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Following an approach due to Guttman the axioms of the classical test theory model are shown to be derivable as constructions from a specified sampling rule and from the assumption that the observed score of an arbitrarily specified or randomly selected person may be considered as an observation of a random variable having finite and positive variance. Without further assumption the reliability of a test is defined. Parallel measurements are then independently defined, and the concept of replication is explicated. The derived axioms of the classical test theory model are then stated in a refined form of Woodbury's stochastic process notation, and the basic results of this model are derived. The assumptions' of experimental independence, homogeneity of error distribution, and conditional independence are related to the classical model and to each other. Finally, a brief sketch of some stronger models assuming the independence of error and true scores or the existence of higher-order moments of error distributions or those making specific distributional assumptions is given.

Following a general approach due to Guttman, coef~icient ex is derived as a lower bound on the reliability of a test. The necessary and sufficient condition under which equality is attained in this inequality and hence that ex is equal to the reliability of the test is derived and shown to be closely related to the recent redefinition of the concept op arallel measurements due to Novick. This condition is then also shown to be closely related to the unit rank assumption originally adopted by Kuder and Richardson in the derivation of their formula 20. The assumption later adopted by Jackson and Fergusqn and·the one adopted by Guliiksen are, shown to be related to the neceasarv and sufficient condition derived here. It is then pointed out that the statement that "coefficient ex is equal to the mean of the split-half reliabilities"·is tru~only 'tinder the condition assumed by Cronbacb, in the body of his derivation of this result.Finally some limitations on the uses of any function of ex as a measure of internal consistency are noted.