Spearman's coefficient of correlation - class-XI

When ranks are tied, Spearman's rank correlation formula requires a correction term: 1 - [6ΣD² + Σ(m³-m)/12]/[n(n²-1)], where m is the number of tied observations in each group. This adjusts for the fact that tied ranks affect the correlation calculation. Option D shows this correction (though there's a typo in 'frac').

Rank correlation is the measure of association or strength between the ranked variables. For example: the rank of this  numerical data 65, 25, 75, 69 would be 3, 4, 1, 2 respectively.

Spearman's rank cofficient : $R=1-\dfrac { 6\sum { { d } _{ i }^{ 2 } }  }{ n({ n }^{ 2 }-1) } $

Fact. If the variables are uncorrelated or independent then covariance
and coefficient of correlation between the variable both are equal to 0
i.e. $r _{xy}=Cov\left ( x, y \right )=0$ 

|
Sl.No
|
Rank Difference $(d)$
|
$d^2$

|
| --- | --- | --- |
|
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
|
-2
-4
-1
3
2
0
-2
3
3
-2
|
4
16
1
9
4
0
4
9
9
4
|

 $\sum d^2=60,\quad n=10$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}$

$r=1-\cfrac{6(60)}{10(10^2-1)}$

$r=1-\cfrac{360}{990}$

$r=0.6363....\approx 0.64$

Descending order arranged data will be as follows:

|
$X$
|
Rank
|
$Y$
|
Rank
|
$|d|$
|
$d^2$

|
| --- | --- | --- | --- | --- | --- |
|
48
33
40
9
16
16
65
25
16
57
|
3
5
4
10
7
7
1
6
7
2
|
13
13
24
6
15
4
20
9
6
19
|
5
5
1
8
4
10
2
7
8
3
|
2
0
3
2
3
3
1
1
1
1
|
4
0
9
4
9
9
1
1
1
1
|

$n=10,\quad \sum d^2=39$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 39}{10(10^2-1)}=1-\cfrac{234}{990}=0.76$

|
Club
|
Attendance
|
Rank
|
Position
|
$|d|$
|
$d^2$

|
| --- | --- | --- | --- | --- | --- |
|
A
B
C
D
E
F
G
H
I
J
K
L
|
27
30
18
25
32
12
19
11
32
12
12
15
|
4
3
7
5
1
9
6
12
1
9
9
8
|
1
2
3
4
5
6
7
8
9
10
11
12
|
3
1
4
1
4
3
1
4
8
1
2
4
|
9
1
16
1
16
9
1
16
64
1
4
16
|

$n=12,\quad \sum d^2=154$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 154}{12(12^2-1)}=1-\cfrac{924}{1716}=0.48$

|
Height(Father)
|
Rank
|
Height(Son)
|
Rank
|
$|d|$
|
$d^2$

|
| --- | --- | --- | --- | --- | --- |
|
65
66
67
67
68
69
70
72
|
8
7
5
5
4
3
2
1
|
67
68
65
68
72
72
69
71
|
7
5
8
5
1
1
4
3
|
1
2
3
0
3
2
2
2
 
|
1
4
9
0
9
4
4
4
|

$n=08,\quad \sum d^2=35$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 35}{8(8^2-1)}=1-\cfrac{210}{504}=0.58$

|
Students
|
History $(H)$
|
Mathematics$(M)$
|
Rank $(H)$
|
Rank$(M)$
|
$|d|$
|
$d^2$

|
| --- | --- | --- | --- | --- | --- | --- |
|
A
B
C
D
E
F
G
H
I
J
K
L
|
69
36
39
71
67
76
40
20
85
65
55
34
 
|
33
52
71
25
79
22
83
81
24
35
46
64
|
4
10
9
3
5
2
8
12
1
6
7
11
|
9
6
4
10
3
12
1
2
11
8
7
5
|
5
4
5
7
2
10
7
10
10
2
0
6
 
|
25
6
25
49
4
100
49
100
100
4
0
36
 
|

$n=12,\quad \sum d^2=508$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 508}{12(12^2-1)}=1-\cfrac{3048}{1716}=-0.77$

|
Statistic $(X)$
|
Mathematics $(Y)$
|
$XY$
|
$X^2$
|
$Y^2$

|
| --- | --- | --- | --- | --- |
|
1
3
3
4
5
6
7
8
9
10
|
2
4
1
5
3
9
7
10
6
8
|
2
12
3
20
15
54
49
80
54
80
|
1
9
9
16
25
6
49
64
81
100
|
4
16
1
25
9
81
49
100
36
64
|

