NCERT Solutions for Class 11 Maths Chapter 15 Statistics

The NCERT Solutions For Class 11 Maths Chapter 15 Statistics are available as a pdf on this page. The NCERT Solutions are authored by the most experienced educators in the teaching industry making the solution of every problem straightforward and justifiable. Every solution written in the pdf given underneath empowers the students to get ready for the test and accomplish it. These solutions assist a Class 11 student with mastering the idea of Limits and Derivatives.

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Get detailed solutions for all the questions listed under the below exercises:

Exercise 15.1 Solutions : 12 Questions

Exercise 15.2 Solutions : 10 Questions

Exercise 15.3 Solutions : 5 Questions

Miscellaneous Exercise Solutions: 7 Questions

Access Solutions for Class 11 Maths Chapter 15 Statistics

Exercise 15.1 Page: 360

Find the mean deviation about the mean for the data in Exercises 1 and 2.

1. 4, 7, 8, 9, 10, 12, 13, 17

Solution:-

First we have to find (x̅) of the given data

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 1

So, the respective values of the deviations from mean,

i.e., xi – x̅ are, 10 – 4 = 6, 10 – 7 = 3, 10 – 8 = 2, 10 – 10 = 0,

10 – 12 = – 2, 10 – 13 = – 3, 10 – 17 = – 7

6, 3, 2, 1, 0, -2, -3, -7

Now absolute values of the deviations,

6, 3, 2, 1, 0, 2, 3, 7

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 2

MD = sum of deviations/ number of observations

= 24/8

= 3

So, the mean deviation for the given data is 3.

2. 38, 70, 48, 40, 42, 55, 63, 46, 54, 44

Solution:-

First we have to find (x̅) of the given data

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 3

So, the respective values of the deviations from mean,

i.e., xi – x̅ are, 50 – 38 = -12, 50 -70 = -20, 50 – 48 = 2, 50 – 40 = 10, 50 – 42 = 8,

50 – 55 = – 5, 50 – 63 = – 13, 50 – 46 = 4, 50 – 54 = -4, 50 – 44 = 6

-12, 20, -2, -10, -8, 5, 13, -4, 4, -6

Now absolute values of the deviations,

12, 20, 2, 10, 8, 5, 13, 4, 4, 6

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 4

MD = sum of deviations/ number of observations

= 84/10

= 8.4

So, the mean deviation for the given data is 8.4.

Find the mean deviation about the median for the data in Exercises 3 and 4.

3. 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

Solution:-

First we have to arrange the given observations into ascending order,

10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18.

The number of observations is 12

Then,

Median = ((12/2)th observation + ((12/2)+ 1)th observation)/2

(12/2)th observation = 6th = 13

(12/2)+ 1)th observation = 6 + 1

= 7th = 14

Median = (13 + 14)/2

= 27/2

= 13.5

So, the absolute values of the respective deviations from the median, i.e., |xi – M| are

3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 5

Mean Deviation,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 6

= (1/12) × 28

= 2.33

So, the mean deviation about the median for the given data is 2.33.

4. 36, 72, 46, 42, 60, 45, 53, 46, 51, 49

Solution:-

First we have to arrange the given observations into ascending order,

36, 42, 45, 46, 46, 49, 51, 53, 60, 72.

The number of observations is 10

Then,

Median = ((10/2)th observation + ((10/2)+ 1)th observation)/2

(10/2)th observation = 5th = 46

(10/2)+ 1)th observation = 5 + 1

= 6th = 49

Median = (46 + 49)/2

= 95

= 47.5

So, the absolute values of the respective deviations from the median, i.e., |xi – M| are

11.5, 5.5, 2.5, 1.5, 1.5, 1.5, 3.5, 5.5, 12.5, 24.5

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 7

Mean Deviation,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 8

= (1/10) × 70

= 7

So, the mean deviation about the median for the given data is 7.

Find the mean deviation about the mean for the data in Exercises 5 and 6.

