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Applied Statistics Handbook
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Spearman Rho Coefficient

For paired observations that are ranked.

 

Used to estimate strength and direction of association between two ordinal level variables.  The Spearman Rho Coefficient presented here can range from a -1.00 to 1.00.  A positive coefficient indicates the values of variable A vary in the same direction as variable B. A negative coefficient indicates the values of variable A and variable B vary in opposite directions.

 

Verify the conditions are appropriate

Scores of two variables are ranks

 

Problem:  Five college students' have the following rankings in math and science courses. Is there an association between the rankings in math and science courses.

 

Student

Alice

Jordan

Dexter

Betty

Corina

Math class rank

1

2

3

4

5

Philosophy class rank

5

3

1

4

2

 

Compute Spearman Rho

        where

n = number of paired ranks

d = difference between the paired ranks

 

Note: When two or more observations of one variable are the same, ranks are assigned by averaging positions occupied in their rank order.

Example:

 

Score

2

3

4

4

5

6

6

6

8

Rank

1

2

3.5

3.5

5

7

7

7

9

 

Math Rank

Philosophy Rank

X-Y

(X-Y)2

X

Y

D

d2

1

5

-4

16

2

3

-1

1

3

1

2

4

4

4

0

0

5

2

3

9

 

 

30

 

      

 

Interpret Coefficient

There is a moderate negative correlation between the math and philosophy course rankings of students. Students who rank high as compared to other students in their math course generally have lower philosophy course ranks and those with low math rankings have higher philosophy course rankings than those with high math rankings.

Note: The formulas for Pearson r and Spearman rho are equivalent when there are no tied ranks.

 


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