-kind risk of 1% of rejecting the null hypothesis of isotropy while it is true, based on the statistics. The horizontal red dashed line is the 10% first-kind risk for the ℬ statistics. The number of systems as a function of distance is given by the black solid line and left-hand scale. Isotropy corresponds to ℬ → 0 and . Lower panel: three eigenvalues of the orientation matrix (Eq. (6)). Isotropy corresponds to λ_1,2,3 = 1 / 3, as seen at large distances. The critical values for λ_1,3 corresponding to α = 1% and 10% as given by Eqs. (11) and (12) are displayed by the dash-dotted lines.