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Class 11 Applied Maths Chapter 13 Solutions
Descriptive Statistics
EXERCISE- 13.7
Q.1 Find the spearman’s rank correlation between marks in Mathematics and Statistics obtained by 10 students:
| Maths in Mathematics | 80 | 38 | 95 | 30 | 74 | 84 | 91 | 60 | 66 | 40 |
| Marks in Statistics | 85 | 50 | 92 | 58 | 70 | 65 | 88 | 56 | 52 | 46 |
Ans.
| Maths in Mathematics | 80 | 38 | 95 | 30 | 74 | 84 | 91 | 60 | 66 | 40 |
| Marks in Statistics | 85 | 50 | 92 | 58 | 70 | 65 | 88 | 56 | 52 | 46 |
| Rank 1 | 4 | 9 | 1 | 10 | 5 | 3 | 2 | 7 | 6 | 8 |
| Rank 2 | 3 | 9 | 1 | 6 | 4 | 5 | 2 | 7 | 8 | 10 |
| Difference (D) | 1 | 0 | 0 | 4 | 1 | -2 | 0 | 0 | -2 | -2 |
| D^2 | 1 | 0 | 0 | 16 | 1 | 4 | 0 | 0 | 4 | 4 |
n = 10, ΣD^2 = 30
r = 1 – [6ΣD^2/n(n^2-1)]
= 1 – [(6×30)/10(10^2-1)]
= 1 – 18/99
= (11 – 2)/11
= 9/11
= +0.82
Q.2 The final positions of twelve clubs in a football league and the average attendances at their home matches were as follows:
| Club | A | B | C | D | E | F | G | H | I | J | K | L |
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Attendance (thousands) | 27 | 30 | 18 | 25 | 32 | 12 | 19 | 11 | 32 | 12 | 12 | 15 |
Calculate a coefficient of correlation by ranks and comment on your result. What other factors do you think might affect the number of spectators apart from the positions of the clubs in the league?
Ans.
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Attendance (thousands) | 27 | 30 | 18 | 25 | 32 | 12 | 19 | 11 | 32 | 12 | 12 | 15 |
| Rank 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Rank 2 | 9 | 10 | 6 | 8 | 11.5 | 3 | 7 | 1 | 11.5 | 3 | 3 | 5 |
| Difference (D) | -8 | -8 | -3 | -4 | -6.5 | 3 | 0 | 7 | -2.5 | 7 | 8 | 7 |
| D^2 | 64 | 64 | 9 | 16 | 42.25 | 9 | 0 | 49 | 6.25 | 49 | 64 | 49 |
n = 12, ΣD^2 = 421.5
r = 1 – [6ΣD^2 + 1/2(m^3-m)+(m^3-m)/n(n^2-1)]
= 1 – [6(421.5 + 2 + 0.5)/12(12^2-1)]
= 1 – 424/(2×143)
= 1 – 212/143
= 1 – 1.49
= -0.49
Moderate negative correlation.
Q.3 The following data relate to the number of vehicles owned and road deaths for the populations of 12 countries:
| Vehicles per 100 population | 30 | 31 | 32 | 30 | 46 | 30 | 19 | 35 | 40 | 46 | 57 | 30 |
| Road deaths per 100000 population | 30 | 14 | 30 | 23 | 32 | 26 | 20 | 21 | 23 | 30 | 35 | 26 |
Calculate Spearman’s rank correlation coefficient and comment on the result.
Ans.
| Vehicles per 100 population | 30 | 31 | 32 | 30 | 46 | 30 | 19 | 35 | 40 | 46 | 57 | 30 |
| Road deaths per 100000 population | 30 | 14 | 30 | 23 | 32 | 26 | 20 | 21 | 23 | 30 | 35 | 26 |
| Rank 1 | 9.5 | 7 | 6 | 9.5 | 2.5 | 9.5 | 12 | 5 | 4 | 2.5 | 1 | 9.5 |
| Rank 2 | 4 | 12 | 4 | 8.5 | 2 | 6.5 | 11 | 10 | 8.5 | 4 | 0 | 6.5 |
| D = R1-R2 | 5.5 | -5 | 2 | 1 | 0.5 | 3 | 1 | -5 | -4.5 | -1.5 | 0 | 3 |
| D^2 | 30.25 | 25 | 4 | 1 | 0.25 | 9 | 1 | 25 | 20.25 | 2.25 | 0 | 9 |
n = 12, ΣD^2 = 127
r = 1 – [6ΣD^2 + 1/2(m^3-m)+(m^3-m)/n(n^2-1)]
= 1 – [6(127 + 1/2(4^3-4) + 0.5 +0.5 + 0.5 + 2)/12(12^2-1)]
= 1 – [(127 + 5 + 2 + 1.5)/2×143]
= 1 – 135.5/286
= (286-135.5)/286
= 150.5/289
= + 0.53
Moderate positive correlation.
Q.4 The following table gives the two kinds of assessment of ten post-graduate students’ performance:
| Marks | Marks | |
| Students | Internal Assessment | 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.
Ans.
| Internal Assessment | External Assessment | Rank 1 | Rank 2 | D = R1-R2 | D^2 |
| 45 | 37 | 6 | 6 | 0 | 0 |
| 62 | 48 | 4 | 4 | 0 | 0 |
| 67 | 65 | 2.5 | 2 | 0.5 | 0.25 |
| 32 | 32 | 9 | 8 | 1 | 1 |
| 12 | 20 | 10 | 10 | 0 | 0 |
| 38 | 35 | 8 | 7 | 1 | 1 |
| 47 | 45 | 5 | 5 | 0 | 0 |
| 67 | 77 | 2.5 | 1 | 1.5 | 2.25 |
| 42 | 30 | 7 | 9 | -2 | 4 |
| 85 | 62 | 1 | 3 | -2 | 4 |
n = 10, ΣD^2 = 12.5
r = 1 – [6ΣD^2 + 1/2(m^3-m)/n(n^2-1)]
= 1 – [6(12.5 + 0.5)/(10×99)]
= 1 – [(6×13)/(10×99)]
= 1 – 13/165
= (165-13)/165
= 152/165
= 0.92
Q.5 Spearman’s coefficient of rank correlation between sales and profits of a group of firms was found to be 0.8. If the sum of the squares of the difference in ranks is 33, find the number of firms in the group.
