## Quick Reference

A non-parametric test of the null hypothesis that two samples of cyclic data have been drawn from the same population. Suppose the samples contain *n*_{1} and *n*_{2} observations. These observations are first arranged in order of magnitude and then their actual values are replaced by coded values that (working in degrees) are multiples of 360/(*n*_{1}+*n*_{2}), so that the *k*th largest value becomes 360*k*/(*n*_{1}+*n*_{2}). Let the coded values for the first sample be *θ*_{1}, *θ*_{2},…, *θ** _{n1}*, and calculate

*R*

^{2}given by The value of

*R*

^{2}for the second sample will be identical. The test statistic is

*T*, given by For large

*n*

_{1}and

*n*

_{2},

*T*has an approximate chi-squared distribution with two degrees of freedom. The test was introduced by Stanley Wheeler and Geoffrey Watson in 1964.

**Wheeler–Watson test.** Two sets of birds are released from the same location. The five birds in set A fly off in the directions 15°, 25°, 28°, 31°, and 45°. The four birds in set B fly off in the directions 40°, 50°, 52°, and 60°. Arranging these data in order, we have AAAABABBB. Since 360/9=40, the coded values for sample B are (in degrees) 200, 280, 320, and 360. The resulting value for *R*^{2} is 4.88 and *T*=3.90. We conclude that there is no significant evidence that the samples have been drawn from different populations.

*Subjects:*
Probability and Statistics.