# Rank Correlation

**ISC – Maths – XII – Important Q.A. **

** Rank Correlation. **

**Q.1. Calculate Spearman’s rank correlation coefficient between the advertisement cost and sales from the following date : **

Advertisement cost Rs. (in thousand) |
39 |
65 |
62 |
90 |
82 |
75 |
25 |
98 |
36 |
78 |

Sales Rs. (in lakhs) |
47 |
53 |
58 |
86 |
62 |
68 |
60 |
91 |
51 |
84 |

** **

**Solution : –**

Advertisement cost Rs. (in thousand) X | Sales Rs. (in lakhs) Y | R_{x} |
R_{y} |
D = R_{x} – R_{y} |
D^{2} |

39 | 47 | 8 | 10 | – 2 | 4 |

65 | 53 | 6 | 8 | – 2 | 4 |

62 | 58 | 7 | 7 | 0 | 0 |

90 | 86 | 2 | 2 | 0 | 0 |

82 | 62 | 3 | 5 | – 2 | 4 |

75 | 68 | 5 | 4 | 1 | 1 |

25 | 60 | 10 | 6 | 4 | 16 |

98 | 91 | 1 | 1 | 0 | 0 |

36 | 51 | 9 | 9 | 0 | 0 |

78 | 84 | 4 | 3 | 1 | 1 |

Total → | 30 |

Spearman’s Rank Correlation Coefficient = r = 1 – [6 ∑ D^{2} ]/[N(N^{2} – 1)]

= 1 – [6 × 30]/[10(10^{2} – 1)]

= 1 – [180/(10 × 99)]

= 1 – (2/11) = 9/11 = 0.818

= 0.82. [**Ans.**]

**Q.2. Internal and External assessment were conducted on a group of 10 students who were studying in a postgraduate class in a college. The following marks were obtained in the assessment : **

Roll no. of students: |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |

Internal Assessment: |
45 |
62 |
67 |
32 |
12 |
38 |
47 |
68 |
42 |
85 |

External Assessment: |
39 |
48 |
65 |
32 |
20 |
35 |
45 |
77 |
30 |
62 |

**Find the Spearman’s Rank Correlation Coefficient and comment on the result. **

**Solution : –**

Roll No. | Internal x | External y | Rank of x | Rank of y | d = x – y | d^{2} |

1 | 45 | 39 | 6 | 6 | 0 | 0 |

2 | 62 | 48 | 4 | 4 | 0 | 0 |

3 | 67 | 65 | 3 | 2 | 1 | 1 |

4 | 32 | 32 | 9 | 8 | 1 | 1 |

5 | 12 | 20 | 10 | 10 | 0 | 0 |

6 | 38 | 35 | 8 | 7 | 1 | 1 |

7 | 47 | 45 | 5 | 5 | 0 | 0 |

8 | 68 | 77 | 2 | 1 | 1 | 1 |

9 | 42 | 30 | 7 | 9 | –2 | 4 |

10 | 85 | 62 | 1 | 3 | –2 | 4 |

Total → | 12 |

Here, n = 10, r = 1 – [6∑d^{2}]/[n(n^{2} – 1)]

= 1 – (6×12)/[10(99)]

= 1 – 0.07 = 0.93 . [**Ans.**]

**Comment :** There is a high positive correlation between the internal and external

assessment.

**Q.3. The mathematical aptitude score (MAS) of ten computer programmers with job performance rating (JPR) is given below. Calculate Spearman’s rating of rank correlation and state whether those who have aptitude for maths are likely to be better programmers. **

Person |
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |

MAS |
2 |
5 |
0 |
4 |
3 |
1 |
6 |
8 |
7 |
9 |

JPR |
8 |
16 |
8 |
9 |
5 |
4 |
3 |
17 |
8 |
12 |

** **

**Solution : –**

Person | MAS | JPR | R_{1} |
R_{2} |
d = R_{1} – R_{2} |
d^{2} |

A | 2 | 8 | 8 | 6 | 2 | 4 |

B | 5 | 16 | 5 | 2 | 3 | 9 |

C | 0 | 8 | 10 | 6 | 4 | 16 |

D | 4 | 9 | 6 | 4 | 2 | 4 |

E | 3 | 5 | 7 | 8 | – 1 | 1 |

F | 1 | 4 | 9 | 9 | 0 | 0 |

G | 6 | 3 | 4 | 10 | – 6 | 36 |

H | 8 | 17 | 2 | 1 | 1 | 1 |

I | 7 | 8 | 3 | 6 | – 3 | 9 |

J | 9 | 12 | 1 | 3 | – 2 | 4 |

Total → | 84 |

Now, r = 1 – 6 [∑d^{2} + 1/12(32 – 3)]/n(n^{2} – 1)

= 1 – 6 (84 + 2)/[10(100) – 1]

= 1 – (6×86)/990

= (990 – 516)/990

= 474/990 = 0.478 = 0.48. [**Ans.**]

**Q.4. The marks obtained by nine students in Physics and Mathematics are given below : **

Physics |
35 |
23 |
47 |
17 |
10 |
43 |
9 |
6 |
28 |

Mathematics |
30 |
33 |
45 |
23 |
8 |
49 |
12 |
4 |
31 |

** Calculate Spearman’s coefficient of rank correlation and interpret the result. **

**Solution : –**

Physics | Mathematics | R_{1} |
R_{2} |
R_{1} – R_{2} = d |
d^{2} |

35 | 30 | 3 | 5 | –2 | 4 |

23 | 33 | 5 | 3 | 2 | 4 |

47 | 45 | 1 | 2 | –1 | 1 |

17 | 23 | 6 | 6 | 0 | 0 |

10 | 8 | 7 | 8 | –1 | 1 |

43 | 49 | 2 | 1 | 1 | 1 |

9 | 12 | 8 | 7 | 1 | 1 |

6 | 4 | 9 | 9 | 0 | 0 |

28 | 31 | 4 | 4 | 0 | 0 |

Total → | 12 |

Here, n = 9,

r = 1 – [6∑d^{2}]/[n(n^{2} – 1)]