5.

xi 5 10 15 20 25
fi 7 4 6 3 5

Solution:-

Let us make the table of the given data and append other columns after calculations.

Xi fi fixi |xi – x̅| fi |xi – x̅|
5 7 35 9 63
10 4 40 4 16
15 6 90 1 6
20 3 60 6 18
25 5 125 11 55
25 350 158

The sum of calculated data,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 9

The absolute values of the deviations from the mean, i.e., |xi – x̅|, as shown in the table.

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 10

6.

xi 10 30 50 70 90
fi 4 24 28 16 8

Solution:-

Let us make the table of the given data and append other columns after calculations.

Xi fi fixi |xi – x̅| fi |xi – x̅|
10 4 40 40 160
30 24 720 20 480
50 28 1400 0 0
70 16 1120 20 320
90 8 720 40 320
80 4000 1280

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 11

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 12

Find the mean deviation about the median for the data in Exercises 7 and 8.

7.

xi 5 7 9 10 12 15
fi 8 6 2 2 2 6

Solution:-

Let us make the table of the given data and append other columns after calculations.

Xi fi c.f. |xi – M| fi |xi – M|
5 8 8 2 16
7 6 14 0 0
9 2 16 2 4
10 2 18 3 6
12 2 20 5 10
15 6 26 8 48

Now, N = 26, which is even.

Median is the mean of the 13th and 14th observations. Both of these observations lie in the cumulative frequency 14, for which the corresponding observation is 7.

Then,

Median = (13th observation + 14th observation)/2

= (7 + 7)/2

= 14/2

= 7

So, the absolute values of the respective deviations from the median, i.e., |xi – M| are shown in the table.

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 13

8.

xi 15 21 27 30 35
fi 3 5 6 7 8

Solution:-

Let us make the table of the given data and append other columns after calculations.

Xi fi c.f. |xi – M| fi |xi – M|
15 3 3 13.5 40.5
21 5 8 7.5 37.5
27 6 14 1.5 9
30 7 21 1.5 10.5
35 8 29 6.5 52

Now, N = 30, which is even.

Median is the mean of the 15th and 16th observations. Both of these observations lie in the cumulative frequency 21, for which the corresponding observation is 30.

Then,

Median = (15th observation + 16th observation)/2

= (30 + 30)/2

= 60/2

= 30

So, the absolute values of the respective deviations from the median, i.e., |xi – M| are shown in the table.

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 14

Find the mean deviation about the mean for the data in Exercises 9 and 10.

9.

Income per day in ₹ 0 – 100 100 – 200 200 – 300 300 – 400 400 – 500 500 – 600 600 – 700 700 – 800
Number of persons 4 8 9 10 7 5 4 3

Solution:-

Let us make the table of the given data and append other columns after calculations.

Income per day in ₹ Number of persons fi Mid – points

xi

fixi |xi – x̅| fi|xi – x̅|
0 – 100 4 50 200 308 1232
100 – 200 8 150 1200 208 1664
200 – 300 9 250 2250 108 972
300 – 400 10 350 3500 8 80
400 – 500 7 450 3150 92 644
500 – 600 5 550 2750 192 960
600 – 700 4 650 2600 292 1160
700 – 800 3 750 2250 392 1176
50 17900 7896

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 15

10.

Height in cms 95 – 105 105 – 115 115 – 125 125 – 135 135 – 145 145 – 155
Number of boys 9 13 26 30 12 10

Solution:-

Let us make the table of the given data and append other columns after calculations.

Height in cms Number of boys fi Mid – points

xi

fixi |xi – x̅| fi|xi – x̅|
95 – 105 9 100 900 25.3 227.7
105 – 115 13 110 1430 15.3 198.9
115 – 125 26 120 3120 5.3 137.8
125 – 135 30 130 3900 4.7 141
135 – 145 12 140 1680 14.7 176.4
145 – 155 10 150 1500 24.7 247
100 12530 1128.8

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 16

11. Find the mean deviation about median for the following data:

Marks 0 -10 10 -20 20 – 30 30 – 40 40 – 50 50 – 60
Number of girls 6 8 14 16 4 2

Solution:-

Let us make the table of the given data and append other columns after calculations.