 $\sum X=56,\quad \sum Y=55,\quad \sum XY=369,\quad \sum X^2=390,\quad \sum Y^2=385$

$N=10$

Cov$(x,y)=\cfrac{\sum XY}{N}-\cfrac{\sum X}{N}.\cfrac{\sum Y}{N}=\cfrac{369}{10}-\cfrac{56}{10}.\cfrac{55}{10}=6.1$

$\sigma _x=\sqrt{\cfrac{\sum X^2}{N}-\left(\cfrac{\sum X^2}{N}\right)^2}=\sqrt{\cfrac{390}{10}-\left(\cfrac{56}{10}\right)^2}=2.76$

$\sigma _y=\sqrt{\cfrac{\sum Y^2}{N}-\left(\cfrac{\sum Y^2}{N}\right)^2}=\sqrt{\cfrac{385}{10}-\left(5.5\right)^2}=2.87$

$r=\cfrac{Cov(x,y)}{\sigma _x.\sigma _y}=\cfrac{6.1}{2.87\times 2.76}=0.77$

The formula for speraman's rank coefficient is

Let the ranks of students obtained in Physics be $x$ and the ranks of students obtained in Chemistry be $y$.

Rank Correlation Coefficient,$r _s=1-\cfrac{6\sum{\rho _i^2}}{n(n^2-1)}$

The term tie is used in connection with rank order statistics. 

Rank correlation is the technique in which ranks are provided to the data after sorting it so sometimes, it becomes very difficult to assign ranks if the variables are large in numbers and also in this, ranks are taken inspite of original values.

Rank correlation coefficient is developed by Charles Spearman, a renowned statistician that measures the strength between the ranked variables.

Rank correlation technique measures the strength and direction between two variables by providing suitable ranks to the concerned variables, e.g. marks of students in a class can be easily ranked.

|
Competition
|
$1st$ Judge
|
Rank
|
$2nd$ Judge
|
Rank
|
$|d|$
|
$d^2$

|
| --- | --- | --- | --- | --- | --- | --- |
|
A
B
C
D
E
|
5.7
5.8
5.9
5.6
5.5
|
3
2
1
4
5
|
5.6
5.7
6.0
5.5
5.8
|
4
3
1
5
2
|
1
1
0
1
3
|
1
1
0
1
9
|

$n=5,\quad \sum d^2=12$

$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 12}{5(5^2-1)}=1-\cfrac{72}{240}=0.44$

|
Marks of Physics $(X)$
|
Marks in Maths$(Y)$
|
$XY$
|
$X^2$
|
$Y^2$

|
| --- | --- | --- | --- | --- |
|
35
23
47
17
10
43
9
6
28
|
30
33
45
23
8
49
12
4
31
|
1050
759
2115
391
80
2107
108
24
868
|
1225
529
2209
289
100
1849
81
35
784
|
900
1089
2025
529
64
2401
144
16
961
|

 $\sum X=218,\quad \sum Y=235,\quad \sum XY=7502,\quad \sum X^2=7102,\quad \sum Y^2=8129$

$N=09$

Cov$(x,y)=\cfrac{\sum XY}{N}-\cfrac{\sum X}{N}.\cfrac{\sum Y}{N}=\cfrac{7502}{09}-\cfrac{218}{9}.\cfrac{235}{9}=201.08$

$\sigma _x=\sqrt{\cfrac{\sum X^2}{N}-\left(\cfrac{\sum X^2}{N}\right)^2}=\sqrt{\cfrac{7102}{09}-\left(\cfrac{218}{09}\right)^2}=14.22$

$\sigma _y=\sqrt{\cfrac{\sum Y^2}{N}-\left(\cfrac{\sum Y^2}{N}\right)^2}=\sqrt{\cfrac{8129}{09}-\left(\cfrac{235}{9}\right)^2}=14.88$

$r=\cfrac{Cov(x,y)}{\sigma _x.\sigma _y}=\cfrac{201.08}{14.22\times 14.88}=0..95$

The correction factor in the rank correlation coefficient is given by $\sum m(m^2-1)$

Let the ranks of students obtained in Physics be $x$ and the ranks of students obtained in Chemistry be $y$.

Spearman's rank correlation r = 1 - (6 * sum(D^2)) / (n * (n^2 - 1)). Given n = 9 and sum(D^2) = 44. r = 1 - (6 * 44) / (9 * 80) = 1 - 264 / 720 = 1 - 0.366 = 0.633.