Cronbach’s alpha (scale reliability)
Measure the internal consistency of questionnaire items with Cronbach’s alpha, alpha if an item is deleted, item-total correlations and a Feldt interval.
A clear answer. Here’s how.
For k items answered by n respondents, alpha = k ÷ (k − 1) × (1 − Σ sᵢ² ÷ s²ₜ), where sᵢ² are the item variances and s²ₜ the variance of the total score, all with n − 1 in the denominator. Standardised alpha, k r̄ ÷ (1 + (k − 1) r̄), uses the mean of the k(k − 1)/2 inter-item correlations and is what alpha would be if every item had the same variance. For each item the table gives its mean and SD, the corrected item–total correlation (with the sum of the other items, so the item is not correlated with itself) and alpha recomputed without it. The interval is Feldt’s (1965): (1 − α) ÷ (1 − α̂) follows an F distribution with n − 1 and (n − 1)(k − 1) degrees of freedom. Reverse-scored items are recoded as lowest + highest − answer before anything else. Respondents with any missing answer are left out (listwise). Alpha can be negative when items are scored in opposite directions. The common thresholds (0.7 acceptable, 0.8 good, 0.9 excellent) are rules of thumb; alpha also rises with the number of items, assumes the items measure one construct with equal loadings (tau-equivalence) and is not evidence that the scale is unidimensional or valid. It does not compute omega, split-half or test–retest reliability. It is a calculator, not a substitute for a statistician when decisions depend on the answer.
How this is checked
5 reference checks passed These 5 cases were run when this page was built, on 20 October 2026, and all reproduce within the stated tolerance. The same cases are part of the automated tests that run on every change to Toolmoz. Engine version 1.
See the checks and where each expected answer comes from
Case 1
- Inputs
- The tool’s default inputs
- Expected, and what Toolmoz gave
- Cronbach’s alpha expected 0.8830313014827018, Toolmoz 0.8830313014827018Standardised alpha expected 0.8676576696316292, Toolmoz 0.8676576696316292Mean inter-item correlation expected 0.5673303743242849, Toolmoz 0.567330374324285Alpha if deleted: q5 expected 0.9604234331769518, Toolmoz 0.9604234331769517Alpha if deleted: q3 expected 0.8105657401931796, Toolmoz 0.810565740193179795% CI lower limit (Feldt) expected 0.730793738231218, Toolmoz 0.730793738231217495% CI upper limit (Feldt) expected 0.9615957535233975, Toolmoz 0.9615957535233977
- Where the expected answer comes from
- The example after recoding q3 as 6 − answer: alpha 536/607, and 7984/8313 without the weak item q5, in exact fractions; Feldt limits 1 − (1 − α) × F with F(11, 44) quantiles 2.30152 and 0.32833 found by bisection on a numerically integrated F density, in an independent Python script
- Allowed difference
- ±0.000001 relative
Case 2
- Inputs
- Reverse-scored items
- Expected, and what Toolmoz gave
- Cronbach’s alpha expected -0.136986301369863, Toolmoz -0.13698630136986328Corrected item–total correlation: q3 expected -0.8986590901054062, Toolmoz -0.8986590901054062Alpha if deleted: q1 expected -2.117353308364544, Toolmoz -2.1173533083645433
- Where the expected answer comes from
- Without reverse scoring, q3 runs against the other items: alpha falls to −10/73 and q3 correlates −0.899 with the rest (exact fractions in an independent Python script)
- Allowed difference
- ±1e-12
Case 3
- Inputs
- Answers: one respondent per line, one item per column 3,4,3,4 4,5,4,4 2,3,3,2 5,5,4,5 3,3,2,3 4,4,5,4Reverse-scored items
- Expected, and what Toolmoz gave
- Cronbach’s alpha expected 0.9183673469387755, Toolmoz 0.9183673469387754Standardised alpha expected 0.9212951397484735, Toolmoz 0.9212951397484735Alpha if deleted: Item 3 expected 0.9551020408163265, Toolmoz 0.955102040816326795% CI lower limit (Feldt) expected 0.7080477265894554, Toolmoz 0.70804772658946495% CI upper limit (Feldt) expected 0.9872999213813479, Toolmoz 0.987299921381348
- Where the expected answer comes from
- Six respondents and four items by hand: item variances 1.1, 0.8, 1.1 and 16/15, total variance 196/15, so alpha = 4/3 × (1 − 4.0667/13.0667) = 45/49; Feldt limits with F(5, 15) quantiles by independent numerical integration in Python
- Allowed difference
- ±0.000001 relative
Case 4
- Inputs
- Answers: one respondent per line, one item per column 1,1,1 2,2,2 3,3,3 5,5,5Reverse-scored items
- Expected, and what Toolmoz gave
- Cronbach’s alpha expected 1, Toolmoz 1Standardised alpha expected 1, Toolmoz 1
- Where the expected answer comes from
- Identical items: the total variance is k² times each item variance, so alpha = 3/2 × (1 − 3/9) = 1
- Allowed difference
- ±1e-12
Case 5
- Inputs
- Answers: one respondent per line, one item per column 1,1,1 1,2,2 2,1,2 2,2,1Reverse-scored items
- Expected, and what Toolmoz gave
- Cronbach’s alpha expected 0, Toolmoz 0Standardised alpha expected 0, Toolmoz 0Mean inter-item correlation expected 0, Toolmoz 0
- Where the expected answer comes from
- Three items with zero covariance between every pair: the total variance equals the sum of the item variances, so alpha = 3/2 × (1 − 1) = 0
- Allowed difference
- ±1e-12
What this shows: the engine reproduces independently worked values for these cases. It does not prove every possible input is right. To test an answer you were given, see how to check a number; to see what is and is not checked across Toolmoz, see the accuracy ledger.
See a worked example
Worked example
A fixed demonstration, separate from the values you enter above.
- Answers: one respondent per line, one item per column
q1,q2,q3,q4,q5 4,5,2,4,3 3,4,3,3,4 5,5,1,4,4 2,3,4,2,3 4,4,2,5,2 3,3,3,3,4 5,4,1,5,3 2,2,5,1,3 4,4,2,4,5 3,4,2,3,3 1,2,4,2,2 4,5,1,4,2
- Column separator
- Detect automatically
- Items to include
- Reverse-scored items
- q3
- Lowest possible answer
- 1
- Highest possible answer
- 5
- Confidence level
- 95%
Example result
| Item | Mean | SD | Corrected item–total r | Alpha if deleted |
|---|---|---|---|---|
| q1 | 3.3333 | 1.2309 | 0.9382 | 0.8011 |
| q2 | 3.75 | 1.0553 | 0.8461 | 0.8313 |
| q3 (reversed) | 3.5 | 1.3143 | 0.8987 | 0.8106 |
| q4 | 3.3333 | 1.2309 | 0.8338 | 0.829 |
| q5 | 3.1667 | 0.9374 | 0.1305 | 0.9604 |
Alpha = k ÷ (k − 1) × (1 − Σ item variances ÷ variance of the total), with sample (n − 1) variances. Standardised alpha uses the mean inter-item correlation r̄: k r̄ ÷ (1 + (k − 1) r̄). The interval is Feldt’s, from the F distribution with 11 and 44 df. Dropping q5 would raise alpha; check the wording and scoring of that item before removing anything. Reverse-scored as 1 + 5 − answer: q3. Thresholds such as 0.7 (“acceptable”) and 0.8 (“good”) are rules of thumb, not standards. Alpha rises with the number of items, assumes the items measure one thing with equal loadings, and does not show unidimensionality.