In this session, we will discuss, how to calculate Spearman's correlation when two or more ranks are the same. We have considered multiple situations, various permutations and combinations to clarify the concept.

1. Spearman’s Rank order Correlation
Part-2: What happens when two or more
ranks are same?
Dr. Rajeev Kumar
M.S.W., (TISS, Mumbai), M.Phil., (CIP, Ranchi), UGC-JRF, Ph.D. (IIT Kharagpur)

2. A Case-1: In a college, 10 students obtained marks in Bangla and English. In Bangla 3 students obtained
94 marks and two students received 52 marks. In English, two students obtained 65 and two secured 52
marks. Here we will learn, how to calculate the Spearman correlation, when marks and ranks are similar.
We also want to see if the proficiency in English and Bangla are associated with each other.
Name of contestants Marks in Bangla (x) Marks in English (y)
Ramesh 52 50
Pooja 94 65
Joseph 72 90
Amir 94 52
Sonu 94 91
Rahul 29 52
Manisha 50 49
Santosh 52 65
Pratibha 60 58
Manoj 70 75

3. Making hypotheses
Alternative hypothesis
 Ha: There will be a significant correlation between the marks in Bangla and
the marks in English.
Null hypothesis
 H0: There will be no significant correlation between the marks in Bangla
and the marks in English.

4. Step-1: Obtain the ranks of all marks
Name of
contesta
nts
Marks in
Bangla
(x)
Marks
in
English
(y)
Ranks
(x)=d1
Revised
Ranks
(x)=d1
Ranks
(y)=d2
Revised
Ranks
(y)=d2
D=
d1- d2
D²
Ramesh 52 50 8 9
Pooja 94 65 1 5
Joseph 72 90 4 2
Amir 94 52 2 8
Sonu 94 91 3 1
Rahul 29 52 10 7
Manisha 50 49 9 10
Santosh 52 65 7 4
Pratibha 60 58 6 6
Manoj 70 75 5 3

5. Step 2: calculate the means of similar
ranks
Similar ranks are: (94, 94, 94) in marks of Bangla. Three students got 94
Calculate the mean of ranks = 1+2+3/ 3 = 6/3 =2
For (94, 94, and 94) the mean rank will be 2
Similarly (52 and 52) and same marks. Two students got 52
Calculate their mean = 8+7/2 15/2= 7.5. In Bangla for 52 marks, the mean
ranks will be 7.5
In English (65 and 65) and (52 and 52) are similar marks. Two students got 65
and 52 respectively
For 65, their mean rank will be = 4+5/2 = 9/2 = 4.5
For 52 the mean rank will be = 7+8/2 = 15/2 = 7.5

6. Step-3: assign the revised ranks
Name of
contesta
nts
Marks in
Bangla
(x)
Marks
in
English
(y)
Ranks
(x)=d1
Revised
Ranks
(x)=d1
Ranks
(y)=d2
Revised
Ranks
(y)=d2
D=
(revised)
(d1- d2)
D²
Ramesh 52 50 8 7.5 9 9
Pooja 94 65 1 2 5 4.5
Joseph 72 90 4 4 2 2
Amir 94 52 2 2 8 7.5
Sonu 94 91 3 2 1 1
Rahul 29 52 10 10 7 7.5
Manisha 50 49 9 9 10 10
Santosh 52 65 7 7.5 4 4.5
Pratibha 60 58 6 6 6 6
Manoj 70 75 5 5 3 3

7. Step-4: subtract the revised ranks and obtain ‘D’
Name of
contesta
nts
Marks in
Bangla
(x)
Marks
in
English
(y)
Ranks
(x)=d1
Revised
Ranks
(x)=d1
Ranks
(y)=d2
Revised
Ranks
(y)=d2
D=
(revised)
(d1- d2)
D²
Ramesh 52 50 8 7.5 9 9 -1.5
Pooja 94 65 1 2 5 4.5 -2.5
Joseph 72 90 4 4 2 2 2
Amir 94 52 2 2 8 7.5 -5.5
Sonu 94 91 3 2 1 1 1
Rahul 29 52 10 10 7 7.5 2.5
Manisha 50 49 9 9 10 10 -1
Santosh 52 65 7 7.5 4 4.5 3
Pratibha 60 58 6 6 6 6 0
Manoj 70 75 5 5 3 3 2

