Modifications of the Rayleigh Test for Uniformity in Analysis of Two-Dimensional Orientation Data

We review critically the Rayleigh test for uniformity as used by Curray. After discussing the test in Curray's form, we propose modifications: when the preferred orientation is known a priori, when the asymptotic formula must be corrected for small sample size, and when the natural range of angles is not 360° but 180°. When the preferred orientation is known, we use the statistic V', which is the component in the preferred direction of the vector sum used by Curray, and we discuss the circumstances under which a test based on V' may be regarded as unequivocally a best test for uniformity. We give formulas, a table, and charts for applying the Rayleigh test and the V-test to samples of all sizes that occur in practice. In an appendix we expose the relations between Rayleigh's test and a test proposed by Tukey.