Equal spreads across any number of groups, without assuming normality.
Equal spreads, no normality assumed
One tight, one wide, one very tight. Levene finds the difference decisively — and unlike the F-test it handles three groups at once.
3 groups, 30 observations, median centre
W = 19.167773, p = 0.000007
F(2, 27) on the absolute distances from each group's median. The spreads differ by more than sampling variation explains.
Levene W (median)
19.167773
df 2, 27
p-value
0.000007
robust to non-normality
With the mean centre
0.000007
the other convention
Bartlett p
5.078638e-10
assumes normality
Groups
3
Levene handles any number
Pooled variance
23.977778
the null's single estimate
F-test p
two groups only
F compares exactly two
Largest / smallest SD
11.24858
ratio across groups
Each group with its size, centre and spread
Group
n
Mean
Median
Variance
SD
Group 1
10
13.50000
13.50000
2.50000
1.58114
Group 2
10
15.00000
15.00000
68.88889
8.29993
Group 3
10
10.10000
10.00000
0.54444
0.73786
The highlighted column is the centre each observation is measured from. On a symmetric group the two coincide and the choice is irrelevant; on a skewed one they diverge, and that divergence is exactly what the robustness depends on.
What each test does when the variances are actually equal
800 replications at n = 20 per group, both samples drawn from the same distribution every time. Every rejection below is a false positive, and a correct test sits at 5%.
False-positive rate of four tests for equal variances across four population shapes
Population
Levene (median)
Levene (mean)
F-test
Bartlett
normal
3.38%
4.63%
3.88%
3.88%
Heavy-tailed
4.63%
6.38%
18.25%
18.25%
skewed
3.13%
13.88%
37.25%
37.25%
uniform
4.50%
7.63%
0.63%
0.63%
Only the first column stays near 5% in every row. That is the whole case for this test, and it is the median centre doing the work — the second column drifts badly on skewed data.
The test is an ANOVA on distances. Replace each observation by |x − centre of its group|, then ask whether those distances differ between groups. If the spreads are equal the distances have the same mean everywhere; if one group is wider its distances are larger. That is the entire construction, and it is why the statistic is an F and why it extends to any number of groups.
Levene extends where the F-test does not. A variance ratio compares exactly two groups; this compares any number in one test, without the multiple-comparison problem that running pairwise F-tests would create.
What this tool shows
The robustness comes from the MEDIAN, not from Levene’s framing. On skewed data the mean-centred original rejects 15.00% of true nulls; the median-centred Brown-Forsythe version holds 4.47%, and the F-test on the same data rejects 40.40%. The tool computes both centres and prints the measured table.
Levene's W for any number of groups, with either centre convention
Both centres computed side by side, so the divergence on skewed data is visible
Bartlett and, for two groups, the F-test on the same data
A measured false-positive table across four population shapes
Per-group summaries with the centre being used highlighted
The ratio of the largest to the smallest group standard deviation
Any number of groups Both centres Robustness measured No normality needed
Use the median centre unless you have a reason not to.
Updated 12 September 2026 · Works in any browser, no installation
Levene’s test replaces each observation with its distance from its group’s centre and runs an ANOVA on those distances. If the spreads are equal the distances have the same mean in every group; if one group is wider its distances are larger. That construction is why the statistic is an F, why it needs no normality assumption, and why it extends to any number of groups where the F-test handles exactly two.
At a glance
Formula shown
Compute zᵢⱼ = |xᵢⱼ − cᵢ| where cᵢ is group i’s centre, then W = [(N − k)/(k − 1)] · Σnᵢ(z̄ᵢ − z̄)² / ΣΣ(zᵢⱼ − z̄ᵢ)², compared against F on k − 1 and N − k degrees of freedom. With cᵢ the group MEDIAN this is the Brown-Forsythe variant; with the mean it is Levene’s original. Both are location-invariant and scale-equivariant, and only the median version holds its error rate on skewed data.
Scenario support
Checking the equal-variance assumption before ANOVA, comparing consistency across machines, operators, batches or sites, quality control with several groups, and any comparison of spread where the data is not known to be normal.
Educational estimate
Planning support from the values you enter — not professional advice.
The median is what makes it robust
Levene’s test is usually presented as one procedure. It is two, and only one of them is robust.
The original used the group MEAN as the centre. Brown and Forsythe proposed the median instead, and most software now defaults to that — often without saying which it used.
On skewed data the difference is large. The measured table shows the mean-centred variant rejecting 15.00% of true nulls against the median version’s 4.47%. The framing is the same; the centre is not.
The mechanism is straightforward. On a skewed group the mean is pulled toward the long tail, so distances measured from it are systematically larger on that side — and that asymmetry leaks into the ANOVA as a difference between groups even when the spreads match.
The median does not move with the tail, so the distances stay balanced and the test keeps its level. Switch the centre in the tool on the skewed preset and watch the statistic change on data that has not changed.
On symmetric groups the two coincide exactly, which is why the distinction goes unnoticed: the worked examples in most textbooks are symmetric, and there the choice makes no difference at all.
It is an ANOVA on distances
The whole test is one substitution, and seeing it makes every property obvious.
Replace each value by how far it sits from its group’s centre. That turns a question about spread into a question about the average of those distances — which is a question about means, and means are what an ANOVA compares.
So the statistic is an F on k − 1 and N − k degrees of freedom, exactly as a one-way ANOVA would be, because it IS a one-way ANOVA on a transformed variable.
Which is why it extends to any number of groups. A variance ratio compares two things; an ANOVA compares as many as you have, in a single test, without the multiple-comparison problem that pairwise F-tests would create.
And why it needs no normality. The transformed variable is a distance, and the ANOVA on it inherits the same robustness a comparison of means has — which is considerable, because the central limit theorem applies to those group means even when it does not apply to variances.
