Some Relationships Between Cronbach's Alpha and the Spearman-Brown Formula

Cronbach's alpha is an estimate of the reliability of a test score if the items are essentially tau-equivalent. Several authors have derived results that provide alternative interpretations of alpha. These interpretations are also valid if essential tau-equivalency does not hold. For example, alpha is the mean of all split-half reliabilities if the test is split into two halves that are equal in size. This note presents several connections between Cronbach's alpha and the Spearman-Brown formula. The results provide new interpretations of Cronbach's alpha, the stepped down alpha, and standardized alpha, that are also valid in the case that essential tau-equivalency or parallel equivalency do not hold. The main result is that the stepped down alpha is a weighted average of the alphas of all subtests of a specific size, where the weights are the denominators of the subtest alphas. Thus, the stepped down alpha can be interpreted as an average subtest alpha. Furthermore, we may calculate the stepped down alpha without using the Spearman-Brown formula.