Marks Number of Girls fi Cumulative frequency (c.f.) Mid – points

xi

|xi – Med| fi|xi – Med|
0 – 10 6 6 5 22.85 137.1
10 – 20 8 14 15 12.85 102.8
20 – 30 14 28 25 2.85 39.9
30 – 40 16 44 35 7.15 114.4
40 – 50 4 48 45 17.15 68.6
50 – 60 2 50 55 27.15 54.3
50 517.1

The class interval containing Nth/2 or 25th item is 20-30

So, 20-30 is the median class.

Then,

Median = l + (((N/2) – c)/f) × h

Where, l = 20, c = 14, f = 14, h = 10 and n = 50

Median = 20 + (((25 – 14))/14) × 10

= 20 + 7.85

= 27.85

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 17

12. Calculate the mean deviation about median age for the age distribution of 100 persons given below:

Age

(in years)

16 – 20 21 – 25 26 – 30 31 – 35 36 – 40 41 – 45 46 – 50 51 – 55
Number 5 6 12 14 26 12 16 9

[Hint Convert the given data into continuous frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval]

Solution:-

The given data is converted into continuous frequency distribution by subtracting 0.5 from the lower limit and adding the 0.5 to the upper limit of each class intervals and append other columns after calculations.

Age Number fi Cumulative frequency (c.f.) Mid – points

xi

|xi – Med| fi|xi – Med|
15.5 – 20.5 5 5 18 20 100
20.5 – 25.5 6 11 23 15 90
25.5 – 30.5 12 23 28 10 120
30.5 – 35.5 14 37 33 5 70
35.5 – 40.5 26 63 38 0 0
40.5 – 45.5 12 75 43 5 60
45.5 – 50.5 16 91 48 10 160
50.5 – 55.5 9 100 53 15 135
100 735

The class interval containing Nth/2 or 50th item is 35.5 – 40.5

So, 35.5 – 40.5 is the median class.

Then,

Median = l + (((N/2) – c)/f) × h

Where, l = 35.5, c = 37, f = 26, h = 5 and N = 100

Median = 35.5 + (((50 – 37))/26) × 5

= 35.5 + 2.5

= 38

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 18


Exercise 15.2 Page: 371

Find the mean and variance for each of the data in Exercise 1 to 5.

1. 6, 7, 10, 12, 13, 4, 8, 12

Solution:-

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 19

So, x̅ = (6 + 7 + 10 + 12 + 13 + 4 + 8 + 12)/8

= 72/8

= 9

Let us make the table of the given data and append other columns after calculations.

Xi Deviations from mean

(xi – x̅)

(xi – x̅)2
6 6 – 9 = -3 9
7 7 – 9 = -2 4
10 10 – 9 = 1 1
12 12 – 9 = 3 9
13 13 – 9 = 4 16
4 4 – 9 = – 5 25
8 8 – 9 = – 1 1
12 12 – 9 = 3 9
74

We know that Variance,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 20

σ2 = (1/8) × 74

= 9.2

∴Mean = 9 and Variance = 9.25

2. First n natural numbers

Solution:-

We know that Mean = Sum of all observations/Number of observations

∴Mean, x̅ = ((n(n + 1))2)/n

= (n + 1)/2

and also WKT Variance,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 21

By substitute that value of x̅ we get,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 22

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 23

WKT, (a + b)(a – b) = a2 – b2

σ2 = (n2 – 1)/12

∴Mean = (n + 1)/2 and Variance = (n2 – 1)/12

3. First 10 multiples of 3

Solution:-

First we have to write the first 10 multiples of 3,

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 24

So, x̅ = (3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30)/10

= 165/10

= 16.5

Let us make the table of the data and append other columns after calculations.

