Tests for the equality/equivalence of means/medians of independent samples and their properties and assumptions.
Assumes the populations are normally distributed. Due to the central limit theorem, the test may still be useful when this assumption is not true if the sample sizes are equal, moderate size, and the distributions have a similar shape. However, in this situation the Wilcoxon-Mann-Whitney test may be more powerful.
Assumes the population variances are equal. This assumption can be tested using the Levene test. The test may still be useful when this assumption is not true if the sample sizes are equal. However, in this situation, the Welch t-test may be preferred.
Does not assume the population variances are equal.
Assumes the populations are normally distributed. Due to the central limit theorem, the test may still be useful when this assumption is not true if the sample sizes are equal, moderate size, and the distributions have a similar shape.
Assumes the populations are normally distributed. Due to the central limit theorem, the test may still be useful when this assumption is not true if the sample sizes are equal and moderate size. However, in this situation the Kruskal-Wallis test is may be more powerful.
Assumes the population variances are equal. This assumption can be tested using the Levene test. The test may still be useful when this assumption is not true if the sample sizes are equal. However, in this situation, the Welch ANOVA may be preferred.
Assumes the populations are normally distributed. Due to the central limit theorem, the test may still be useful when this assumption is not true when the sample sizes are equal and moderate size. The Kruskal-Wallis test may be preferable as it is more powerful than Welch's ANOVA.
When the population distributions are identically shaped, except for a possible shift in central location, the hypotheses can be stated in terms of a difference between means/medians.
When the population distributions are not identically shaped, the hypotheses can be stated as a test of whether the samples come from populations such that the probability is 0.5 that a random observation from one group is greater than a random observation from another group.
Assumes the population distributions are identically shaped, except for a possible shift in central location.