Comparing more than two means

A t-test (or non-parametric equivalent) can be used to determine if there is a statistically significant difference between two data sets. However, in some cases there are more than two groups of interest. For example, imagine you would like to determine if there is a difference in anxiety levels at exam time between first, second, third, and fourth year university students. One approach would be to conduct a t-test between every possible combination of years (e.g. first vs second, first vs third, etc.). However, conducting the tests separately increases the probability of a type I error.

Analysis of Variance (ANOVA) is a family of statistical tests that are useful when comparing several sets of scores. A common application of ANOVA is to test if the means of three or more groups are equal. The basic idea behind ANOVA is a comparison of the variance between the groups and the variance within the groups.

Tests available online

See to discussions about experiment design and parametric vs non-parametric for help selecting an appropriate test for your data.

ParametricNon-parametric
Independent groupsOne-way ANOVAOne-way Kruskal-Wallis
Same subjectsRepeated measures ANOVA (rANOVA)Friedman test
(and Kendall's W)

Test assumptions

The parametric tests rely on the following assumptions:

One-way ANOVA
  1. The (within group) scores are normally distributed. The Shapiro–Wilk normality test is automatically applied by the calculator below for small data sets.
  2. The variances of the populations are approximately equal (this is sometimes known as homogeneity of variance or homoscedasticity). Levene's test for equality of variance is used by the calculator below to test this assumption.
rANOVA
  1. The (within group) scores are normally distributed. The Shapiro–Wilk normality test is automatically applied by the calculator below for small data sets.
  2. A property known as sphericity. Sphericity means that the subject-wise differences between variables for each data set have approximately the same variance. Mauchly's sphericity test is used by the calculator below to test this assumption. Furthermore, the Greenhouse-Geisser, Huynh-Feldt and lower-bound adjustments are included in the results.
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