Definition
Cronbach's is defined as
where is the number of components (K-items or testlets), the variance of the observed total test scores, and the variance of component i for the current sample of persons. See Develles (1991).
Alternatively, the Cronbach's can also be defined as
where is as above, the average variance of each component (item), and the average of all covariances between the components across the current sample of persons (that is, without including the variances of each component).
The standardized Cronbach's alpha can be defined as
where is as above and the mean of the non-redundant correlation coefficients (i.e., the mean of an upper triangular, or lower triangular, correlation matrix).
Cronbach's is related conceptually to the Spearman–Brown prediction formula. Both arise from the basic classical test theory result that the reliability of test scores can be expressed as the ratio of the true-score and total-score (error plus true score) variances:
Theoretically, alpha varies from zero to 1, since it is the ratio of two variances. Empirically, however, alpha can take on any value less than or equal to 1, including negative values, although only positive values make sense. Higher values of alpha are more desirable. Some professionals as a rule of thumb, require a reliability of 0.70 or higher (obtained on a substantial sample) before they will use an instrument. Obviously, this rule should be applied with caution when has been computed from items that systematically violate its assumptions. Furthermore, the appropriate degree of reliability depends upon the use of the instrument. For example, an instrument designed to be used as part of a battery of tests may be intentionally designed to be as short as possible, and therefore somewhat less reliable. Other situations may require extremely precise measures with very high reliabilities. In the extreme case of a two-item test, the Spearman–Brown prediction formula is more appropriate than Cronbach's alpha.
This has resulted in a wide variance of test reliability. In the case of psychometric tests, most fall within the range of 0.75 to 0.83 with at least one claiming a Cronbach's alpha above 0.90 (Nunnally 1978, page 245–246).
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