# 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]

## Definition

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

${\displaystyle \pi _{i+}=\sum _{j=1}^{c}\pi _{ij}}$ and ${\displaystyle \pi _{+j}=\sum _{i=1}^{r}\pi _{ij}.}$

Then the mean square contingency is given as

${\displaystyle \phi ^{2}=\sum _{i=1}^{r}\sum _{j=1}^{c}{\frac {(\pi _{ij}-\pi _{i+}\pi _{+j})^{2}}{\pi _{i+}\pi _{+j}}},}$

and Tschuprow's T as

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

### Properties

T equals zero if and only if independence holds in the table, i.e., if and only if ${\displaystyle \pi _{ij}=\pi _{i+}\pi _{+j}}$. 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 ${\displaystyle \pi _{ij}>0}$ 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.

### Estimation

If we have a multinomial sample of size n, the usual way to estimate T from the data is via the formula

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

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

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

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