The Randomization Bases of the Problem of the Amalgamation of Weighted Means

SUMMARY A generalization is given of a theorem of Miles on the probability distribution of the number of resultant means arising from a process of amalgamation of a set of weighted values, under an hypothesis of randomization. Two proofs are given, one inductive and one combinatorial. These are based on a geometrical representation. The theorem is used to provide a non-parametric test for trend. The connection of this probability distribution with that of the number of records (in the sense of Foster and Stuart) is explored and an explicit formula for the variance of their round trip test is obtained. Two tests, based on the number of records and on the number of resultant means, are applied to two sets of data. Three standard tests are also applied for comparison.