A Note on Calculating Tau and Average Tau and on the Sampling Distribution of Average Tau with a Criterion Ranking

Abstract

Calculational short cuts for Kendall's tau and average tau and for Hays' average tau with a criterion ranking (τ_C ) are shown. The short cuts are based mainly on counting “inversions” in the ordering of n objects by a given judge, and summing over m judges, rather than counting agreements for a given object-pair and summing over object-pairs. The sum of the number of inversions thus counted for τ_C has a limiting normal distribution under the null hypothesis. From this is derived a statistic which approximates the sampling distribution of τ _C . The approximation is shown to be close even for small m and n.