Tests for equality of means/medians (independent samples)

Tests for the equality of means/medians of independent samples and their properties and assumptions.

Test Purpose
Z Test if the difference between means is equal to a hypothesized value when the population standard deviation is known.
Student's t Test if the difference between means is equal to a hypothesized value.  

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.

Welch t Test if the difference between means is equal to a hypothesized value.  

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.

Does not assume the population variances are equal.

ANOVA Test if two or means 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 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.

Welch ANOVA Test if two or more means 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 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.

Does not assume the population variances are equal.

Wilcoxon-Mann-Whitney Test if there is a shift in location.  

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.

Kruskal-Wallis Test if two or more medians are equal.  

Assumes the population distributions are identically shaped, except for a possible shift in central location.