Let us suppose that a group of n individuals is arranged in order of merit or proficiency in possession of two characteristics A and B. these ranks in two characteristics will, in general, be different. For example, if we consider the relation between intelligence and beauty, it is not necessary that a beautiful individual is intelligent also. Let (xi,yi); i=1,2,………,n be the ranks of the ith individual in two characteristics A and B respectively. Pearson coefficient of correlation between the ranks xi’s and yi ‘s is called the rank correlation coefficient between A and B for that group of individuals.
Assuming that no two individuals are bracketed equal in either classification, each of the variables X and Y takes the values 1,2,………..,n
In general xi ≠ yi . Let di =xi -yi
Squaring and summin over I form 1 to n, we get
Dividing both side by n, we get
Where is the rank correlation coefficient between A and B.
Which is the spearman’s formula for the rank correlation coefficient.