Let us consider certain examples before we discuss the independence in a formal manner.
Example:
Consider the following sample data on the liking of males and females for fish.
Gender 


Males 
Females 
Total 
Like Fish 



Do not like Fish 



Total 



Discussion: There are 100 males out of which 80 like fish and out of 100 females 80 like fish. Males and females have the same liking for fish. We say that there is independence between gender and liking or disliking of fish. Another way of saying the same thing is that there is no relation between the gender and liking for fish.
Example:
Consider the following sample data on smoking by adult males and females:
Gender



Males 
Females 
Total 
Smokers 



NonSmokers 



Total 



There are 100 males out of which 20 are smokers and out of 100 females there is only one smoker. It means that the smokers in males are 20 times more than the smokers among females. Males have a strong relation or association with smoking. Thus males and smoking are strongly associated. We say that there is positive association between males and smoking. There are 99 females who are nonsmokers as compared to 80 malenon smokers. Thus females are inclined towards nonsmoking. The association between females and nonsmoking is also of positive type. There is only 1 female smoker as compared to 20 male smokers. Thus there is negative association between females and smoking and there is also negative association between males and non  smoking. Thus in a certain contingency table, when there is positive association between two attributes, then in the same table there exists the negative association between some other pairs of attributes. If there is positive association between and , then and are also positively associated. In this case there is negative association between and , and between and .
The data may be written as

Males 
Females 
Total 
NonSmokers, 



Smokers, 



Total 



In this table 80 is less than 99 and 20 is greater than 1 (or 1 is less than 20). There is negative association between and and between and . There is positive association between and and between and . If the attributes in the one diagonal have positive association, then the attributes in the other diagonal have negative association.