A New Logical Basis for Variance and Covariance

There is a fundamental duality that extends through the exact sciences. This paper briefly develops the relatively new logic of partitions dual to the Boolean logic of subsets. At a more basic level, the duality is between the elements of a subset and the distinctions (pairs of elements in different blocks) of a partition. Probability theory and information theory (based on logical entropy) start as the respective quantitative versions of the two dual logics of subsets and partitions. The subset side of the duality uses a one-sample (or one element) approach; the partition side uses a two-sample (or pair of elements) approach. The paper gives a new derivation of the variance and covariance based on that two-sample approach which positions those concepts on the partition and information-theory side of the duality.