RD Sharma Solutions for Class 11 Maths Chapter 32 Statistics 1. Calculate the mean deviation about the median of the following observation : i 3011, 2780, 3020, 2354, 3541, 4150, 5000 ii 38, 70, 48, 34, 42, 55, 63, 46, 54, 44 iii 34, 66, 30, 38, 44, 50, 40, 60, 42, 51 iv 22, 24, 30, 27, 29, 31, 25, 28, 41, 42 v 38, 70, 48, 34, 63, 42, 55, 44, 53, 47 Solution: i 3011, 2780, 3020, 2354, 3541, 4150, 5000 To calculate the Median M, let us arrange the numbers in ascending order. Median is the middle number of all the observation. 2354, 2780, 3011, 3020, 3541, 4150, 5000 So, Median = 3020 and n = 7 By using the formula to calculate Mean Deviation, \(MD = \frac{1}{n}\sum_{i=1}^{n}|d_{i}|\) xi |di| = |xi – 3020| 3011 9 2780 240 3020 0 2354 666 3541 521 4150 1130 5000 1980 Total 4546 \(MD = \frac{1}{n}\sum_{i=1}^{n}|d_{i}|\) = 1/7 × 4546 = 649.42 ∴ The Mean Deviation is 649.42. ii 38, 70, 48, 34, 42, 55, 63, 46, 54, 44 To calculate the Median M, let us arrange the numbers in ascending order. Median is the middle number of all the observation. Aakash Institute
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RD Sharma Solutions for Class 11 Maths Chapter 32 Statistics
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RD Sharma Solutions for Class 11 Maths Chapter 32 Statistics
1. Calculate the mean deviation about the median of the following observation :i 3011, 2780, 3020, 2354, 3541, 4150, 5000 ii 38, 70, 48, 34, 42, 55, 63, 46, 54, 44 iii 34, 66, 30, 38, 44, 50, 40, 60, 42, 51 iv 22, 24, 30, 27, 29, 31, 25, 28, 41, 42 v 38, 70, 48, 34, 63, 42, 55, 44, 53, 47 Solution:
i 3011, 2780, 3020, 2354, 3541, 4150, 5000 To calculate the Median M, let us arrange the numbers in ascending order. Median is the middle number of all the observation.
ii 38, 70, 48, 34, 42, 55, 63, 46, 54, 44 To calculate the Median M, let us arrange the numbers in ascending order. Median is the middle number of all the observation.
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34, 38, 42, 44, 46, 48, 54, 55, 63, 70
Here the Number of observations are Even then Median = 46+48/2 = 47 Median = 47 and n = 10
iii 34, 66, 30, 38, 44, 50, 40, 60, 42, 51 To calculate the Median M, let us arrange the numbers in ascending order. Median is the middle number of all the observation.
30, 34, 38, 40, 42, 44, 50, 51, 60, 66
Here the Number of observations are Even then Median = 42+44/2 = 43 Median = 43 and n = 10
iv 22, 24, 30, 27, 29, 31, 25, 28, 41, 42 To calculate the Median M, let us arrange the numbers in ascending order. Median is the middle number of all the observation.
22, 24, 25, 27, 28, 29, 30, 31, 41, 42
Here the Number of observations are Even then Median = 28+29/2 = 28.5 Median = 28.5 and n = 10
v 38, 70, 48, 34, 63, 42, 55, 44, 53, 47 To calculate the Median M, let us arrange the numbers in ascending order. Median is the middle number of all the observation.
34, 38, 43, 44, 47, 48, 53, 55, 63, 70
Here the Number of observations are Even then Median = 47+48/2 = 47.5 Median = 47.5 and n = 10
2. Calculate the mean deviation from the mean for the following data : i 4, 7, 8, 9, 10, 12, 13, 17 ii 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17 iii 38, 70, 48, 40, 42, 55, 63, 46, 54, 44 iv 36, 72, 46, 42, 60, 45, 53, 46, 51, 49 v 57, 64, 43, 67, 49, 59, 44, 47, 61, 59 Solution:
∴ The Mean Deviation of set 1 is 240 and set 2 is 457.14
4. The lengths incm of 10 rods in a shop are given below: 40.0, 52.3, 55.2, 72.9, 52.8, 79.0, 32.5, 15.2, 27.9, 30.2 i Find the mean deviation from the median. ii Find the mean deviation from the mean also. Solution:
i Find the mean deviation from the median Let us arrange the data in ascending order,
5. In question 1iii, iv, v find the number of observations lying between \(\overline{X} – M.D\) and \(\overline{X} + M.D\), where M.D. is the mean deviation from the mean. Solution:
iii 34, 66, 30, 38, 44, 50, 40, 60, 42, 51 We know that,
5. Find the mean deviation from the mean and from a median of the following distribution:
Marks 0-10 10-20 20-30 30-40 40-50
No. of students 5 8 15 16 6
Solution:
To find the mean deviation from the median, firstly let us calculate the median.
