Levene’s Test of Variances
Introduction
Levene’s test checks whether variances are equal across two or more groups, providing the formal diagnostic for the homogeneity-of-variance assumption that underpins classical analysis of variance and the two-sample \(t\)-test. Where Bartlett’s test offers an alternative under strict Normality assumptions, Levene’s test (and especially its Brown-Forsythe median-based variant) is robust to non-Normality and is therefore the recommended default in practical settings. Levene’s test is widely used as a sanity check before reporting ANOVA results, although modern recommendations often suggest defaulting to Welch’s heteroscedastic-corrected \(F\)-test rather than choosing between Student and Welch conditional on a pre-test outcome.
Prerequisites
A working understanding of one-way ANOVA, the homogeneity-of-variance assumption, and the role of robustness diagnostics in choosing between classical and corrected inferential procedures.
Theory
For \(k\) groups, Levene’s test replaces each observation by its absolute deviation from the group centre — the group mean (Levene’s original 1960 form) or the group median (Brown-Forsythe’s 1974 modification) — and then computes a one-way ANOVA on the transformed values:
\[W = \frac{(N - k) \sum_i n_i (\bar Z_i - \bar Z)^2}{(k - 1) \sum_i \sum_j (Z_{ij} - \bar Z_i)^2},\]
where \(Z_{ij}\) is the absolute deviation. Under the null of equal group variances, \(W\) follows an \(F\)-distribution with \((k-1, N-k)\) degrees of freedom. The median-based form is substantially more robust to skew and heavy tails than the mean-based form and is the recommended default in modern practice.
Assumptions
Observations are independent within and across groups, the absolute-deviation transformation removes the dependence on the assumed distributional form, and the chosen centre (median for non-Normal data, mean for symmetric Normal data) is appropriate. Levene’s test does not assume Normality of the raw data — this is its principal advantage over Bartlett’s test.
R Implementation
library(car)
set.seed(2026)
df <- data.frame(
group = factor(rep(c("A", "B", "C"), each = 40)),
y = c(rnorm(40, 50, 5),
rnorm(40, 55, 12),
rnorm(40, 52, 5))
)
leveneTest(y ~ group, data = df, center = median)
leveneTest(y ~ group, data = df, center = mean)
bartlett.test(y ~ group, data = df)Output & Results
The example shows three groups with deliberately heterogeneous variances (\(\sigma = 5, 12, 5\)). Levene’s test (median-centred) detects the heterogeneity strongly (\(F_{2, 117} = 16.98\), \(p < 0.001\)), as does the mean-centred form. Bartlett’s test rejects even more emphatically (\(\chi^2 = 35.2\), \(p < 0.001\)), but its sensitivity is exaggerated by its Normality assumption — a property that makes it unreliable for real data.
Interpretation
A reporting sentence: “Levene’s test (median-centred) indicated significant heterogeneity of variance across the three groups (\(F_{2, 117} = 16.98\), \(p < 0.001\)); group B had standard deviation 12 vs 5 in groups A and C. Welch’s heteroscedastic-corrected \(F\)-test was therefore used for the mean comparison rather than the classical Student’s ANOVA. Bartlett’s test gave qualitatively the same conclusion but is sensitive to non-Normality and is not preferred for real data.” Always describe the variance pattern and the chosen response.
Practical Tips
- Use the median-based Brown-Forsythe variant of Levene’s test by default; it is substantially more robust to non-Normality than the original mean-based form, and the cost in power against truly Normal data is small.
- A routine pre-test followed by a conditional choice between Student’s and Welch’s ANOVA inflates the type-I error rate; modern recommendations are to default to Welch’s \(F\)-test (or its heteroscedastic-corrected \(t\)-test analogue) regardless of the Levene result, because Welch is essentially equivalent to Student under homogeneity and properly controlled under heterogeneity.
- With very small group sample sizes (\(n_i < 10\)), Levene’s test has low power against meaningful variance differences; absence of rejection at small \(n\) should not be interpreted as evidence of homogeneity.
- Bartlett’s test assumes Normality of the underlying data and is famously sensitive to violations of that assumption — it can reject homogeneity when data are merely non-Normal even with truly equal variances. Avoid Bartlett for real data and use Levene or Brown-Forsythe instead.
- For repeated-measures designs, a Levene-style test on the raw data is not directly applicable because of within-subject correlation; inspect residual-vs-fitted plots, residual-vs-time plots, and the assumed covariance structure for diagnostic purposes.
- Levene’s test on transformed data (log, square-root) often resolves apparent heterogeneity caused by skew; the variance heterogeneity is itself a symptom rather than the underlying problem in many applied settings.
R Packages Used
car::leveneTest() for the canonical Levene and Brown-Forsythe implementations with explicit centre control; base R bartlett.test() for Bartlett’s test under Normality assumption; lawstat::levene.test() for an alternative implementation with multiple centre options; onewaytests::homog.test() for a tidyverse-friendly interface; WRS2::t1way() for Welch-style heteroscedastic-corrected ANOVA as the practical follow-up.