What Are the Mathematical Bounds for Coefficient α?

Abstract

Coefficient α, although ubiquitous in the research literature, is frequently criticized for being a poor estimate of test reliability. In this note, we consider the range of α and prove that it has no lower bound (i.e., α ∈ ( −∞, 1]). While outlining our proofs, we present algorithms for generating data sets that will yield any fixed value of α in its range. We also prove that for some data sets—even those with appreciable item correlations—α is undefined. Although α is a putative estimate of the correlation between parallel forms, it is not a correlation as α can assume any value below −1 (and α values below 0 are nonsensical reliability estimates). In the online supplemental materials, we provide R code for replicating our empirical findings and for generating data sets with user-defined α values. We hope that researchers will use this code to better understand the limitations of α as an index of scale reliability.