Tests for equality of means/medians (related samples)

Tests for the equality of means/medians of related 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.

Assumes the population differences are normally distributed. Due to the central limit theorem, the test may still be useful when this assumption is not true if the sample size is moderate. However, in this case, the Wilcoxon test may be more powerful.

Student's t Test if the difference between means is equal to a hypothesized value.

Assumes the population differences are normally distributed. Due to the central limit theorem, the test may still be useful when this assumption is not true if the sample size is moderate. However, in this case, the Wilcoxon test may be more powerful.

Wilcoxon Test if there is a shift in location equal to the hypothesized value.  

Under the assumption that the population distribution of the differences is symmetric, the hypotheses can be stated in terms of a difference between means/medians.

Under the less strict hypotheses, requiring no distributional assumptions, the hypotheses can be stated as the probability that the sum of a randomly chosen pair of differences exceeds zero is 0.5.

Sign Test if the median of the differences is equal to a hypothesized value.  

Under the more general hypotheses, tests if given a random pair of observations (xi, yi), that xi and yi are equally likely to be larger than the other.

Has few assumptions, but lacks power compared to the Wilcoxon and Student's t test.

ANOVA Test if the two or means are are equal.

Assumes the populations are normally distributed. Due to the central limit theorem, the test may still be useful when the assumption is violated if the sample sizes are equal and moderate size. However, in this situation the Friedman test is may be more powerful.

Friedman Test if two or medians are equal.

Has few assumptions, and is equivalent to a two-sided Sign test in the case of two samples.