Summary: This article presents relations between the degree of symmetry of the gamma density and some quotients of expectations of some functions of order statistics. The main approach depends on two steps. First, the theoretical derivation of some inequalities as finite series in the gamma function, is an intermediate step. Second, using the lower and upper boundaries of these inequalities a set of quotients to be computationally tested in behavior is formulated. This leads to discover the relations that represent the main purpose of the article. The results are presented for some selected values of the shape parameter of the gamma distribution which is assumed to be a positive integer. An application is given to illustrate the possibility of applying the suggested approach in measuring the inequality degree of income distributions.