0.777778
0.3
0.7
7.071068
3.333333
77.78
%
0.777778
0.3
0.7
7.071068
3.333333
77.78
%
The Cronbach's Alpha Calculator computes the most widely used measure of internal consistency reliability for multi-item scales and questionnaires. Cronbach's alpha (α) estimates how closely related a set of items are as a group, indicating whether the items in a test or survey consistently measure the same underlying construct. Values range from 0 to 1, with higher values indicating greater internal consistency.
Whether you are developing a psychological test, validating a survey instrument, or evaluating the reliability of a composite score, Cronbach's alpha is the standard metric reported in research publications. Enter the number of items, the sum of individual item variances, and the total scale variance to compute alpha along with an interpretation and the standard error of measurement.
Cronbach's alpha is calculated using the following formula:
$$\alpha = \frac{k}{k-1}\left(1 - \frac{\sum_{i=1}^{k} s_i^2}{s_t^2}\right)$$
Where:
The formula has an elegant interpretation. The ratio Σsi²/st² represents the proportion of total variance attributable to individual item variability (error). Subtracting this from 1 gives the proportion of variance due to the common construct (true score variance). The factor k/(k-1) is a correction that adjusts for the number of items.
The Standard Error of Measurement (SEM) estimates the precision of individual scores:
$$SEM = s_t \cdot \sqrt{1 - \alpha}$$
A smaller SEM means more precise individual measurement. The SEM can be used to construct confidence intervals around individual scores: Score ± 1.96 × SEM for a 95% confidence interval.
Cronbach's alpha can be understood as the expected correlation between two random halves of the test (averaged over all possible split-half partitions). It is equivalent to the mean of all possible split-half reliabilities computed from the same set of items, adjusted by the Spearman-Brown prophecy formula.
The value of Cronbach's alpha is interpreted using widely accepted thresholds from George and Mallery (2003):
| Alpha Range | Interpretation |
|---|---|
| α ≥ 0.90 | Excellent |
| 0.80 ≤ α < 0.90 | Good |
| 0.70 ≤ α < 0.80 | Acceptable |
| 0.60 ≤ α < 0.70 | Questionable |
| 0.50 ≤ α < 0.60 | Poor |
| α < 0.50 | Unacceptable |
The SEM provides a complementary perspective on measurement precision at the individual level, which alpha alone does not convey.
Inputs
Results
A 10-item questionnaire has Σs²ᵢ = 15 and s²ₜ = 50. α = (10/9)(1 - 15/50) = 1.111 × 0.70 = 0.778. This is acceptable for research purposes. SEM = √50 × √(1-0.778) = 7.07 × 0.471 = 3.33 points.
Inputs
Results
A 20-item test with Σs²ᵢ = 8, s²ₜ = 64. α = (20/19)(1 - 8/64) = 1.053 × 0.875 = 0.921. Excellent reliability. SEM = 8 × √(1-0.921) = 8 × 0.281 = 2.25. A student scoring 75 has a 95% CI of approximately 75 ± 4.4.
Cronbach's alpha is a coefficient of internal consistency reliability that estimates how well a set of items measures a single underlying construct. It ranges from 0 to 1 (and can be negative if items are negatively correlated). Higher values indicate that the items are more consistent with each other. It is the most commonly reported reliability statistic in psychology, education, healthcare, and social science research. Published by Lee Cronbach in 1951, it has been cited over 50,000 times.
You need three values: (1) k -- the number of items in your scale; (2) Sum of item variances (Σs²ᵢ) -- compute the variance of each item across all respondents, then sum them; (3) Total scale variance (s²ₜ) -- compute the total score for each respondent (sum of all items), then calculate the variance of these total scores. Most statistical software (SPSS, R, SAS, Stata) calculates these automatically when you request reliability analysis.
Yes, alpha can be negative when items are negatively correlated on average. This typically happens when some items need to be reverse-scored but were not. For example, if a satisfaction survey includes both positively worded items ("I am satisfied") and negatively worded items ("I am dissatisfied"), the negative items must be reverse-coded before computing alpha. A negative alpha always indicates a problem with the scoring or item selection, not a legitimate measurement result.
Adding more items generally increases alpha, even if the new items have the same average inter-item correlation. This is because the k/(k-1) factor in the formula grows with k. The Spearman-Brown prophecy formula predicts: doubling the test length increases alpha from a level determined by the average inter-item correlation. This means a long test can have high alpha even with weak inter-item correlations, which is why alpha should always be interpreted alongside the number of items and the average inter-item correlation (ideally 0.15-0.50).
The SEM estimates the precision of an individual's observed score. It represents the standard deviation of measurement errors -- if a person took the test many times, their scores would fluctuate around their true score with a standard deviation equal to the SEM. A 95% confidence interval for a person's true score is: Observed Score ± 1.96 × SEM. Lower SEM means more precise measurement. SEM = SD × √(1 - α), so it decreases as reliability increases.
Key alternatives include: (1) McDonald's omega (ω) -- does not assume tau-equivalence (equal factor loadings); generally preferred by psychometricians; (2) Split-half reliability -- correlates two halves of the test; (3) Test-retest reliability -- correlates scores from two administrations; (4) Composite reliability (CR) -- based on factor loadings from CFA; (5) Kuder-Richardson 20 (KR-20) -- equivalent to alpha for dichotomous (yes/no) items. Alpha remains dominant due to its simplicity, but omega is gaining adoption as the more theoretically sound measure.
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