What is Levene’s test used for?

Levene’s test assesses whether several groups have equal population variances. It is typically used before applying methods such as the independent samples t test or one-way ANOVA, which assume homogeneity of variances.

Should I use the mean or median version of Levene’s test?

The original Levene’s test is based on deviations from the group mean and works well under approximate normality. The Brown–Forsythe modification uses the median and is more robust to outliers and heavy tails. Many analysts prefer the median version unless normality is well supported.

What are the assumptions of Levene’s test?

Levene’s test assumes independent observations and a reasonably continuous scale. It is less sensitive to departures from normality than tests like Bartlett’s test, especially in the median (Brown–Forsythe) form, but very small samples or extreme outliers can still distort the results.

How do I interpret the p-value in Levene’s test?

The null hypothesis states that all group variances are equal. A small p-value (typically below 0.05) indicates evidence against homogeneity of variances. A large p-value suggests that the data are compatible with equal variances, but it does not prove that the variances are exactly the same.

Levene’s Test for Homogeneity of Variances

F-test

Run Levene’s test (and its Brown–Forsythe median-based variant) to check whether multiple groups have equal variances. Paste your data for 2 or more groups and get the F statistic, degrees of freedom, p-value and an interpretation at your chosen significance level.

Designed for teaching, lab work and quick checks of the equal-variance assumption before t-tests and ANOVA. For critical decisions, consult a statistician and your field’s reporting standards.

Interactive Levene’s test calculator

Center (Z_ij deviations from)

Median is recommended for robustness to outliers and non-normality.

Significance level α

Used to decide whether to reject equal variances.

Group count

3 groups

Between 2 and 8 groups. Each needs at least 2 observations.

Paste one group per box. You can use commas, semicolons, spaces, or line breaks (e.g. 12.4, 11.9, 13.2 12.8).

Levene’s test results

Component	df	SS	MS	F

F statistic computed from an ANOVA on absolute deviations \(Z_{ij} = |X_{ij} - T_i|\), where \(T_i\) is the chosen group center (mean or median). p-value is based on the F distribution with the reported degrees of freedom.

What is Levene’s test?

Levene’s test is a widely used omnibus test of equality of variances across two or more groups. It tests the null hypothesis

\[ H_0: \sigma_1^2 = \sigma_2^2 = \dots = \sigma_k^2 \] \[ H_A: \text{at least two group variances differ.} \]

It works by transforming the original observations \(X_{ij}\) into absolute deviations from a group-specific center \(T_i\):

\[ Z_{ij} = |X_{ij} - T_i| \]

Then a standard one-way ANOVA is run on the \(Z_{ij}\). If the mean (or median) absolute deviations differ substantially between groups, the test statistic becomes large and the p-value becomes small.

Mean vs. median (Brown–Forsythe) Levene’s test

The choice of center \(T_i\) defines several variants:

Mean-based Levene’s test: \(T_i = \bar{X}_i\), the sample mean of group \(i\). This is the original version and is efficient when data are approximately normal.
Median-based (Brown–Forsythe) test: \(T_i = \tilde{X}_i\), the sample median of group \(i\). This version is more robust to outliers and skewed distributions.

Many contemporary references recommend the median-based variant whenever normality is questionable. This calculator offers both, with the median version selected by default.

How the F statistic is computed

After computing the deviations \(Z_{ij}\) for each observation in group \(i\), let \(\bar{Z}_i\) be the mean of group \(i\), and \(\bar{Z}\) the overall mean:

Between-group sum of squares:

\[ SS_B = \sum_{i=1}^k n_i(\bar{Z}_i - \bar{Z})^2 \]

Within-group sum of squares:

\[ SS_W = \sum_{i=1}^k \sum_{j=1}^{n_i} (Z_{ij} - \bar{Z}_i)^2 \]

Degrees of freedom:

\[ df_B = k - 1, \qquad df_W = N - k \]

Mean squares and Levene’s F statistic:

\[ MS_B = \frac{SS_B}{df_B}, \quad MS_W = \frac{SS_W}{df_W}, \quad F = \frac{MS_B}{MS_W}. \]

Under \(H_0\) (equal variances), and assuming independent observations and mild regularity conditions, the F statistic approximately follows an \(F(df_B, df_W)\) distribution. The p-value is then:

\[ p = P(F_{df_B, df_W} \ge F_{\text{obs}}). \]

When should you use Levene’s test?

Before running a t-test or ANOVA, to assess whether the assumption of equal variances across groups is reasonable.
In experimental design and quality control, to check stability of variability across conditions, treatments, or production lines.
As part of model diagnostics when comparing residual spreads across categories.

Remember that a non-significant Levene’s test does not prove that the variances are exactly equal—it only indicates that the data are compatible with homogeneity at the chosen significance level.

Practical tips and caveats

Sample size: with very small \(n_i\) per group, the test has low power. With very large samples, trivial differences in variance may become statistically significant.
Outliers: outliers can inflate variance and distort tests; the median-based variant mitigates this but does not eliminate it.
Multiple testing: if you run Levene’s test repeatedly across many outcomes or subsets, consider multiplicity adjustments.
Complementary plots: always inspect standard deviations, boxplots, and residual plots alongside the numerical test.