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 
XY 
(XY)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.
