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)