# Two Attributes

Tall and short people may further be divided into intelligent and unintelligent people. High intelligence may be denoted by $B$ and $\beta$ may be used for low intelligence. The following table shows different attributes and their combinations. When two attributes are involved, the division of the sample as below is called two-way classification.

 Two-Way Classification $A$ $\alpha$ Total $B$ $\left( {AB} \right)$ $\left( {\alpha B} \right)$ $\left( B \right)$ $\beta$ $\left( {A\beta } \right)$ $\left( {\alpha \beta } \right)$ $\left( \beta \right)$ Total $\left( A \right)$ $\left( \alpha \right)$ $n$

The column totals are denoted by $\left( A \right)$ and $\left( \alpha \right)$ and the row totals are denoted by $\left( B \right)$ and$\left( \beta \right)$. The above table contains 2 rows and 2 columns and is therefore called a 2 x 2 contingency table or a 2 x 2 cross-tabulation, briefly written as 2 x 2 cross-tables.

There may be more than two attributes. The symbols $A,B,C$ are used for the attributes and $\alpha ,\beta ,\gamma$ are used for the absence of the attributes $A,B,C$. Thus $\alpha$ means not $A$ and $\beta$ means not $\beta$ and $\gamma$ means not $C$.

Suppose that out of 60 tall people 30 are intelligent, and out of 40 short people 20 are intelligent. We can write these frequencies in the following 2 x 2 contingency table:

 2 x 2 Contingency Table $A$ $\alpha$ Total $B$ $\left( {AB} \right) = 30$ $\left( {\alpha B} \right) = 20$ $\left( B \right) = 50$ $\beta$ $\left( {A\beta } \right) = 30$ $\left( {\alpha \beta } \right) = 20$ $\left( \beta \right) = 50$ Total $\left( A \right) = 60$ $\left( \alpha \right) = 40$ $n = 100$

From the table we can write some relations:

(1) $\left( A \right) + \left( \alpha \right) = n$

(2) $\left( B \right) + \left( \beta \right) = n$

(3) $\left( A \right) = \left( {AB} \right) + \left( {A\beta } \right)$

(4) $\left( \alpha \right) = \left( {\alpha B} \right) + \left( {\alpha \beta } \right)$

(5) $\left( B \right) = \left( {AB} \right) + \left( {\alpha B} \right)$

(6) $\left( \beta \right) = \left( {A\beta } \right) + \left( {\alpha \beta } \right)$