Any topological framework requires the development of a theory of errors of characteristic and appropriate mathematical form. The paper develops a form of theory which appears to be appropriate to measurements of position on a sphere. The primary problems of estimation as applied to the true direction, and the precision of observations, are discussed in the subcases which arise. The simultaneous distribution of the amplitude and direction of the vector sum of a number of random unit vectors of given precision, is demonstrated. From this is derived the test of significance appropriate to a worker whose knowledge of precision lies entirely in the internal evidence of the sample. This is the analogue of 'Student's' test in the Gaussian theory of errors. The general formulae obtained are illustrated using measurements of the direction of remanent magnetization in the directly and inversely magnetized lava flows obtained in Iceland by Mr J. Hospers.