TOPIC 17

Spearman’s Rank Correlation Co-efficient

Spearman’s Rank Correlation Co-efficientIn some situations it is simply the order or rank that is important,

for example, judges at a dance competition might rank the dances in

order rather than awarding marks for each dance.

The formula for Spearman’s Rank Correlation Co-efficient is:

r = 1 - 6 ∑d2

n(n2 – 1)

Where: d = difference between the ranking of each item

n = the number of paired observations

Spearman’s Rank Correlation Co-efficientEXAMPLE 1

Eight dances are performed at a local dance competition. After seeing all the

dances, two judges rank the dances in order so the dance each judge likes best has

rank 1.

Calculate Spearman’s rank correlation co-efficient for this data.

Dance Judge 1, x Judge 2, y d=x–y d2

A 1 4 -3 9

B 7 3 4 16

C 2 6 -4 16

D 3 7 -4 16

E 5 1 4 16

F 4 8 -4 16

G 6 5 1 1

H 8 2 6 36

∑d2=126

Spearman’s Rank Correlation Co-efficientANSWER 1 r = 1 - 6 ∑d2

n(n2 – 1)= 1 – 6 x 126

8(82 – 1)= 1 – 756

8(64 – 1)= 1 – 756

8 x 63= 1 – 756

504= 1 – 1.5= -0.5

Spearman’s Rank Correlation Co-efficientSpearman’s Rank Correlation Co-efficient can take any value between +1 and -1.+1 = Perfect positive correlation 0 = No correlation-1 = Perfect negative correlation

So for Example 1, there is some negative correlation between the judges’ scores.

Dealing With Tied RanksSometimes more than one item is given the same rank. In this case new ranks are allocated to the items that have tied ranks.

When asked to calculate Spearman’s Rank Correlation Co-efficient of data given as numerical values, it is important to rank the data first.

Example 2

The marks of 12 pupils in geography and history essays as follows:

Geography, x History, y Rank, x Rank, y d = x - y d2

15 10 9.5 9 0.5 0.25

16 12 7.5 4 3.5 12.25

19 12 1 4 -3 9

17 13 5.5 1.5 4 16

17 11 5.5 7 -1.5 2.25

15 9 9.5 10 -0.5 0.25

18 11 3 7 -4 16

16 13 7.5 1.5 6 36

18 11 3 7 -4 16

18 12 3 4 -1 1

14 8 11 11 0 0

10 7 12 12 0 0

∑d2=109

Calculate Spearman’s Rank Correlation Co-efficient

ANSWER 2

Geography History

19 = 1 13 = ½(1 + 2) = 1.5

18 = ⅓(2 + 3 + 4) = 3 12 = ⅓(3 + 4 + 5) = 4

17 = ½(5 + 6) = 5.5 11 = ⅓(6 + 7 + 8) = 7

16 = ½(7 + 8) = 7.5 10 = 9

15 = ½(9 + 10) = 9.5 9 = 10

14 = 11 8 = 11

10 = 12 7 = 12

Spearman’s Rank Correlation Co-efficientr = 1 - 6 ∑d2

n(n2 – 1)= 1 – 6 x 109

12(122 – 1)= 1 – 654

12(144 – 1)= 1 – 654

12 x 143= 1 – 654

1716= 1 – 0.381= 0.619

Some positive correlation between the geography and history results.