Xi Deviations from mean

(xi – x̅)

(xi – x̅)2
3 3 – 16.5 = -13.5 182.25
6 6 – 16.5 = -10.5 110.25
9 9 – 16.5 = -7.5 56.25
12 12 – 16.5 = -4.5 20.25
15 15 – 16.5 = -1.5 2.25
18 18 – 16.5 = 1.5 2.25
21 21 – 16.5 = – 4.5 20.25
24 24 – 16.5 = 7.5 56.25
27 27 – 16.5 = 10.5 110.25
30 30 – 16.5 = 13.5 182.25
742.5

Then, Variance

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 25

= (1/10) × 742.5

= 74.25

∴Mean = 16.5 and Variance = 74.25

4.

xi 6 10 14 18 24 28 30
fi 2 4 7 12 8 4 3

Solution:-

Let us make the table of the given data and append other columns after calculations.

Xi fi fixi Deviations from mean

(xi – x̅)

(xi – x̅)2 fi(xi – x̅)2
6 2 12 6 – 19 = 13 169 338
10 4 40 10 – 19 = -9 81 324
14 7 98 14 – 19 = -5 25 175
18 12 216 18 – 19 = -1 1 12
24 8 192 24 – 19 = 5 25 200
28 4 112 28 – 19 = 9 81 324
30 3 90 30 – 19 = 11 121 363
N = 40 760 1736

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 26

5.

xi 92 93 97 98 102 104 109
fi 3 2 3 2 6 3 3

Solution:-

Let us make the table of the given data and append other columns after calculations.

Xi fi fixi Deviations from mean

(xi – x̅)

(xi – x̅)2 fi(xi – x̅)2
92 3 276 92 – 100 = -8 64 192
93 2 186 93 – 100 = -7 49 98
97 3 291 97 – 100 = -3 9 27
98 2 196 98 – 100 = -2 4 8
102 6 612 102 – 100 = 2 4 24
104 3 312 104 – 100 = 4 16 48
109 3 327 109 – 100 = 9 81 243
N = 22 2200 640

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 27

6. Find the mean and standard deviation using short-cut method.

xi 60 61 62 63 64 65 66 67 68
fi 2 1 12 29 25 12 10 4 5

Solution:-

Let the assumed mean A = 64. Here h = 1

We obtain the following table from the given data.

Xi Frequency fi Yi = (xi – A)/h Yi2 fiyi fiyi2
60 2 -4 16 -8 32
61 1 -3 9 -3 9
62 12 -2 4 -24 48
63 29 -1 1 -29 29
64 25 0 0 0 0
65 12 1 1 12 12
66 10 2 4 20 40
67 4 3 9 12 36
68 5 4 16 20 80
0 286

Mean,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 28

Where A = 64, h = 1

So, x̅ = 64 + ((0/100) × 1)

= 64 + 0

= 64

Then, variance,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 29

σ2 = (12/1002) [100(286) – 02]

= (1/10000) [28600 – 0]

= 28600/10000

= 2.86

Hence, standard deviation = σ = √2.886

= 1.691

∴ Mean = 64 and Standard Deviation = 1.691

Find the mean and variance for the following frequency distributions in Exercises 7 and 8.

7.

Classes 0 – 30 30 – 60 60 – 90 90 – 120 120 – 150 150 – 180 180 – 210
Frequencies 2 3 5 10 3 5 2

Solution:-

Let us make the table of the given data and append other columns after calculations.

Classes Frequency fi Mid – points

xi

fixi (xi – x̅) (xi – x̅)2 fi(xi – x̅)2
0 – 30 2 15 30 -92 8464 16928
30 – 60 3 45 135 -62 3844 11532
60 – 90 5 75 375 -32 1024 5120
90 – 120 10 105 1050 -2 4 40
120 – 150 3 135 405 28 784 2352
150 – 180 5 165 825 58 3364 16820
180 – 210 2 195 390 88 7744 15488
N = 30 3210 68280

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 30

8.