8. Step-5: Square the D and obtain the D² and do the summation of D² and obtain ΣD²
Name of
contesta
nts
Marks in
Bangla
(x)
Marks
in
English
(y)
Ranks
(x)=d1
Revised
Ranks
(x)=d1
Ranks
(y)=d2
Revised
Ranks
(y)=d2
D=
(revised)
(d1- d2)
D²
Ramesh 52 50 8 7.5 9 9 -1.5 2.25
Pooja 94 65 1 2 5 4.5 -2.5 6.25
Joseph 72 90 4 4 2 2 2 4
Amir 94 52 2 2 8 7.5 -5.5 30.25
Sonu 94 91 3 2 1 1 1 1
Rahul 29 52 10 10 7 7.5 2.5 6.25
Manisha 50 49 9 9 10 10 -1 1
Santosh 52 65 7 7.5 4 4.5 3 9
Pratibha 60 58 6 6 6 6 0 0
Manoj 70 75 5 5 3 3 2 4
ΣD²=64

9. Step-6: obtain the value of (m1, m2,
m3, and m4)
Marks in Bangla (94) repeated 3 times, so (m1=3)
Marks in Bangla (52) repeated 2 times, so (m2=2)
Marks in English (65) repeated 2 times, so (m3=2)
Marks in English (52) repeated 2 times, so (m4=2)
So here, we have 4 tied ranks

10. The formula for Spearman correlation
with tied ranks

11. Step-7: according to the number of
tied ranks, expand the equation

12. Step-8: put the values and calculate
the spearman correlation

13. Step-9: obtain the coefficient of
spearman correlation

14. Step-10: see the table of critical values

15. Step-11: Interpret the result
 The critical value of Spearman correlation for n=10 and alpha value =0.05
(p≤ 0.05) = 0.648
 Our test value of Spearman correlation is 0.59. Our test value is less than
the critical value (p≤ 0.05) =0.648.
 Therefore the coefficient of Spearman correlation is not significant at 0.05
(p≤ 0.05). Also, we should not check the value at 0.01.
Alternative hypothesis
 Ha: There will be a significant correlation between the marks in Bangla and
the marks in English.
Null hypothesis
 H0: There will be no significant correlation between the marks in Bangla
and the marks in English.

16. Step-12: the final conclusion
 There is no significant correlation between the marks in Bangla and marks
in English. It means, it is not necessary that those who are good in Bangla
will be good in English also. Therefore the alternative hypothesis is
rejected and null hypothesis is accepted
 In the graph also the points are very far from the best fit line, which shows
the weak correlation.

17. A story of dots who are away from the line of best fit
Alternative ways of explaining data and graph
X axis represent marks in Bangla and Y axis marks in
English. Here we can see, Joseph and Rahul scored more
in English than Bangla. Joseph and Rahul and displayed
along Y axis. Puja and Amir scored higher in Bangla than
English, they are also away from line of best fit.
Name of
contesta
nts
Marks in
Bangla
(x)
Marks
in
English
(y)
Ranks
(x)=d1
Ramesh 52 50 8
Pooja 94 65 1
Joseph 72 90 4
Amir 94 52 2
Sonu 94 91 3
Rahul 29 52 10
Manisha 50 49 9
Santosh 52 65 7
Pratibha 60 58 6
Manoj 70 75 5
Joseph
Rahul
Puja
Amir

18. Situation-2: The alternative way of calculating correlation with tied ranks. Start from step 3. Take
the revised ranks as (x) and (y)
Name
contesta
nts
Marks in
Bangla
(x)
Marks
in
English
(y)
Ranks
(x)=d1
Revised
Ranks
(x)=d1
Ranks
(y)=d2
Revised
Ranks
(y)=d2
D=
(revised)
(d1- d2)
D²
Ramesh 52 50 8 7.5 9 9
Pooja 94 65 1 2 5 4.5
Joseph 72 90 4 4 2 2
Amir 94 52 2 2 8 7.5
Sonu 94 91 3 2 1 1
Rahul 29 52 10 10 7 7.5
Manisha 50 49 9 9 10 10
Santosh 52 65 7 7.5 4 4.5
Pratibha 60 58 6 6 6 6
Manoj 70 75 5 5 3 3

