Bivariate inductive statistics - ordinal variables

Kendall's tau, Gamma coefficient

Kendall's rank correlation provides a distribution free test of independence and a measure of the strength of dependence between two ordinal variables.

You are advised to use Kendall's rank correlation as this reflects the strength of the relationship between the variables and copes with ties much better than more frequently used Spearman's method. Spearman's rank correlation is calculated as Pearson's coeficient with the difference that values are replaced by their ranks. As a test of independence Kendall's method is also superior because it is sensitive to some types of dependence which can not be detected using Spearman's method.

Kendall tau represents the difference between the probability that the observed data are in the same order for the two variables versus the probability that the observed data are in different orders for the two variables.

Gamma statistic is preferable to Spearman R or Kendall tau when the data contain many tied observations.
Gamma is also a probability; specifically, it is computed as the difference between the probability that the rank ordering of the two variables agree minus the probability that they disagree.

The correlations ranges between plus and minus one. 0 is no correlation, 1 is perfect positive correlation, −1 is perfect negative.

Example: Is related person's income level to education achieved?