N = 50
So, N/2 = 50/2 = 25
The cumulative frequency just greater than 25 is 58, and the corresponding value of x is 28
So, Median = 28
By using the formula to calculate Mean,
= 1350/50
= 27
Class Interval
xi fi Cumulative Frequency
|di| = |xi – Median|
fi |di| FiXi |Xi – Mean|
Fi |Xi – Mean|
0-10 5 5 5 23 115 25 22 110
10-20 15 8 13 13 104 120 12 96
20-30 25 15 28 3 45 375 2 30
30-40 35 16 44 7 112 560 8 128
40-50 45 6 50 17 102 270 18 108
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N = 50
Total = 478
Total = 1350
Total = 472
Mean deviation from Median = 478/50 = 9.56
And, Mean deviation from Median = 472/50 = 9.44
∴ The Mean Deviation from the median is 9.56 and from mean is 9.44.
EXERCISE 32.4 PAGE NO: 32.28 1. Find the mean, variance and standard deviation for the following data: i 2, 4, 5, 6, 8, 17
ii 6, 7, 10, 12, 13, 4, 8, 12
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2. The variance of 20 observations is 4. If each observation is multiplied by 2, find the variance of the resulting observations.
Solution:
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3. The variance of 15 observations is 4. If each observation is increased by 9, find the variance of the resulting observations.
Solution:
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4. The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.
Solution:
5. The mean and standard deviation of 6 observations are 8 and 4 respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.
Solution:
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∴ The mean of new observation is 24 and Standard deviation of new observation is 12.
6. The mean and variance of 8 observations are 9 and 9.25 respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.
Solution:
EXERCISE 32.5 PAGE NO: 32.37 1. Find the standard deviation for the following distribution:
x: 4.5 14.5 24.5 34.5 44.5 54.5 64.5
f: 1 5 12 22 17 9 4
Solution: Aak
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= 100
1.857–0.0987
= 100
1.7583
Var X = 175.83 Aak
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2. Table below shows the frequency f with which ‘x’ alpha particles were radiated from a diskette
x: 0 1 2 3 4 5 6 7 8 9 10 11 12
f: 51 203 383 525 532 408 273 139 43 27 10 4 2
Calculate the mean and variance.
Solution:
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3. Find the mean, and standard deviation for the following data: i
Year render: 10 20 30 40 50 60
No. of persons cumulative 15 32 51 78 97 109
Solution:
ii
Marks: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
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Frequency: 1 6 6 8 8 2 2 3 0 2 1 0 0 0 1
Solution:
4. Find the standard deviation for the following data:
i
x: 3 8 13 18 23
f: 7 10 15 10 6
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Solution:
ii
x: 2 3 4 5 6 7
f: 4 9 16 14 11 6
Solution:
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EXERCISE 32.6 PAGE NO: 32.41 1. Calculate the mean and S.D. for the following data:
Expenditure in₹: 0-10 10-20 20-30 30-40 40-50
Frequency: 14 13 27 21 15
Solution:
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2. Calculate the standard deviation for the following data:
4. A student obtained the mean and standard deviation of 100 observations as 40 and 5.1 respectively. It was later found that one observation was wrongly copied as 50, the correct figure is 40. Find the correct mean and S.D.
Solution:
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5. Calculate the mean, median and standard deviation of the following distribution
EXERCISE 32.7 PAGE NO: 32.47 1. Two plants A and B of a factory show the following results about the number of workers and the wages paid to them
Plant A Plant B
No. of workers 5000 6000
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Average monthly wages ₹2500 ₹2500
The variance of distribution of wages 81 100
In which plant A or B is there greater variability in individual wages?
Solution:
Variation of the distribution of wages in plant A (σ2 =18)
So, Standard deviation of the distribution A σ–9 Similarly, the Variation of the distribution of wages in plant B (σ2 =100)
So, Standard deviation of the distribution B σ–10 And, Average monthly wages in both the plants is 2500,
Since, the plant with a greater value of SD will have more variability in salary.
∴ Plant B has more variability in individual wages than plant A
2. The means and standard deviations of heights and weights of 50 students in a class are as follows:
Weights Heights
Mean 63.2 kg 63.2 inch
Standard deviation 5.6 kg 11.5 inch
Which shows more variability, heights or weights?
Solution:
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3. The coefficient of variation of two distribution are 60% and 70%, and their standard deviations are 21 and 16 respectively. What is their arithmetic means?
Solution:
= 22.86
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∴ Means are 35 and 22.86
4. Calculate coefficient of variation from the following data:
Income in₹: 1000-1700 1700-2400 2400-3100 3100-3800 3800-4500 4500-5200
No. of families: 12 18 20 25 35 10
Solution:
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5. An analysis of the weekly wages paid to workers in two firms A and B, belonging to the same industry gives the following results:
Firm A Firm B
No. of wage earners 586 648
Average weekly wages ₹52.5 ₹47.5
The variance of the distribution of wages 100 121
i Which firm A or B pays out the larger amount as weekly wages? ii Which firm A or B has greater variability in individual wages? Solution:
6. The following are some particulars of the distribution of weights of boys and girls in a class: