Internal consistency, with the part that is only length separated out.
Alpha, and what each item does to it
Eight respondents, five items. Alpha 0.9644 with a mean inter-item correlation of 0.8611 — high because the items agree, not because there are many of them.
5 items, 8 respondents
α = 0.964395
Mean inter-item correlation 0.8611. No single item is holding alpha down; every deletion lowers it.
Cronbach's α
0.964395
5 items
Mean inter-item r
0.86110
does not grow with length
Total-score variance
34.8571
sum of items, per respondent
Weakest item
Item 2
r with total 0.8473
Per-item variance, item-total correlation and alpha with that item removed
Item
Variance
r with total
α if dropped
Verdict
Item 1
1.9821
0.9798
0.9425
earns its place
Item 2
0.8571
0.8473
0.9689
earns its place
Item 3
2.2857
0.9577
0.9493
earns its place
Item 4
1.4107
0.8883
0.9578
earns its place
Item 5
1.4286
0.8944
0.9568
earns its place
What length alone would have bought you
Alpha predicted by the Spearman–Brown relation at a fixed mean inter-item correlation, holding item quality constant and changing only how many there are.
Alpha at 3, 5, 10, 20, 40 and 100 items for four levels of mean inter-item correlation
Mean inter-item r
3 items
5 items
10 items
20 items
40 items
100 items
0.05
0.1364
0.2083
0.3448
0.5128
0.6780
0.8403
0.10
0.2500
0.3571
0.5263
0.6897
0.8163
0.9174
0.20
0.4286
0.5556
0.7143
0.8333
0.9091
0.9615
0.30
0.5625
0.6818
0.8108
0.8955
0.9449
0.9772
Yours (0.861)
0.9490
0.9687
0.9841
0.9920
0.9960
0.9984
At r = 0.05 the items are very nearly unrelated, and 100 of them still report α = 0.8403. The conventional 0.7 threshold is cleared by length alone at r = 0.05 from 40 items, and at r = 0.2 from 10.
Alpha is a lower bound on reliability, not an estimate of it It does not test whether the scale is one-dimensional Reverse-score before you compute it
What this tool shows
At a mean inter-item correlation of 0.05 — items that are very nearly unrelated to one another — a 100-item scale still reports α = 0.8403. Forty items at 0.1 report 0.8163. Alpha rises with the number of items whether or not they measure the same thing, so the tool prints your scale’s mean inter-item correlation and what alpha would be at other lengths beside the headline number.
Cronbach’s alpha from a respondent-by-item matrix
Alpha with each item dropped in turn, and the item that most improves the scale
Item-total correlations, with items pointing the wrong way flagged
The mean inter-item correlation — the part of alpha that is not length
A length table: alpha at 3, 5, 10, 20, 40 and 100 items at fixed item quality
Negative alpha reported as it is, rather than floored at zero
Item analysis α if dropped Mean inter-item r Length table
Alpha is a lower bound on reliability, not a measure of dimensionality.
Updated 12 September 2026 · Works in any browser, no installation
Cronbach’s alpha is the proportion of total-score variance that the items share, rescaled by how many items there are. It is the mean of every possible split-half reliability of the scale, which is why it is reported as internal consistency. It is a lower bound on reliability under its assumptions, and an overestimate when they fail — both facts matter more than the conventional 0.7 cutoff.
At a glance
Formula shown
α = [k/(k−1)]·(1 − Σσ²ᵢ/σ²T), where k is the number of items, σ²ᵢ is item i’s variance and σ²T the variance of the total score. Written with the mean inter-item correlation r̄ it becomes α = kr̄/(1 + (k−1)r̄), the Spearman–Brown form — which shows directly that alpha is determined by item quality and item count together, and that either one alone can carry it.
Scenario support
Psychometric scales and questionnaires, survey batteries meant to measure one construct, exam item analysis, patient-reported outcome measures, employee engagement instruments, and any composite score built by summing items that are supposed to belong together.
Educational estimate
Planning support from the values you enter — not professional advice.
Most of a high alpha can be item count
Alpha is usually read as a statement about the items. Half of it is a statement about how many there are.
Hold item quality fixed at a mean inter-item correlation of 0.05 and lengthen the scale: alpha runs 0.1364 at 3 items, 0.3448 at 10, 0.6780 at 40 and 0.8403 at 100. Nothing about the items changed. Only the count did.
At r̄ = 0.1 the conventional 0.7 threshold is cleared at 20 items (0.6897 at 20, 0.8163 at 40). At r̄ = 0.2 it is cleared at 10 (0.7143). A long scale of mediocre items outscores a short scale of good ones on this metric, every time.
So “α = 0.92” on a 60-item instrument says almost nothing. The number that carries information is the mean inter-item correlation, which the tool prints beside it — and which most papers do not report.
The healthy range for r̄ is roughly 0.15 to 0.50. Below that the items are not measuring a common thing; above about 0.7 they are close to redundant restatements of each other, and the scale would lose nothing by being shorter.
Alpha-if-dropped is the diagnostic, not alpha
A single number for a whole scale hides which item is the problem. Removing each item in turn does not.
The tool’s second example reports α = 0.7057 — respectable by convention, and nothing in that number suggests a defect. Dropping item 4 raises it to 0.9877.
That item’s correlation with the total is −0.5260. It moves in the opposite direction to the rest of the scale, which is what a reverse-worded item looks like when nobody reverse-scored it.
An item-total correlation below about 0.3 is the usual flag, and a negative one is not a borderline case. The tool marks both.
Dropping items to chase alpha is its own trap, though. Each deletion narrows what the scale measures, and deleting until alpha is high produces a short, highly redundant instrument that covers less of the construct than the one you started with.