19. Step-3: Lets try the same correlation with Pearson method.
Students X
(revised
rank of
marks in
Bangla)
(x-xˉ) Y
(revised
ranks of
marks in
English)
(Y-Yˉ) (x-xˉ) (Y-Yˉ) (x-xˉ)² (Y-Yˉ)²
Ramesh 7.5 9
Pooja 2 4.5
Joseph 4 2
Amir 2 7.5
Sonu 2 1
Rahul 10 7.5
Manisha 9 10
Santosh 7.5 4.5
Pratibha 6 6
Manoj 5 3
Xˉ= 5.5 Yˉ=5.5 = 48.75
Σ(x-xˉ) (Y-Yˉ)
=80
Σ(x-xˉ)²
=81.5
Σ(Y-Yˉ)²

20. Apply the formula of Pearson
correlation

21. Situation-4: lets see, what will happen, if we calculate the Pearson’s correlation using (x)=
marks in Bangla and (y) = marks in English. The Pearson’s correlation coefficient (r=0.48),
which is lower than previous two correlation values.
Name of contestants Marks in Bangla (x) Marks in English (y)
Ramesh 52 50
Pooja 94 65
Joseph 72 90
Amir 94 52
Sonu 94 91
Rahul 29 52
Manisha 50 49
Santosh 52 65
Pratibha 60 58
Manoj 70 75

22. Situation-4: lets see, what will happen, if there is only tied rank (52 and 52) in marks of Bangla.
Name of
contestan
ts
Marks in
Bangla (x)
Marks in
English
(y)
Ranks
(x)=d1
Revised
Ranks
(x)=d1
Ranks
(y)=d2
D=
(revised)
(d1- d2)
D²
Ramesh 52 50 8 7.5 9 -1.5 2.25
Pooja 94 65 1 1 5 -4 16
Joseph 72 90 4 4 2 2 4
Amir 95 52 2 2 8 -6 36
Sonu 96 91 3 3 1 2 4
Rahul 29 53 10 10 7 3 9
Manisha 50 49 9 9 10 -1 1
Santosh 52 66 7 7.5 4 3.5 12.25
Pratibha 60 58 6 6 6 0 0
Manoj 70 75 5 5 3 2 4
ΣD²=

23. What does result say?
Correlated coefficient decreased from previous
situations also. It means, there was no impact of
decreasing tied ranks.

24. Situation-5: lets see, what will happen, if we remove all the repeated ranks?
The calculated value r=0.50 shows that there is no impact of removing all the tied ranks.
Name of
contestant
s
Marks in
Bangla (x)
Marks in
English
(y)
Ranks
(x)=d1
Ranks
(y)=d2
D=
(revised)
(d1- d2)
D²
Ramesh 53 50 8 9 -1 1
Pooja 94 65 1 5 -4 16
Joseph 72 90 4 2 2 4
Amir 95 52 2 8 -6 36
Sonu 96 91 3 1 2 4
Rahul 29 53 10 7 3 9
Manisha 50 49 9 10 -1 1
Santosh 52 66 7 4 3 9
Pratibha 60 58 6 6 0 0
Manoj 70 75 5 3 2 4
ΣD²= 84

25. The real case study based on the true data. There are two rankings of
SAARC countries. HDI ranking conducted by UNDP and corruption
rating done by transparency international in 2018 among 176
countries.
Countries HDI
ranking
(X)
TI ranking
(y)
Ranking
ranking
d1
Ranking
ranking
d2
D= d1-d2 D²
India 129 79 6 7 -1 1
Nepal 147 131 3 3 0 0
Bhutan 134 27 5 8 -3 9
Bangladesh 135 145 4 2 2 4
Pakistan 152 116 2 4 -2 4
Afghanistan 170 169 1 1 0 0
Maldives 104 95 7 5 -2 4
Shri Lanka 71 96 8 6 2 4
ΣD²=26

26. What does result say?
The calculated (r=0.70) which is less than the critical value of Spearman’ s at 0.05
(P≤0.05) for n=8 (0.73). Though there is no significant correlation, but it is tend to be
significant. There is no statistical significance, but there is practical significance.
Developed countries are less corrupt and honest countries are more developed.
Therefore, therefore is a positive correlation between honesty and development.

27. What does graph say? The data are little far from the best
fit line, but it a rising trend in positive direction. There is
further chance that dots can come closure to the line.

28. Thanks for your kind attention. Keep
learning. In the next session, we will
learn the partial correlation.