Tschuprow's T

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 T = \sqrt{ \frac{\phi^2}{\sqrt{(r-1)(c-1)}} } 

Tschuprow's T

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.[1]


For an r × c contingency table with r rows and c columns, let \pi_{ij} be the proportion of the population in cell (i,j) and let

\pi_{i+}=\sum_{j=1}^c\pi_{ij} and \pi_{+j}=\sum_{i=1}^r\pi_{ij}.

Then the mean square contingency is given as

 \phi^2 = \sum_{i=1}^r\sum_{j=1}^c\frac{(\pi_{ij}-\pi_{i+}\pi_{+j})^2}{\pi_{i+}\pi_{+j}} ,

and Tschuprow's T as

T = \sqrt{\frac{\phi^2}{\sqrt{(r-1)(c-1)}}} .


T equals zero if and only if independence holds in the table, i.e., if and only if \pi_{ij}=\pi_{i+}\pi_{+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 \pi_{ij}>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

\hat T = \sqrt{ \frac{\sum_{i=1}^r\sum_{j=1}^c\frac{(p_{ij}-p_{i+}p_{+j})^2}{p_{i+}p_{+j}}}{\sqrt{(r-1)(c-1)}} } ,

where p_{ij}=n_{ij}/n is the proportion of the sample in cell (i,j). This is the empirical value of T. With \chi^2 the Pearson chi-square statistic, this formula can also be written as

\hat T = \sqrt{ \frac{\chi^2/n}{\sqrt{(r-1)(c-1)}} } .

See also[edit]

Other measures of correlation for nominal data:

Other related articles:


  1. ^ Tschuprow, A. A. (1939) Principles of the Mathematical Theory of Correlation; translated by M. Kantorowitsch. W. Hodge & Co.
  • Liebetrau, A. (1983). Measures of Association (Quantitative Applications in the Social Sciences). Sage Publications