Ans. Given, r = 0.8, n = ?, ΣD^2 = 33
r = 1 – [6ΣD^2/n(n^2-1)]
0.8 = 1 – [(6×33)/n(n^2-1)]
(6x3x11)/n(n^2-1) = 1 – 0.8
(2x3x3x11)/n(n^2-1) = 0.2
(2x9x11)/n(n^2-1 = 2/10
10x9x11 = n(n-1)(n+1)
Therefore, n = 10
Q.6 In a beauty contest, the spearman’s coefficient of rank correlation between rankings given by two judges of 10 contestants was found to be 0.5. Later, it was discovered that difference in rankings in one case was wrongly taken as 3 instead of 7. Find the correct coefficient cost and sales correlation.
Ans. n = 10, r = 0.5
r = 1 – [6ΣD^2/n(n^2-1)]
0.5 = 1 – [(6xΣD^2)/10(10^2-1)]
(6xΣD^2)/(10×99) = 1 – 0.5
(2ΣD^2)/(10×33) = 0.5
ΣD^2 = (0.5x10x33)/2
ΣD^2 = (5×33)/2
ΣD^2 = 165/2 = 82.5
Incorrect ΣD^2 = 82.5
Correct ΣD^2 = 82.5 – 9 + 49
= 122.5
Correct r = 1 – [(6×122.5)/10(10^2-1)]
= 1 – 367.5/(5×99)
= (495 – 367.5)/495
= 127.5/495
= 0.257
Q.7 Calculate Spearman’s rank correlation coefficient between the advertisement cost and sales from the following data:
| Advertisement cost (Rs. in thousand) | 39 | 65 | 62 | 90 | 82 | 75 | 25 | 98 | 36 | 78 |
| Sales (Rs. in lakh) | 47 | 53 | 58 | 86 | 62 | 68 | 60 | 91 | 51 | 84 |
Ans.
| Advertisement cost (Rs. in thousand) | 39 | 65 | 62 | 90 | 82 | 75 | 25 | 98 | 36 | 78 |
| Sales (Rs. in lakh) | 47 | 53 | 58 | 86 | 62 | 68 | 60 | 91 | 51 | 84 |
| Rank 1 | 8 | 6 | 7 | 2 | 3 | 5 | 10 | 1 | 9 | 4 |
| Rank 2 | 10 | 8 | 7 | 2 | 5 | 4 | 6 | 1 | 9 | 3 |
| D = R1-R2 | -2 | -2 | 0 | 0 | -2 | 1 | 4 | 0 | 0 | 1 |
| D^2 | 4 | 4 | 0 | 0 | 4 | 1 | 16 | 0 | 0 | 1 |
n = 10, ΣD^2 = 30
r = 1 – [6ΣD^2/n(n^2-1)]
= 1 – [(6×30)/(10×99)]
= 1 – 2/11
= (11 – 2)/11
= 9/11
= 0.828 = 0.83
Q.8 The mathematical aptitude test score (MAS) of ten computer programmers 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 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 |
Ans.
| MAS | 2 | 5 | 0 | 4 | 3 | 1 | 6 | 8 | 7 | 9 |
| JPR | 8 | 16 | 8 | 9 | 5 | 4 | 3 | 17 | 8 | 12 |
| Rank 1 | 8 | 5 | 10 | 6 | 7 | 9 | 4 | 2 | 3 | 1 |
| Rank 2 | 6 | 2 | 6 | 4 | 8 | 9 | 10 | 1 | 6 | 3 |
| D = R1-R2 | 2 | 3 | 4 | 2 | -1 | 0 | -6 | 1 | -3 | -2 |
| D^2 | 4 | 9 | 16 | 4 | 1 | 0 | 36 | 1 | 9 | 4 |
n = 10, ΣD^2 = 84
r = 1 – [6ΣD^2/n(n^2-1)]
= 1 – 6[84+2]/10(10^2-1)]
= 1 – (6×86)/(10×99)
= (165 – 86)/165
= 79/165
= 0.48
FAQ’s related to Class 11 Applied Maths Chapter 13 on Descriptive Statistics:
Q.1 What is Descriptive Statistics?
Ans. Descriptive Statistics involves methods for summarizing and organizing the information in a data set. This includes measures of central tendency (mean, median, mode) and measures of variability (range, variance, standard deviation).
Q.2 What are Measures of Central Tendency?
Ans. Measures of central tendency are statistical metrics used to determine the center or typical value of a data set. They include:
- Mean: The average of all data points.
- Median: The middle value when data points are ordered.
- Mode: The most frequently occurring value(s) in the data set.
These are a few Frequently Asked Questions relating to Class 11 Applied Maths Chapter 13
In Class 11 Applied Maths chapter 13, you will explore fascinating topics that form the backbone of practical problem-solving techniques. Through clear explanations, illustrative examples, and step-by-step solutions, you’ll grasp complex concepts effortlessly. Whether you’re preparing for exams or simply eager to deepen your mathematical understanding, Class 11 Applied Maths Chapter 13 promises an enriching learning experience that will set you on the path to success. Class 11 Applied Maths Chapter 13, we delve deep into advanced mathematical concepts that are crucial for understandin
