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
$$\alpha $$
$$\left( {AB} \right)$$
$$\left( {\alpha B} \right)$$
$$\left( B \right)$$
$$\beta $$
$$\left( {A\beta } \right)$$
$$\left( {\alpha \beta } \right)$$
$$\left( \beta \right)$$
$$\left( A \right)$$
$$\left( \alpha \right)$$

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


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