Cronbach’s Alpha
Internal consistency of a scale, from item variances or the average inter-item correlation.
Internal consistency across 10 items. Report as α = .802 in APA style, without the leading zero.
Cronbach’s alpha measures how consistently the items on a scale measure the same thing. With 10 items whose variances sum to 12.5 and a total-score variance of 45, alpha is (10 ÷ 9) × (1 − 12.5 ÷ 45) = .80 — generally considered good.
What alpha actually measures
Alpha asks a narrow question: do the items on this scale move together? If everyone who agrees strongly with item 1 also agrees strongly with items 2 through 10, the items share variance, the total score is stable, and alpha is high. If responses to each item are unrelated, the total is mostly noise and alpha falls towards zero.
The formula compares two quantities. The sum of the individual item variances captures how much each item varies on its own; the variance of the total score captures how much the summed scale varies. When items correlate, the total-score variance is inflated well beyond the sum of the parts, and that gap is what alpha converts into a reliability estimate.
Two ways in
With raw data you have both variances and can compute raw alpha directly. Without it, the standardised form needs only the number of items and their average inter-item correlation, which papers often report. The two agree closely when items have similar variances and diverge when one item varies far more than the rest.
k is the number of items, Σσᵢ² the sum of the item variances, σₜ² the variance of the total score, and r̄ the average correlation between item pairs.
Worked example: 10 items, Σσᵢ² = 12.5, σₜ² = 45
Sum the parts, compare with the whole, then rescale by the item count:
- 1 Count the items. The scale has 10 items, so k = 10.
- 2 Sum the item variances. Adding the ten individual variances gives Σσᵢ² = 12.5.
- 3 Find the total-score variance. The variance of each respondent’s summed score across all ten items is 45.
- 4 Take the ratio. 12.5 ÷ 45 = 0.2778. The smaller this is, the more the items share.
- 5 Apply the formula. (10 ÷ 9) × (1 − 0.2778) = 1.1111 × 0.7222 = .80.
- 6 Report it in APA style. Write α = .80, dropping the leading zero because alpha cannot exceed 1.
Commonly cited interpretation bands
Rules of thumb widely quoted in the methods literature. They are conventions, not thresholds with a statistical basis — a scale is not ‘unreliable’ at .69 and ‘reliable’ at .71.
| Alpha | Common description |
|---|---|
| .90 and above | Excellent — but check for redundant items |
| .80 – .89 | Good |
| .70 – .79 | Acceptable |
| .60 – .69 | Questionable |
| .50 – .59 | Poor |
| Below .50 | Unacceptable |
Where alpha misleads
Alpha rises with the number of items regardless of quality, so a 40-item scale can reach .90 on mediocre items simply by length. That is why a very high alpha is worth inspecting rather than celebrating: it often signals near-duplicate items that add words without adding information.
Alpha also assumes the scale is unidimensional — that all items tap one construct. A questionnaire measuring two distinct things can still return a respectable alpha, which is why factor analysis belongs before the reliability check rather than after it. And alpha assumes items contribute equally; when they do not, McDonald’s omega is the better estimate and is increasingly expected in psychology journals.
A last practical point: reverse-scored items must be recoded before computing alpha. Forgetting to do so usually produces a startlingly low or even negative value, which is a symptom of the coding rather than of the scale.