Tschuprow's T


In statistics, Tschuprow's T is a measure of association between two nominal variables, giving a value between 0 and 1. It is closely related to Cramér's V, coinciding with it for square contingency tables.
It was published by Alexander Tschuprow in 1939.

Definition

For an r × c contingency table with r rows and c columns, let be the proportion of the population in cell and let
Then the mean square contingency is given as
and Tschuprow's T as

Properties

T equals zero if and only if independence holds in the table, i.e., if and only if. T equals one if and only there is perfect dependence in the table, i.e., if and only if for each i there is only one j such that and vice versa. Hence, it can only equal 1 for square tables. In this it differs from Cramér's V, which can be equal to 1 for any rectangular table.

Estimation

If we have a multinomial sample of size n, the usual way to estimate T from the data is via the formula
where is the proportion of the sample in cell. This is the empirical value of T. With the Pearson chi-square statistic, this formula can also be written as