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 twoway classification.
TwoWay 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 crosstabulation, briefly written as 2 x 2 crosstables.
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) $$\left( \beta \right) = \left( {A\beta } \right) + \left( {\alpha \beta } \right)$$