Appraising the central tendency of distributions of a cardinal and an ordinal variable

This paper provides a simple uniÖed axiomatic framework for appraising the central tendency of distributions of a single attribute (pie) among a collection of individuals depending upon the available measurement of the attribute. Two types of measurement are considered: cardinal and ordinal. For each of them, three properties are posited on an ordering of distributions of numbers among individuals. The two first properties are the anonymity requirement that permutations of the same list of numbers be equivalent and the weak Pareto requirement that a strict increase in the value of the variable for all individuals be favorably appraised. The third property requires that inverting the numerical measurement of the variable leads to an inversion of the ranking of the any two distributions to which the inversion is applied. The mean of a distribution is shown to be the only ordering of distributions consistent with cardinal measurability that satisÖes those three requirements in the cardinal context while the median is the only such ranking consistent with ordinal measurability of the variable that satisÖes those same requirement if the number of individuals is odd. If the number of individuals is even, then those three requirements applied to the ordinal context are shown to be inconsistent.