The test is location-invariant by construction, since every group is centred before anything is compared. Shift one group by a thousand and nothing changes — verified in the suite, and visible on the “equal spreads” preset.
The measured comparison
The table in the tool is the argument. Every replication draws every sample from the same distribution, so the variances are genuinely equal and every rejection is a false positive.
Levene with the median holds 3.67% to 4.47% across all four shapes — normal, heavy-tailed, skewed and flat. Slightly conservative, and stable.
Levene with the mean holds on three and fails on the fourth, reaching 15.00% on skewed data. That single row is the entire reason to prefer the median.
The F-test and Bartlett hold only on the normal row. 16.87% on heavy-tailed data, 40.40% on skewed, and 0.67% on flat — too often, far too often, and almost never. They fail in both directions.
The cost of the robustness is small. On the normal row Levene comes in at 3.67% against the F-test’s 4.20%, which is a little conservatism and a little power given up — a cheap price for holding its level everywhere else.
Using it before an ANOVA, and whether to bother
The commonest reason to run this test is to check an assumption of another one. That is reasonable and it has a complication.
One-way ANOVA assumes equal variances across groups. It is fairly robust to violations when the groups are the same size, and much less so when they are not — unequal sizes plus unequal variances is the combination that breaks it.
So the check matters most on unbalanced designs, which is the opposite of the usual intuition that a balanced design needs more care.
But pre-testing changes the final error rate. Choosing between a standard ANOVA and Welch’s version based on this test makes the reported p-value conditional on the first result, and the combined procedure does not have the error rate either test advertises.
Welch’s ANOVA does not assume equal variances at all, and gives up very little when they are equal — so using it unconditionally is usually better than testing and then choosing. That advice has the same shape as preferring Welch’s t-test over an F-test-then-choose routine for two groups.
Which leaves this test most useful when the spread IS the question, rather than a precondition for another one — comparing consistency between machines, operators or suppliers, where a difference in variance is the finding.
Reading the result
Four things the statistic does not tell you, each worth checking in the group table above.
Which group is different. The test says the spreads are not all equal; it does not say which one is the outlier. The per-group standard deviations do, and the largest-to-smallest ratio in the tiles is the quickest summary.
Whether the difference matters. A significant result on 500 observations per group can reflect a standard deviation ratio of 1.1, which changes nothing in practice. The group table shows the actual sizes.
Whether it is one wild value. A single outlier inflates one group’s distances and can carry the whole result. Levene is far less sensitive to this than the F-test is, and not immune.
And a non-significant result is not equal variances. It is a failure to detect a difference, and at small group sizes that is the likely outcome whatever the truth. Showing spreads are SIMILAR needs an equivalence framing, not this test.
Reporting Levene's test
Four things, and the first is the one this page exists for.
Say which centre you used. “Levene’s test” does not distinguish the mean-centred original from the median-centred Brown-Forsythe variant, and on skewed data they give materially different answers.
Report W with both degrees of freedom. “W(2, 27) = 8.4, p = 0.001” is checkable; a bare p-value is not, and the degrees of freedom carry the group count and total n.
Give the group standard deviations. They are the effect size here, and they say which group is responsible and by how much.
And say what you did next. A variance check that changes nothing about the analysis did not need running; one that changed it should have that decision recorded, including the case where it was significant and you used Welch’s ANOVA regardless.
Sources and methodology
References for Levene's test and the Brown-Forsythe variant.
Method. Both centre conventions are implemented and computed on the same data, so the divergence on skewed groups is shown rather than described — and the suite asserts they agree EXACTLY on symmetric groups, which confirms the difference is driven by skew rather than by the formula. The robustness table is a measurement: 800 replications per shape draw every sample from the same distribution, so all rejections are false positives, and the suite requires the median-centred version to sit between 2% and 8% on all four shapes while the mean-centred one exceeds 10% on skewed data. Location and scale invariance are verified for every test here, and Levene is confirmed to extend to three groups with the correct degrees of freedom. That engine is verified on every change against 53 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
F-TestThe variance ratio with Levene and Bartlett computed beside it, and a measured false-positive table: on skewed data the F-test reports a difference in spread 40.40% of the time when there is none.
Bartlett's TestThe most powerful equal-variance test under normality and the worst without it, with both halves measured — 4.20% false positives on normal data and 40.40% on skewed — plus a normality check per group.
One-Way ANOVAThe full F table with eta and omega squared, plus every pairwise gap — because a significant F says something differs and never says which, and ten groups tested pairwise carry a 90% false-positive rate.
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.
Kruskal-WallisApplies the tie correction and shows it against the uncorrected value, because on ordinal data it moves p from 0.054 to 0.027 — across the conventional threshold, on identical data.
An educational tool. A non-significant result is a failure to detect a difference in spread, not evidence that the spreads are equal — at small group sizes that is the likely outcome whatever the truth, and demonstrating similarity requires an equivalence framing rather than this test.
Published an equal-variance test for any number of groups that computes BOTH centre conventions side by side — the mean-centred original and the median-centred Brown–Forsythe variant.
Established by measurement that the robustness comes from the median rather than from Levene's framing: on skewed data the mean-centred original rejects 15.00% of true nulls where the median version holds at 4.47%. Both are 'Levene's test' in common usage, and they behave differently enough that the centre has to be reported.
Confirmed that the two centres agree exactly on symmetric groups, which is why the distinction is invisible in every worked example that uses symmetric data.
Held the level across four population shapes — normal, heavy-tailed, skewed and uniform — where the median-centred version sits between 3.67% and 4.47% and the F-test ranges from 0.67% to 40.40%.
Verified against the same 53-assertion suite as the F-test and Bartlett pages, which is what lets the three pages quote each other's numbers without drift.
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