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Arrow's impossibility theorem can be viewed as requiring that each subset of two social alternatives be a potential feasible subset or environment, with transitive and complete social choices over these subsets for each profile of individual preference orders. The feasibility assumption for every two-alternative subset is relaxed with consequent changes in the social ordering condition. An Arrow-type impossibility result still obtains when the set of social alternatives is the union of two disjoint sets, each of which has two or more elements, and when {x, y} is feasible whenever x is from one set and y is from the other. Variants of the basic theorem are included, one of which requires that strict binary social choices be acyclic. [ABSTRACT FROM AUTHOR]

This article focuses on the range of order statistics. It presents several mathematical formulas regarding the statistics. It further says that P.A. Samuelson has shown that one of the given theorems yields an inequality sharper than that of Chebyshev. It is of interest to note that there is a discrepancy between Samuelson's second theorem and the corresponding theorem of this article.