A negative alpha is information, not an error
Alpha has no lower bound of zero. It can and does come out negative, and a tool that clamps it is hiding the most useful result it will ever produce.
Negative alpha means the average covariance between items is below zero. The items are, on balance, disagreeing with each other — and the formula reports that faithfully.
The overwhelmingly common cause is unscored reverse items. “I enjoy my work” and “I dread Mondays” measure the same construct with opposite signs; summed raw, they cancel.
The second cause is that there is no common construct. Items assembled because they seemed related, rather than because they load on one factor, produce exactly this.
Either way the fix is upstream of the statistic. Reverse-score, or accept that the battery is not a scale and should not be summed into one score.
Alpha does not test whether the scale measures one thing
This is the most-repeated misreading of the statistic, and Cronbach himself spent later papers correcting it.
A scale with two distinct, internally coherent subscales can report a high alpha, because alpha responds to average covariance and a two-factor structure still has plenty of it.
Unidimensionality is a separate question with a separate method. A factor analysis or a correlation matrix between items answers it; alpha does not, at any value.
Alpha also assumes tau-equivalence — that every item relates to the underlying construct with the same weight. When they do not, alpha understates reliability, which is why it is called a lower bound.
And when errors are correlated, alpha OVERSTATES reliability instead. Items placed next to each other, sharing a stem or a response format, pick up shared error that alpha reads as shared construct. Both directions of bias are live at once in most real instruments.
When to reach for something other than alpha
Alpha is the default because it is easy to compute from a single administration, not because it is the best available answer.
McDonald’s omega drops the tau-equivalence assumption by estimating each item’s loading rather than assuming they are equal. Where loadings genuinely differ, omega is the better estimate and is usually higher.
For agreement between raters rather than items, the question is different. An intraclass correlation handles continuous ratings and Cohen’s kappa categorical ones.
For stability over time, alpha says nothing at all. Test-retest reliability needs two administrations, and a scale can be internally consistent and completely unstable across a fortnight.
For a two-item scale, the Spearman–Brown corrected correlation is the standard choice. Alpha on two items is defined but badly behaved, and the pairwise correlation is more interpretable.
Reporting alpha so a reader can judge it
Three numbers instead of one, and the scale length is the one that changes the reading.
Report alpha with the number of items and the sample size. “α = 0.91 (12 items, n = 214)” is interpretable; “α = 0.91” is not, because 0.91 from 60 items and 0.91 from 4 items describe very different instruments.
Report the mean inter-item correlation. It is the item-quality half of alpha with the length removed, and it is one line.
Say whether you deleted items, and on what basis. Alpha computed after dropping the worst items on the same data is optimistic, in the same way any statistic selected on its own sample is.
And give a confidence interval if the sample is small. Alpha from 20 respondents is an unstable estimate; the point value alone implies a precision that is not there.
Sources and methodology
References for alpha, its assumptions and its misreadings.
Method. Alpha is computed from the item variances and the total-score variance, and the mean inter-item correlation is computed independently from the correlation matrix rather than inverted out of alpha — so the Spearman–Brown identity between them is a check rather than a definition. The length table is generated by that identity at a fixed correlation, which is what makes the length claim exact rather than illustrative: α = 0.8403 at 100 items and r̄ = 0.05 is arithmetic, not a simulation. Alpha-if-dropped recomputes the whole statistic on the reduced matrix each time. Negative alpha is returned as computed and never clamped. Ragged rows, a single item and a single respondent all return no result rather than a number. That engine is verified on every change against 69 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Intraclass CorrelationAll six ICC forms from one subject-by-rater matrix, with the rater means that drive them apart: one 8x3 matrix gives ICC(1,1) = 0.1277 and ICC(3,1) = 0.9852.
Cohen's KappaKappa with the two figures that explain it: the maximum the marginals permit, and PABAK. Two built-in tables with identical 85% agreement give kappas of 0.6995 and 0.3219, and a third with 94.4% agreement gives −0.0234.
Correlation CoefficientReports Pearson, Spearman and Kendall together with the scatter plot, and ships Anscombe's quartet built in — four datasets with an identical r of 0.816 that Spearman tells apart.
Point-Biserial Correlationr with Cohen's d, t and Pearson's r on the same columns computed independently, plus the split table: hold d at 1.9799 and r falls from 0.707107 at 50/50 to 0.269630 at 2/98.
VarianceSample and population variance from your data, with a live simulation that shows exactly how much the wrong divisor costs — 20% low at n = 5, closing as the sample grows.
Standard DeviationSample and population standard deviation, plus variance, mean, median, quartiles, z-scores, outliers, and confidence intervals.
An educational tool. Alpha is a lower bound on reliability under tau-equivalence and an overestimate when item errors are correlated, and it rises with the number of items whether or not they measure the same construct — so a high value is not evidence that a scale is unidimensional, and this tool does not test that.
Published an internal-consistency tool that separates the two things alpha confounds: item quality and item count.
Made the length effect exact rather than cautionary. At a mean inter-item correlation of 0.05 — items very nearly unrelated to each other — a 100-item scale still reports alpha = 0.8403, and 40 items at 0.1 report 0.8163. Both clear every conventional threshold on length alone.
Added alpha-if-item-dropped and item-total correlations, with a worked case where alpha of 0.7057 looks acceptable and dropping one item raises it to 0.9877 — that item's correlation with the total is -0.5260.
Reported negative alpha as computed rather than flooring it at zero, since a negative value is the most informative result the statistic produces and usually means an unscored reverse item.
Verified against 69 assertions covering alpha, the drop-one recomputation, the Spearman–Brown identity that generates the length table, and the degenerate matrices that must return no result.
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