Rank correlation

1. NADEEM UDDIN
ASSOCIATE PROFESSOR
OF
STATISTICS

2. Rank Correlation:-
There are many situations where
numerical values are not available,
but data have been assembled in a
relative order, the best being in the
first rank, the second best next and
so on.
OR
Some time it is not possible to
measure certain variable, but it is
possible to arrange them in order.

3. For Example
If two coffee flavor experts were asked
to place 5 coffee flavor in order of
preference, they would rank the five
coffee flavor in order, using the
number 1,2,3,4,5.
The flavor they liked best would be
ranked 1.
The flavor they liked least would be
ranked 5.

4. The formula for correlation between
ranking of two sets of data is called
Rank Correlation or Spearman’s
Coefficient of Rank Correlation.
It is denoted by rs.
𝑟𝑠 = 1 −
6∑𝑑2
𝑛 𝑛2−1
Where ‘d’ is the difference in ranking
between the two sets of observations
and ‘n’ is the number of data pairs.

5. Example-1:
The following table shows ten students
were ranked according to their
performance in their class work and
their final examinations.
We want to find out whether there is a
relationship between the
accomplishment of the students during
the whole year and their performance in
their exams.

6. Solution:
Students Ranking
based on
class work
(x)
Ranking
based on
exam
marks (y)
Difference
d
d2
A 2 1 1 1
B 5 6 -1 1
C 6 4 2 4
D 1 2 -1 1
E 4 3 1 1
F 10 7 3 9
G 7 8 -1 1
H 9 10 -1 1
I 3 5 -2 4
J 8 9 -1 1
SUM ∑ d2=24

7. The rank correlation coefficient is:
𝑟𝑠 = 1 −
6∑𝑑2
𝑛 𝑛2−1
= 1−
6 24
10 102−1
= 1 −
144
10 100−1
= 1 −
144
10 99
= 1 −
144
990
= 1 − 0 ⋅ 145
𝑟𝑠 = 0 ⋅ 855

8. Comments:
The high value of the rank
correlation coefficient indicates
that there is a close relationship
between class work and exam
performance.

9. Example-2:
The marks of eight candidates in Accounting
and Statistics are :
Candidate 1 2 3 4 5 6 7 8
Accounting 50 58 35 86 76 43 40 60
Statistics 65 72 54 82 32 74 40 53

10. Solution:
In this question ranking is not given. So we take
highest mark as rank 1 and next highest mark as
rank 2 and so on.
Candi-
dates
Accoun-
ting
Statistics Rank
Accounting
(R1)
Rank
Statistics
(R2)
Difference
d =R1-R2 d2
1 50 65 5 4 1 1
2 58 72 4 3 1 1
3 35 54 8 5 3 9
4 86 82 1 1 0 0
5 76 32 2 8 -6 36
6 43 74 6 2 4 16
7 40 40 7 7 0 0
8 60 53 3 6 -3 9
SUM ∑d2=72

11. 𝑟𝑠 = 1 −
6∑𝑑2
𝑛 𝑛2 − 1
= 1 −
6 72
8 82−1
= 1 −
432
8 64−1
= 1 −
432
504
= 1 − 0 ⋅ 86
𝑟𝑠 = 0 ⋅ 14
Very low degree of positive correlation.