A method to distinguish sequences with same mean and same variance

Suppose that in four tests Alice´s scores are 85, 90, 90, 95, and Bob´s scores are 85, 95, 90, 90. How to evaluate their scores? The common method by computing their mean and variance is not appropriate for this case because they do not depend on the order relationship of elements, although the relationship is a key factor in economic and social sciences. In this paper, we introduce the concept of ordered probability mass function which can be used to distinguish sequences of positive numbers with the same mean and same variance. To the best of our knowledge, it is the first time to introduce such a method to cope with the statistical problem.