Classes 0 – 10 10 – 20 20 – 30 30 – 40 40 –50
Frequencies 5 8 15 16 6

Solution:-

Let us make the table of the given data and append other columns after calculations.

Classes Frequency fi Mid – points

xi

fixi (xi – x̅) (xi – x̅)2 fi(xi – x̅)2
0 – 10 5 5 25 -22 484 2420
10 – 20 8 15 120 -12 144 1152
20 – 30 15 25 375 -2 4 60
30 – 40 16 35 560 8 64 1024
40 –50 6 45 270 18 324 1944
N = 50 1350 6600

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 31

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 32

9. Find the mean, variance and standard deviation using short-cut method

Height in cms 70 – 75 75 – 80 80 – 85 85 – 90 90 – 95 95 – 100 100 – 105 105 – 110 110 – 115
Frequencies 3 4 7 7 15 9 6 6 3

Solution:-

Let the assumed mean, A = 92.5 and h = 5

Let us make the table of the given data and append other columns after calculations.

Height (class) Number of children

Frequency fi

Midpoint

Xi

Yi = (xi – A)/h Yi2 fiyi fiyi2
70 – 75 2 72.5 -4 16 -12 48
75 – 80 1 77.5 -3 9 -12 36
80 – 85 12 82.5 -2 4 -14 28
85 – 90 29 87.5 -1 1 -7 7
90 – 95 25 92.5 0 0 0 0
95 – 100 12 97.5 1 1 9 9
100 – 105 10 102.5 2 4 12 24
105 – 110 4 107.5 3 9 18 54
110 – 115 5 112.5 4 16 12 48
N = 60 6 254

Mean,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 33

Where, A = 92.5, h = 5

So, x̅ = 92.5 + ((6/60) × 5)

= 92.5 + ½

= 92.5 + 0.5

= 93

Then, Variance,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 34

σ2 = (52/602) [60(254) – 62]

= (1/144) [15240 – 36]

= 15204/144

= 1267/12

= 105.583

Hence, standard deviation = σ = √105.583

= 10.275

∴ Mean = 93, variance = 105.583 and Standard Deviation = 10.275

10. The diameters of circles (in mm) drawn in a design are given below:

Diameters 33 – 36 37 – 40 41 – 44 45 – 48 49 – 52
No. of circles 15 17 21 22 25

Calculate the standard deviation and mean diameter of the circles.

[Hint first make the data continuous by making the classes as 32.5-36.5, 36.5-40.5, 40.5-44.5, 44.5 – 48.5, 48.5 – 52.5 and then proceed.]

Solution:-

Let the assumed mean, A = 42.5 and h = 4

Let us make the table of the given data and append other columns after calculations.

Height (class) Number of children

(Frequency fi)

Midpoint

Xi

Yi = (xi – A)/h Yi2 fiyi fiyi2
32.5 – 36.5 15 34.5 -2 4 -30 60
36.5 – 40.5 17 38.5 -1 1 -17 17
40.5 – 44.5 21 42.5 0 0 0 0
44.5 – 48.5 22 46.5 1 1 22 22
48.5 – 52.5 25 50.5 2 4 50 100
N = 100 25 199

Mean,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 35

Where, A = 42.5, h = 4

So, x̅ = 42.5 + (25/100) × 4

= 42.5 + 1

= 43.5

Then, Variance,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 34

σ2 = (42/1002)[100(199) – 252]

= (1/625) [19900 – 625]

= 19275/625

= 771/25

= 30.84

Hence, standard deviation = σ = √30.84

= 5.553

∴ Mean = 43.5, variance = 30.84 and Standard Deviation = 5.553.


Exercise 15.3 Page: 375

1. From the data given below state which group is more variable, A or B?

Marks 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80
Group A 9 17 32 33 40 10 9
Group B 10 20 30 25 43 15 7

Solution:-

For comparing the variability or dispersion of two series, we calculate the coefficient of variance for each series. The series having greater C.V. is said to be more variable than the other. The series having lesser C.V. is said to be more consistent than the other.

