Levene's test

Equality of variances

Calculates the homogeneity of variances assumption for two or more groups.
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R code.

Tukey HSD / Tukey Kramer

Usually the Tukey HSD compare the averages, but the following test run over the differences from the averages.
The following tables compares the mean absolute deviations (MAD), similar to comparing the standard deviations.

Absolute differences from the group's center

Target: To check if the difference between the variances of two or more groups is significant, using a sample data
The Levene's tests perform an ANOVA test over the absolute deviations from each group's average or the absolute deviations from each group's median.

When performing ANOVA test, we try to determine if the difference between the averages of the absolute residuals reflects a real difference between the groups, or is due to the random noise inside each group. The F statistic represents the ratio of the variance between the groups and the variance inside the groups. Unlike many other statistic tests, the smaller the F statistic the more likely the averages are equal.

Right-tailed F test, for ANOVA test you can use only the right tail. Why?
H0: σ1 = .. = σk
H1: not(σ1 = .. = σk)
Test statistic
F distribution
F distribution right tailed


IndependentIndependent samples.
Normal DistributionNormal population distribution or symmetrical distribution.

Required Sample Data

averageSample data from all compared groups.


k - Number of groups.
ni - Sample side of group i.
n - Overall sample side, includes all the groups (Σni, i=1 to k).
xi - Average of group i.
x - Overall average (Σxi,j / n, i=1 to k, j=1 to ni).
Si - Standard deviation of group i.

Results calculations

SourceDegrees of FreedomSum of SquaresMean SquareF statisticp-value
(between groups)
k - 1averageMSG = SSG / (k - 1)F = MSG / MSEP(x > F)
(within groups)
n - kaverageMSE = SSE / (n - k)
Totaln - 1SS(total) = SSG + SSESample Variance = SS(total) / (n - 1)

R Code

The following R code should produce the same results