In statistics, Tschuprow's T is a measure of association between two nominal variables, giving a value between 0 and 1 (inclusive). It is closely related to Cramér's V, coinciding with it for square contingency tables. It was published by Alexander Tschuprow (alternative spelling: Chuprov) in 1939.
For an r × c contingency table with r rows and c columns, let π i j be the proportion of the population in cell ( i , j ) and let
π i + = ∑ j = 1 c π i j and
π + j = ∑ i = 1 r π i j . Then the mean square contingency is given as
ϕ 2 = ∑ i = 1 r ∑ j = 1 c ( π i j − π i + π + j ) 2 π i + π + j , and Tschuprow's T as
T = ϕ 2 ( r − 1 ) ( c − 1 ) . T equals zero if and only if independence holds in the table, i.e., if and only if π i j = π i + π + j . 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 π i j > 0 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.
If we have a multinomial sample of size n, the usual way to estimate T from the data is via the formula
T ^ = ∑ i = 1 r ∑ j = 1 c ( p i j − p i + p + j ) 2 p i + p + j ( r − 1 ) ( c − 1 ) , where p i j = n i j / n is the proportion of the sample in cell ( i , j ) . This is the empirical value of T. With χ 2 the Pearson chi-square statistic, this formula can also be written as
T ^ = χ 2 / n ( r − 1 ) ( c − 1 ) . 