= 1 – [6×12]/[9(81 – 1)]

= 1 – 72/[9×80]

= 1 – 1/10 = 9/10 = +0.9. [**Ans.**]

**Q.5. The following table gives the two kinds of assessment of ten post-graduate students performance : **

Students |
Marks in Internal Assessment |
Marks in External Assessment |

1 |
45 |
39 |

2 |
62 |
48 |

3 |
67 |
65 |

4 |
32 |
32 |

5 |
12 |
20 |

6 |
38 |
35 |

7 |
47 |
45 |

8 |
67 |
77 |

9 |
42 |
30 |

10 |
85 |
62 |

**Find Spearman’s coefficient of rank correlation and interpret the result. **

**Solution :–**

Students S.No. | Marks in Internal Assessment x | Marks in External Assessment y | Rank of x | Rank of y | x – y = d | d^{2} |

1 | 45 | 39 | 6 | 6 | 0 | 0 |

2 | 62 | 48 | 4 | 4 | 0 | 0 |

3 | 67 | 65 | 2.5 | 2 | 0.5 | 0.25 |

4 | 32 | 32 | 9 | 8 | 1 | 1 |

5 | 12 | 20 | 10 | 10 | 0 | 0 |

6 | 38 | 35 | 8 | 7 | 1 | 1 |

7 | 47 | 45 | 5 | 5 | 0 | 0 |

8 | 67 | 77 | 2.5 | 1 | 1.5 | 2.25 |

9 | 42 | 30 | 7 | 9 | – 2 | 4 |

10 | 85 | 62 | 1 | 3 | – 2 | 4 |

Total → | 12.50 |

Here n = 10, r = 1 – [6∑d^{2}]/[∑n(n^{2} – 1)]

= 1 – [6×12.50]/[10×99]

= 1 – 0.008 = 0.992 [**Ans.**]

**Q.6. A psychologist selected a random sample of 22 students. He grouped them in 11 pairs so that students in a pair have nearly equal scores in an intelligent test. In each pair one student was taught by method A and the other by method B and examined after the course. The marks obtained by them are tabulated below : **

Pair |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |

Method A(marks) |
24 |
29 |
19 |
14 |
30 |
19 |
27 |
30 |
20 |
28 |
11 |

Method B(marks) |
37 |
35 |
16 |
26 |
23 |
27 |
19 |
20 |
16 |
11 |
21 |

**Find the rank correlation coefficient. **

**Solution : –**

Pair | Method A | Method B | Rank x | Rank y | d = x – y | d^{2} |

1 | 24 | 37 | 6 | 1 | 5 | 25 |

2 | 29 | 35 | 3 | 2 | 1 | 1 |

3 | 19 | 16 | 8.5 | 9.5 | –1 | 1 |

4 | 14 | 26 | 10 | 4 | 6 | 36 |

5 | 30 | 23 | 1.5 | 5 | –3.5 | 12.2 |

6 | 19 | 27 | 8.5 | 3 | 5.5 | 30.2 |

7 | 27 | 19 | 5 | 8 | –3 | 9.2 |

8 | 30 | 20 | 1.5 | 7 | –5.5 | 30.2 |

9 | 20 | 16 | 7 | 9.5 | –2.5 | 6.2 |

10 | 28 | 11 | 4 | 11 | –7 | 49 |

11 | 11 | 21 | 11 | 6 | 5 | 25 |

Total → | 225 |

Correlation Coefficient r = 1 – (6∑d^{2})/[n(n^{2} – 1)]

= 1 – (6×225)/(11×120)

= 1 – 45/44 = 1 – 1.0227 = – 0.0227. [**Ans.**]

**Q.7. An examination of 8 applicants for a clerical post was taken by a firm. The marks obtained by the applicants in the Reasoning and Aptitude tests are given below : **

Applicants |
A |
B |
C |
D |
E |
F |
G |
H |

Reasoning Test |
20 |
28 |
15 |
60 |
40 |
80 |
20 |
12 |

Aptitude Test |
30 |
50 |
40 |
20 |
10 |
60 |
30 |
30 |

** Calculate the Spearman’s coefficient of rank correlation from the data given above. **

**Solution : –**

Reasoning Test | Aptitude Test | R_{1} |
R_{2} |
D = R_{1} – R_{2} |
D^{2} |

20 | 30 | 5.5 | 5 | 0.5 | 0.25 |

28 | 50 | 4 | 2 | 2 | 4 |

15 | 40 | 7 | 3 | 4 | 16 |

60 | 20 | 2 | 7 | – 5 | 25 |

40 | 10 | 3 | 8 | – 5 | 25 |

80 | 60 | 1 | 1 | 0 | 0 |

20 | 30 | 5.5 | 5 | 0.5 | 0.25 |

12 | 30 | 8 | 5 | 3 | 9 |

ΣD^{2} = 79.5 |

Using , r = 1 – 6[ΣD2 + 1/12(m3 – m) + 1/12(m3 – m)]/[n(n2 – 1)]

= 1 – 6[79.5 + 6/12 + 24/12]/(8 × 63)

= 1 – 6[82]/(8 × 63)

= 1 – 3/126 = 0.02. [**Ans.**]