Levene's Test Calculator

Levene's test asks whether two or more groups have the same variance, the assumption of homogeneity of variance (homoscedasticity) behind the pooled two-sample t-test, the classical one-way ANOVA and Tukey's HSD test. The null hypothesis is H₀: σ₁² = σ₂² = … = σₖ²; the alternative is that at least one pair of variances differs.

The idea is simple. Replace every observation by its absolute distance from the centre of its group, then run a one-way ANOVA on those distances. A group with a larger spread has larger distances, so the F ratio of that ANOVA, called W here, is large when the variances differ. Levene (1960) measured the distance from the group mean; Brown and Forsythe (1974) proposed the group median and a trimmed mean, which makes the test much less sensitive to skewed or heavy-tailed data. Bartlett's test is more powerful when the data really are normal but far more sensitive to departures from normality.

1. Enter each group's observations separated by commas, spaces or new lines, at least 2 per group (at least 3 are needed for a usable statistic). Add another group gives you up to 10 groups. An empty group is left out and the others keep their own numbers in the results.
2. Choose the center: the median (Brown-Forsythe, the default), the mean (Levene's original) or the 10% trimmed mean.
3. Set the significance level α, usually 0.05. It fixes the F critical value and the wording of the verdict.
4. Press Calculate Levene Test. Read W, the two degrees of freedom, the p-value, the group table and the ANOVA table on the absolute deviations that W comes from.
5. Load example enters the PlantGrowth data, Load InsectSprays example six groups with clearly unequal spread and Load NIST gear example the ten batches of the NIST handbook. Copy link to this calculation shares your exact analysis.

The 10% trimmed mean drops the ⌊0.1 · Nᵢ⌋ smallest and the same number of largest values of a group before averaging, so a group with fewer than 10 observations loses nothing and its trimmed mean is the ordinary mean. Only the centre is trimmed: all N distances still enter the ANOVA. Large values of W mean unequal spread, so the p-value is the upper tail of the F distribution, but the test is two-sided in the variances: it detects any difference in spread, not a direction.

Brown and Forsythe compared the three in Monte Carlo studies: the trimmed mean performed best for a heavy-tailed Cauchy distribution, the median for a skewed χ² distribution with 4 degrees of freedom, and the mean had the best power for symmetric, moderate-tailed distributions. The NIST/SEMATECH handbook recommends the median as the choice that keeps good robustness against many kinds of non-normal data while retaining good power.

The choice can change the verdict. For the handbook's GEAR.DAT data (ten batches of ten gear diameters) the median version gives W = 1.7059 (p = 0.0991), and the handbook concludes that there is insufficient evidence of unequal variances. The mean version gives W = 2.1595 (p = 0.0322) and the 10% trimmed version W = 2.1537 (p = 0.0327), both below 0.05. Decide on the center before you look at the results, and report it.

The groups 23, 25, 24, 26, 22 (group 1), 20, 30, 15, 35, 25 (group 2) and 24, 24, 25, 23, 24 (group 3) are what the calculator holds when the page opens. Group 2 is clearly the most spread out: the sample standard deviations are 1.5811, 7.9057 and 0.7071.

The PlantGrowth data set that ships with R holds the dry weight of 30 plants, ten in each of three groups (a control and two treatments). The sample standard deviations are 0.5831, 0.7937 and 0.4426, a ratio of 1.7933. Press Load example: with the median as center W = 1.1192 on 2 and 27 degrees of freedom, below the F critical value of 3.3541, with p = 0.341227. R's car::leveneTest(weight ~ group, data = PlantGrowth) prints the same F value, 1.1192, and Pr(>F) = 0.3412. No version of the test finds evidence of unequal variances, so the pooled analysis in the ANOVA calculator is not contradicted.

The InsectSprays data set holds the insect counts in 72 plots treated with six sprays (A to F), twelve plots per spray. The standard deviations run from 1.7321 (spray E) to 6.2134 (spray F), a ratio of 3.5873, far above the rule of thumb of 2. Press Load InsectSprays example: all three centers reject at 0.05 and only the size of W depends on the center. R's car::leveneTest(count ~ spray, data = InsectSprays) prints F = 3.8214 with Pr(>F) = 0.004223, and F = 6.4554 with Pr(>F) = 6.1e-05 for center = mean. Counts often have a variance that grows with the mean, and taking square roots, the transformation R's documentation applies to these data, evens the spread out: the median-centered W falls to 0.8836 (p = 0.497135).

• W and the p-value. W is the F ratio of the ANOVA on the absolute deviations. A p-value below α, or a W above the F critical value, means the variances differ significantly. It does not say which group is more variable: compare the standard deviations in the group table.
• Not significant is not equal. With small groups the test has little power, so unequal variances can go undetected; with very large groups it flags differences too small to matter.
• Max SD / Min SD. A rough yardstick that does not depend on the sample size: textbooks such as Moore and McCabe accept the classical procedures when the largest sample standard deviation is less than twice the smallest.
• Group table. It lists each group's center, standard deviation and mean absolute deviation from the center. W grows with the differences among those mean deviations relative to the variation of the distances within the groups.
• Undefined statistic. When every group holds only two values, or the distances inside every group are all equal, there is no variation within the groups to compare against and W is undefined. The calculator says so instead of printing a number.

Many courses teach a two-step routine: run Levene's test, then choose the pooled or the Welch version of the t-test or ANOVA by its p-value. Statisticians increasingly argue against it. Zimmerman (2004) reviewed the evidence, advised against preliminary tests of equal variances and, for unequal group sizes, recommended the separate-variances test unconditionally. Delacre, Lakens and Leys (2017) recommend Welch's t-test by default for two groups, and Delacre and colleagues (2019) Welch's F-test in one-way ANOVA: it costs almost nothing when the variances happen to be equal and protects the error rate when they are not. Both Welch versions are options in the ANOVA calculator and the t-test calculator.

Levene's test remains the right tool when the variances are the question, for example the consistency of two machines, methods or raters, when you need to check an assumption that has no Welch version, such as the pooled error of Tukey's HSD test and of the two-way ANOVA, and as a diagnostic next to a plot of the residuals.

For exactly two normal samples, the F-test for two variances is the classical choice. Which assumptions the other procedures make is summarised in parametric vs nonparametric tests, and how to check normality in normality tests explained.