Cronbach's Alpha
Cronbach's alpha (α) is the most widely reported estimator of internal consistency reliability. Introduced by Lee Cronbach in his 1951 Psychometrika paper as a generalisation of the Kuder–Richardson 20 formula, it extends KR-20 from dichotomous to polytomously scored items and has since become the default reliability index in social-science and language-testing research.
Formula and assumptions
Alpha is computed as
α = (k / (k − 1)) × (1 − Σσᵢ² / σₜ²)
where k is the number of items, σᵢ² is the variance of item i, and σₜ² is the variance of total scores. Values run from 0 to 1, with negative values indicating either coding errors or items that correlate negatively with the rest. Conventional cut-offs — .70 for research instruments, .80 or higher for high-stakes decisions — appear in the Standards and in Bachman's treatment of language-test reliability, though they are guidelines, not absolutes.
The coefficient assumes essential tau-equivalence — that all items measure the same construct on the same scale, differing only by an additive constant. Under that assumption alpha equals true-score reliability; when violated, alpha is a lower bound rather than an exact estimate. It does not require unidimensionality, despite a common misreading: a multidimensional test can return a high alpha simply because its items are numerous and intercorrelated.
Limitations
Sijtsma and others have shown that alpha can underestimate reliability on multidimensional tests and that McDonald's omega is preferable when factor structure is known. Alpha rises mechanically with test length, so high values on long instruments offer weak evidence of homogeneity. Sample heterogeneity also inflates the coefficient: reliability is a property of scores in a population, not of the test.
In language testing, alpha is routinely reported for reading and listening sections and for objectively scored vocabulary and grammar measures, while inter-rater and generalizability analyses handle productive-skill scoring.
References
- Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297–334.
- Bachman, L. F. (1990). Fundamental Considerations in Language Testing. Oxford University Press.
- AERA, APA, & NCME (2014). Standards for Educational and Psychological Testing. American Educational Research Association.