What Cronbach’s alpha measures
Cronbach’s alpha estimates the internal consistency of a questionnaire or test: how strongly its items measure the same underlying construct, such as job satisfaction, anxiety or reading ability. It is the most widely reported reliability coefficient in psychology, education, marketing and health research.
The formulas
α = k ÷ (k − 1) × (1 − Σsᵢ² ÷ s²ₜ)
- k: number of items;
- Σsᵢ²: sum of the variances of the individual items;
- s²ₜ: variance of the total score (the sum of the items).
If the items are highly correlated, the total score varies much more than the items separately, and alpha approaches 1.
Standardized alpha, from the mean inter-item correlation r̄: α = k r̄ ÷ (1 + (k − 1) r̄).
Example
A 10-item scale has item variances summing to 12 and a total-score variance of 40:
- α = 10 ÷ 9 × (1 − 12 ÷ 40) = 1.111 × 0.70 ≈ 0.78: acceptable;
- this corresponds to a mean inter-item correlation of about 0.26;
- to reach α = 0.80, the Spearman–Brown formula gives a length factor of 1.14: about 12 items of the same quality.
Interpreting alpha
| α | Usual reading |
|---|---|
| 0.9 and above | excellent (above 0.95, items may be redundant) |
| 0.8 to 0.9 | good |
| 0.7 to 0.8 | acceptable |
| 0.6 to 0.7 | questionable |
| 0.5 to 0.6 | poor |
| below 0.5 | unacceptable |
A threshold of 0.70 is common for research instruments; decisions about individuals, such as clinical diagnosis, usually call for 0.90 or more.
Pitfalls
- Alpha rises with the number of items, even if they are only weakly related. A long scale can show a high alpha without measuring a single construct.
- A high alpha does not prove unidimensionality; use factor analysis for that.
- Reverse-scored items must be recoded before computing alpha, or they will drag it down, sometimes below zero.
- Alpha assumes that all items contribute equally. When they do not, McDonald’s omega is a better estimate of reliability.
The factor √(1 − α), multiplied by the SD of total scores, gives the standard error of measurement of an individual score.