The spearman’s rank coefficient of correlation was developed by Charles Edward Spearman.
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. Pearsonian coefficient of correlation between the ranks xi’s and yi ‘s is called the Spearman rank correlation coefficient between A and B for that group of individuals.
rk = 1 – [6 ∑D2 / N3 – N]
Where D = R1 – R2, between the paired items in the two rank series.
The value of rank correlation coefficient tells us about the degree of agreement between the 2 ranks.
Assuming that no two individuals are bracketed equal in either classification, each of the variables X and Y takes the values 1,2,………..,n
Hence
Which is the spearman's formula for the rank correlation coefficient.
-1 ≤ rk ≤ +1
Rank Correlation Coefficient Example
Calculate the Rank Correlation Coefficient in each of the following cases:
X | R1 | Y | R2 |
---|---|---|---|
10 | 1 | 5 | 1 |
20 | 2 | 6 | 2 |
30 | 3 | 7 | 3 |
To calculate the rank correlation coefficient, first we will determine the value of D = R1 – R2 in each of the entries:
X | R1 | Y | R2 | D | D2 |
---|---|---|---|---|---|
10 | 1 | 5 | 1 | 0 | 0 |
20 | 2 | 6 | 2 | 0 | 0 |
30 | 3 | 7 | 3 | 0 | 0 |
Then the Spearman’s rank correlation coefficient is calculated using the formula as:
rk = 1 – [6 ∑D2 / N3 – N]
= 1- 6(0)
= +1
Thus the value of rank correlation coefficient equal to +1 implies that there is complete agreement in the order of ranks and the ranks are in the same direction.
Let us calculate the rank correlation coefficient in another example:
X | R1 | Y | R2 |
---|---|---|---|
10 | 1 | 7 | 3 |
20 | 2 | 6 | 2 |
30 | 3 | 5 | 1 |
To calculate the rank correlation coefficient, first we will determine the value of D = R1 – R2 in each of the entries:
X | R1 | Y | R2 | D | D2 |
---|---|---|---|---|---|
10 | 1 | 7 | 3 | -2 | 4 |
20 | 2 | 6 | 2 | 0 | 0 |
30 | 3 | 5 | 1 | +2 | 4 |
Then the Spearman’s rank correlation coefficient is calculated using the formula as:
rk = 1 – [6 ∑D2 / N3 – N]
= 1 – (6*8)/9-3
= -1
Thus the value of rank correlation coefficient equal to -1 implies that there is complete agreement in the order of ranks and the ranks are in opposite direction
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