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Spearman Correlation Coefficient

Spearman Correlation Coefficient

04/25/2021
Spearman Correlation Coefficient with Example
Edubomb Edubomb

Spearman rank correlation

Karl person method is especially useful only when given data are purely quantitative but when given data are qualitative then we use Spearman rank correlation.  For example, if our data are related to quality like beauty, character, morality, honesty, etc then we use the Spearman rank correlation coefficient. In this method, all qualitative data are arranged in series according to their rank and corresponding correlation is calculated. Hence rank correlation gives the degree of the linear relationship between the two or more than two ranks or grade of characteristics. Spearman rank Correlation coefficient is denoted by the R and given by the flowing formula

 ... (1)

where d=R1-R2=diffrence of rank and 

R1=rank of the first characteristics 

R2=rank of the second characteristics 

n=nos. of observation 

Interpretation:

  1. When R= +1 perfectly positive correlation
  2. When R= -1 then the perfectly negative correlation
  3. When R= 0 no correlation
  4. When R= +ve then said to a positive correlation
  5. When R= -ve then said to negative correlation
  6. When R= 0.7 to 0.999 or – (0.7 to -0.99) then said to be a high degree of positive (negative) correlation.
  7. When R= 0.5 to 0.699 then said to be a moderate degree of positive correlation.
  8. When R is less than 0.5 then said to be a low degree of correlation

 

Method of calculating the rank correlation coefficient

We can calculate spearman rank correlation in the following cases

When actual rank is given

In this case formula for the calculation of the spearman rank is given by the general formula as given above

 ……………………………... (1)

where d=R1-R2=diffrence of rank and 

R1=rank of the first characteristics 

R2=rank of the second characteristics 

n=nos. of observation 

Examples:

Q.No.1 calculate rank correlation coefficient of flowing data

Types of vitamin I II III IV V Vi
The rank of the weight of pig A 3 4 6 5 2 1
The rank of the weight of Pig B 2 5 6 3 4 1

Solution: Calculation of rank correlation coefficient

R1 R2 d=R1-R2 d2
3 2 1 1
4 5 -1 1
6 6 0 0
5 3 2 4
2 4 -2 4
1 1 0 0
    ∑ d = 0 ∑ d2=10

Now,

Here, n = 6,  ∑d2=10  

 

 

 

QNO.2 Suppose we have ranks of 8 students of B.Sc. in Statistics and Mathematics. On the basis of rank, we would like to know that to what extent the knowledge of the student in Statistics and Mathematics is related. 

Rank in Statistics 

1

2

3

4

5

6

7

8

Rank in Mathematics

2

4

1

5

3

8

7

6

 

Solution:  Spearman’s rank correlation coefficient formula is 

Now,

Let us denote the rank of students in Statistics by Rx and rank in Mathematics by Ry for the calculation of rank correlation coefficient, we have to find ∑ d2

Now,

Rank in stat Rx

rank in maths Ry the difference in rank d=Rx-Ry d2
1 2 -1 1
2 4 -2 4
3 1 2 4
4 5 -1 1
5 3 2 4
6 8 -2 4
7 7 0 0
8 6 2 4
    ∑d = 0 ∑d2 = 22

Now,

Here, n = 8,  ∑d2=22  

   R=0.74

Thus, there is a positive association between the ranks of Statistics and Mathematics.

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