Explanation
Physics$$(P)$$
Mathematics $$(M)$$
Rank $$(P)$$
$$|d|$$
$$d^2$$
48
60
72
62
56
40
39
52
30
78
65
70
38
54
32
31
6
3
1
2
4
7
8
5
9
0
$$n=9,\quad \sum d^2=40$$
$$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{40\times 6}{9(9^2-1)}=1-\cfrac{240}{720}=0.66$$
Sl.No
Rank Difference $$(d)$$
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
-2
-4
-1
16
$$\sum d^2=60,\quad n=10$$
$$r=1-\cfrac{6\sum d^2}{n(n^2-1)}$$
$$r=1-\cfrac{6(60)}{10(10^2-1)}$$
$$r=1-\cfrac{360}{990}$$
$$r=0.6363....\approx 0.64$$
Students
History $$(H)$$
Mathematics$$(M)$$
Rank $$(H)$$
Rank$$(M)$$
A
B
C
D
E
F
G
H
I
J
K
L
69
36
71
67
76
20
85
55
34
33
25
79
22
83
81
24
35
46
64
10
12
11
49
100
$$n=12,\quad \sum d^2=508$$
$$r=1-\cfrac{6\sum d^2}{n(n^2-1)}=1-\cfrac{6\times 508}{12(12^2-1)}=1-\cfrac{3048}{1716}=-0.77$$
Since $$r<0,$$ we can say that a very good student of history is a very bad student in mathematics.
Statistic $$(X)$$
Mathematics $$(Y)$$
$$XY$$
$$X^2$$
$$Y^2$$
15
80
$$\sum X=56,\quad \sum Y=55,\quad \sum XY=369,\quad \sum X^2=390,\quad \sum Y^2=385$$
$$N=10$$
Cov$$(x,y)=\cfrac{\sum XY}{N}-\cfrac{\sum X}{N}.\cfrac{\sum Y}{N}=\cfrac{369}{10}-\cfrac{56}{10}.\cfrac{55}{10}=6.1$$
$$\sigma_x=\sqrt{\cfrac{\sum X^2}{N}-\left(\cfrac{\sum X^2}{N}\right)^2}=\sqrt{\cfrac{390}{10}-\left(\cfrac{56}{10}\right)^2}=2.76$$
$$\sigma_y=\sqrt{\cfrac{\sum Y^2}{N}-\left(\cfrac{\sum Y^2}{N}\right)^2}=\sqrt{\cfrac{385}{10}-\left(5.5\right)^2}=2.87$$
$$r=\cfrac{Cov(x,y)}{\sigma_x.\sigma_y}=\cfrac{6.1}{2.87\times 2.76}=0.77$$
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