Cronbach's alpha
Also known as: Coefficient alpha
A number summarizing how consistently a scale's items hang together.
What it means
Cronbach's alpha is a coefficient of internal-consistency reliability that estimates how closely related a set of items are as a group, based on their average inter-item correlation and the number of items. Ranging up to 1, it is widely reported as evidence that a multi-item scale measures a single coherent attribute, with values around 0.7 to 0.9 often deemed acceptable. Its conveniences hide important caveats: alpha rises mechanically with more items, assumes the items are essentially equivalent (tau-equivalence), and a high value does not establish unidimensionality. These limitations have prompted recommendations to report alternatives such as McDonald's omega alongside or instead of it.
How the number is built
Alpha compares the variance shared across a scale's items to the total variance of the summed score. In its common form it takes the number of items, k, times one minus the ratio of summed item variances to the total-score variance, all scaled by k over k-minus-one. An equivalent and more intuitive reading is that alpha equals the average of every possible split-half reliability the scale could produce (via the Rulon/Flanagan formula). This is why splitting a questionnaire in two and correlating the halves lands in alpha's neighbourhood. Reading it as an averaged split-half coefficient also explains its quirks: it rewards items that covary and is indifferent to what any single item actually measures, so long as the items move together.
What a high alpha does not tell you
A high alpha is routinely read as proof that a scale measures one thing. It does not. Cortina showed that a battery built from two entirely unrelated six-item subtests can still post an alpha near 0.85, because alpha responds to the sheer number of moderately correlated items, not to whether they share a single underlying dimension. Under classical assumptions alpha is a lower bound on reliability, so it usually understates rather than overstates how reliable a score is; a modest alpha is not automatically a broken scale. At the other extreme, a value above roughly 0.95 is a warning rather than a triumph. It typically signals redundant, near-duplicate items that inflate the coefficient while adding no real information.
Assumptions, and where they fail
Alpha equals reliability only if the items are tau-equivalent, each contributing the same amount of true-score variance. Real items rarely are; when their loadings differ, alpha understates reliability, sometimes badly. It also assumes uncorrelated errors, and any shared method or wording that correlates them inflates the coefficient. The classical formula presumes continuous, roughly normal items, so for binary or ordinal responses the ordinary version (equivalent to Kuder-Richardson 20 for right-or-wrong items) tends to be biased downward, and polychoric-based or categorical estimates are preferred. A negative alpha is not a paradox but a diagnostic: it almost always means reverse-scored items were left un-recoded, pushing inter-item correlations below zero. Each of these failures is common enough that a reported alpha should never be read without knowing the items behind it.
From alpha to omega
The reform argument, made forcefully by Sijtsma and sharpened by McNeish, is that alpha's assumptions are so rarely met that it should give way to model-based coefficients. McDonald's omega is estimated from a fitted factor model and allows items to load unequally, so it tracks reliability more faithfully when tau-equivalence fails; Dunn and colleagues and Flora provide practical recipes for computing it. Omega is not a free lunch, though. It inherits the assumptions of whatever factor model produced it, and critics note that a mis-specified model leaves omega no safer than the alpha it replaced. The defensible practice is to state which coefficient was used, the measurement model behind it, and a confidence interval, rather than reporting a single naked number.
Examples
A ten-item anxiety questionnaire reports an alpha of 0.88, indicating its items respond in a mutually consistent way.
A team adds fifteen near-identical questions to a customer-satisfaction survey and alpha climbs to 0.95 — not because the scale improved, but because alpha rises mechanically as items pile up.
A job-satisfaction scale mixes questions about pay with questions about colleagues and still reports an alpha of 0.8; the tidy number hides that it is measuring two things, not one.
A depression-screening tool returns an alpha near zero in a new sample; two reverse-worded items were never recoded, so their negative correlations with the rest dragged the coefficient down.
A physics instructor's thirty-item final exam yields a KR-20 of 0.55, the dichotomous-item version of alpha; the modest value reflects questions sampling several loosely related topics rather than one ability.
First described in Lee Cronbach (1951); building on Kuder & Richardson (1937).
Key references
- Flora, D. B. (2020). Your coefficient alpha is probably wrong, but which coefficient omega is right? A tutorial on using R to obtain better reliability estimates. Advances in Methods and Practices in Psychological Science, 3(4), 484-501. doi.org/10.1177/2515245920951747
- McNeish, D. (2018). Thanks coefficient alpha, we'll take it from here. Psychological Methods, 23(3), 412-433. doi.org/10.1037/met0000144
- Dunn, T. J., Baguley, T., & Brunsden, V. (2014). From alpha to omega: A practical solution to the pervasive problem of internal consistency estimation. British Journal of Psychology, 105(3), 399-412. doi.org/10.1111/bjop.12046
- Sijtsma, K. (2009). On the use, the misuse, and the very limited usefulness of Cronbach's alpha. Psychometrika, 74(1), 107-120. doi.org/10.1007/s11336-008-9101-0
- Cortina, J. M. (1993). What is coefficient alpha? An examination of theory and applications. Journal of Applied Psychology, 78(1), 98-104. doi.org/10.1037/0021-9010.78.1.98
- Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297-334. doi.org/10.1007/BF02310555