It always helps to go back to the basic question: *what*. What question is it that Kendall's \tau is intended to answer?

This statistic puts a number on the **ordinal** association of two vectors of equal length in which equal index positions of one and the other are paired, such as movie ratings in terms of thumbs up/down in the one and tomatoes in the other, for example. As you can't directly compare thumbs to apples, it uses comparative rank order.

The value of \tau is in the range -1\dots1 corresponding to total disagreement and total agreement, with the 0 neutral ground as the basis for the **null hypothesis** H_O indicating **independence** of the two variables.

Characterizing the relationship between two variables based on \tau's specific value as good, bad, or indifferent depends on some stipulated domain convention if the appearance of motivated reasoning is to be avoided. For example, a dataset as a whole may have lurking parametric collinearity among variables and there may be subject matter learning that some categorical variable pairing can be helpful somehow to shed light based on experience that \tau <= -0.1 tends to tease out how some of the possible continuous variables can be handled.

Other than that, the number is what the number is. Except.

There are further refinements to be made to handle ranking involving having to break ties. \tau_a, \tau_b \ \& \ \tau_c test provide different strategies to deal with these cases.

The standard text for ordinal categorical data is Agresti A. Analysis of ordinal categorical data. John Wiley & Sons; 2010 Apr 19.