An objective general index for multivariate ordered data

Multivariate quantitative data are often summarized into a general index as a weighted sum when the variates have a prescribed order. Although the sum of standardized scores is a sensible choice of index, it may have negative correlation with some of the variates. In this paper, a general index that has positive correlation with all the variates is constructed. The index is applied to study the fairness of decathlon scoring. Quantification of ordered categorical data is also discussed. The limit of quantification characterizes the Gaussian distribution.