Statistics on a Sphere

Summary Expectations deduced from the probability density functions of Fisher are used to develop further the statistics of points on a sphere. The paper presents unbiased estimators of the precision parameter k in terms of vector deviations both for cases when the true direction μ is known and unknown. On the basis of a one way random effect vector model, the scatter of various sampling distributions of means are derived as functions of the within ( k w ) and between ( k b ) sites scatter. The relations take the curvature of the sphere into account and extend the analysis of dispersion on a sphere to include highly‐scattered distributions of palaeomagnetic data. In addition to the mean square method, which is modified by a new expression for the expectation of the mean square between sites, two alternative ways of estimating k w and k b are described. The latter statistics contribute to determine the confidence circle of the overall mean direction with unit weight to samples and sites respectively. Finally, the theory is applied to palaeomagnetic results from the Kaoko lavas of South‐West Africa.