Cronbach's alpha, published by Lee Cronbach in 1951, provides a lower bound on the reliability of a composite score. It has become the most commonly reported reliability statistic in the social sciences, despite ongoing debates about its interpretation and limitations.
k = number of items
σ²ᵢ = variance of item i
σ²ₓ = variance of total scores
Interpretation
Alpha ranges from 0 to 1, with higher values indicating greater internal consistency. Conventional guidelines suggest α ≥ 0.70 for research use and α ≥ 0.90 for individual diagnostic decisions. However, alpha is a function of both the average inter-item correlation and the number of items: adding more items always increases alpha, even if the new items are of poor quality.
Common Misconceptions
Alpha is often misinterpreted as measuring unidimensionality, but a high alpha does not guarantee that items measure a single construct. A multidimensional test can have a high alpha if the subscales are correlated. For this reason, factor analysis should always accompany reliability assessment. McDonald's omega (ω) is increasingly recommended as a superior alternative that makes fewer assumptions about the test's factor structure.