Levene’s Test for Equality of Variances

Levene’s test asks a single, narrow question: do two or more groups have the same population variance? It matters because so many of the standard parametric tests researchers reach for — the independent-samples t-test, one-way ANOVA, and their relatives — assume homogeneity of variance (equal spread across groups) as a precondition for the p-values they report to be trustworthy. Levene’s test is the diagnostic most commonly used to check that precondition before trusting the main result. This guide covers the logic behind how it works, why the mean-based and median-based (Brown-Forsythe) versions can disagree, and what a significant result should actually change about your analysis.

The Logic: Testing Absolute Deviations, Not the Scores Themselves

Levene’s test doesn’t compare variances directly with a ratio, the way an F-test for two variances does. Instead, it reframes a question about spread as a question about means, which lets it reuse ordinary ANOVA machinery. The procedure:

1. For each group, calculate the group’s center — the mean in the original Levene (1960) formulation.
2. For every observation, compute its absolute deviation from that group’s center: |xij − centerj|.
3. Run a standard one-way ANOVA on these absolute deviations, treating group membership exactly as it would in a normal ANOVA on the raw scores.
4. If that ANOVA is significant, the groups differ in how far, on average, their observations sit from their own center — which is precisely what unequal variance means. A significant Levene’s test is really an ANOVA on dispersion, not on location.

This is why Levene’s test is often described as converting a variance-comparison problem into a mean-comparison problem: once you’re working with absolute deviations, any test that compares group means — and its assumptions — can be reused, even though the original question was about spread.

Why the Center Choice Matters: Levene vs. Brown-Forsythe

The step researchers most often skip past — and the one this guide is really about — is that “center” isn’t fixed. Levene’s original 1960 test uses the group mean. Brown and Forsythe (1974) proposed the same procedure using the group median instead, and some software also offers a trimmed mean or a median with adjusted degrees of freedom as further variants. All four are legitimate members of the same test family; they differ only in which center the absolute deviations are measured from.

That choice isn’t cosmetic. The mean is sensitive to outliers and to skew: in a skewed or heavy-tailed distribution, deviations from the mean are inflated by whichever tail is doing the pulling, which can make Levene’s original version over-detect unequal variance — flagging groups as heteroscedastic when the real issue is non-normal shape, not unequal spread. The median is far less sensitive to outliers and skew, so absolute deviations from the median are a cleaner, more robust signal of dispersion on its own. This is the entire rationale for the Brown-Forsythe variant: it keeps Levene’s mechanism (ANOVA on absolute deviations) but swaps in a center statistic that doesn’t itself get distorted by the same non-normality the test is trying to look past.

In practice this means the mean-based and median-based versions can genuinely disagree on the same dataset — not because one is calculated wrong, but because they’re measuring dispersion around two different anchors. When a distribution is symmetric and reasonably well-behaved, they typically agree closely. When it’s skewed, has heavy tails, or contains a handful of outliers, they can diverge, and the Brown-Forsythe (median-based) result is generally the one to trust: simulation-based comparisons have consistently found it holds its nominal Type I error rate better than the mean-based version under non-normality, which is exactly the condition where you’re most likely to need a variance-equality check in the first place. Most current statistical software defaults to, or at least offers, the Brown-Forsythe version for this reason.

What a Significant Levene’s Test Should Actually Change

A significant Levene’s test is a diagnostic result, not an endpoint — it tells you a specific assumption behind your planned test doesn’t hold, and the correct response depends on which test you were about to run:

• Independent-samples t-test: switch from the pooled-variance (Student’s) t-test to Welch’s t-test, which doesn’t assume equal variances and adjusts the degrees of freedom to compensate. Most statistical software (SPSS, R’s default t.test(), Stata) already reports both versions or defaults to Welch’s, precisely so a significant Levene’s result has a ready-made fix sitting next to it.
• One-way ANOVA: use a variance-robust alternative to the F-test, most commonly the Welch ANOVA (or the Brown-Forsythe F-test, a related correction), and follow up with a post hoc procedure that doesn’t assume equal variances, such as the Games-Howell test, rather than Tukey’s HSD.
• Reaching for a data transformation instead: this is a common but often misdirected response. Transforming the outcome (a log or square-root transform, for example) can sometimes stabilize variance, but it also changes what the analysis is actually testing — the results are now about the transformed scale, which complicates interpretation and reporting. Because robust alternatives like Welch’s correction exist specifically to handle unequal variance without altering the outcome scale, routing a significant Levene’s result to Welch’s test (or its ANOVA equivalent) rather than to a transformation is usually the more defensible default. Reserve transformation for cases where non-normality itself, not just unequal variance, is the primary problem being addressed.

What a significant result should not trigger is abandoning the comparison altogether or treating unequal variance as disqualifying. Homogeneity of variance is a condition that affects which version of a test to run and how its degrees of freedom are calculated — it’s a routing decision, not a stop sign.

Assumptions and Limits of Levene’s Test Itself

Levene’s test is commonly chosen because, unlike the older F-test for two variances, it doesn’t assume the underlying data are normally distributed — that robustness is the whole reason it displaced the F-test as the default variance-equality check for two or more groups. But it isn’t assumption-free:

• It still assumes independence of observations, both within and across groups — the same requirement nearly every parametric group-comparison test carries.
• With very small group sizes, its power to detect real variance differences is limited, the same way any test’s power depends on sample size.
• It tests equality of variance, not any particular ratio or magnitude of difference — a significant result tells you variances differ, not by how much in a way that’s directly interpretable without further calculation.

These limits are also why the choice of center (mean vs. median) is worth understanding rather than treating as a software default to accept blindly: picking the Brown-Forsythe median-based version when your data are plausibly skewed is a defensible, deliberate methodological decision, not a minor technical footnote.

Frequently Asked Questions

Is Levene’s test the same as the Brown-Forsythe test?

They’re closely related but not identical. Brown-Forsythe is a variant of Levene’s test that substitutes the group median for the group mean as the center used to compute absolute deviations. Some software labels the median-based version “Levene’s test (based on median)” rather than naming it separately as Brown-Forsythe, which is why the two names are often used almost interchangeably in practice — but strictly, Brown-Forsythe is one specific version within the Levene family.

Which version should I report: mean-based or median-based?

When in doubt, the median-based (Brown-Forsythe) version is the safer default, particularly if your data show any skew or potential outliers, since it holds its Type I error rate better under non-normality. If your data are close to symmetric with no notable outliers, the two versions will usually agree closely and the choice matters less.

What should I do if Levene’s test is significant before running an independent-samples t-test?

Report and interpret Welch’s t-test result instead of the standard (pooled-variance) Student’s t-test. This is the standard routing decision most statistical packages already build in, rather than transforming the data or abandoning the comparison.

Does a non-significant Levene’s test prove the variances are equal?

No — like any null-hypothesis significance test, a non-significant result means there wasn’t enough evidence to detect a variance difference, not proof that none exists. With a small sample, a real difference in variance can easily go undetected simply because the test lacks the power to find it.

For the mechanics of running Levene’s test and reading its output in specific software, see Levene’s Test in SPSS: Where It Appears and What to Do When It’s Significant, which covers the automatic-vs-opt-in behavior across the t-test, one-way ANOVA, and two-way ANOVA procedures. Related background: the t-test, one-way ANOVA, the Kruskal-Wallis test (a non-parametric alternative when both normality and variance assumptions are in doubt), degrees of freedom, and standard deviation.