Co-efficient of variation (C.V.) = (σ/ x̅) × 100

Where, σ = standard deviation, x̅ = mean

For Group A

Marks Group A

fi

Mid-point

Xi

Yi = (xi – A)/h (Yi)2 fiyi fi(yi)2
10 – 20 9 15 ((15 – 45)/10) = -3 (-3)2

= 9

– 27 81
20 – 30 17 25 ((25 – 45)/10) = -2 (-2)2

= 4

– 34 68
30 – 40 32 35 ((35 – 45)/10) = – 1 (-1)2

= 1

– 32 32
40 – 50 33 45 ((45 – 45)/10) = 0 02 0 0
50 – 60 40 55 ((55 – 45)/10) = 1 12

= 1

40 40
60 – 70 10 65 ((65 – 45)/10) = 2 22

= 4

20 40
70 – 80 9 75 ((75 – 45)/10) = 3 32

= 9

27 81
Total 150 -6 342

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 37

Where A = 45,

and yi = (xi – A)/h

Here h = class size = 20 – 10

h = 10

So, x̅ = 45 + ((-6/150) × 10)

= 45 – 0.4

= 44.6

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 38

σ2 = (102/1502) [150(342) – (-6)2]

= (100/22500) [51,300 – 36]

= (100/22500) × 51264

= 227.84

Hence, standard deviation = σ = √227.84

= 15.09

∴ C.V for group A = (σ/ x̅) × 100

= (15.09/44.6) × 100

= 33.83

Now, for group B.

Marks Group B

fi

Mid-point

Xi

Yi = (xi – A)/h (Yi)2 fiyi fi(yi)2
10 – 20 10 15 ((15 – 45)/10) = -3 (-3)2

= 9

– 30 90
20 – 30 20 25 ((25 – 45)/10) = -2 (-2)2

= 4

– 40 80
30 – 40 30 35 ((35 – 45)/10) = – 1 (-1)2

= 1

– 30 30
40 – 50 25 45 ((45 – 45)/10) = 0 02 0 0
50 – 60 43 55 ((55 – 45)/10) = 1 12

= 1

43 43
60 – 70 15 65 ((65 – 45)/10) = 2 22

= 4

30 160
70 – 80 7 75 ((75 – 45)/10) = 3 32

= 9

21 189
Total 150 -6 592

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 39

Where A = 45,

h = 10

So, x̅ = 45 + ((-6/150) × 10)

= 45 – 0.4

= 44.6

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 40

σ2 = (102/1502) [150(592) – (-6)2]

= (100/22500) [88,800 – 36]

= (100/22500) × 88,764

= 394.50

Hence, standard deviation = σ = √394.50

= 19.86

∴ C.V for group B = (σ/ x̅) × 100

= (19.86/44.6) × 100

= 44.53

By comparing C.V. of group A and group B.

C.V of Group B > C.V. of Group A

So, Group B is more variable.

2. From the prices of shares X and Y below, find out which is more stable in value:

X 35 54 52 53 56 58 52 50 51 49
Y 108 107 105 105 106 107 104 103 104 101

Solution:-

From the given data,

Let us make the table of the given data and append other columns after calculations.

X (xi) Y (yi) Xi2 Yi2
35 108 1225 11664
54 107 2916 11449
52 105 2704 11025
53 105 2809 11025
56 106 8136 11236
58 107 3364 11449
52 104 2704 10816
50 103 2500 10609
51 104 2601 10816
49 101 2401 10201
Total = 510 1050 26360 110290

We have to calculate Mean for x,

Mean x̅ = ∑xi/n

Where, n = number of terms

= 510/10

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 41= 51

Then, Variance for x =

= (1/102)[(10 × 26360) – 5102]

= (1/100) (263600 – 260100)

= 3500/100

= 35

WKT Standard deviation = √variance

= √35

= 5.91

So, co-efficient of variation = (σ/ x̅) × 100

= (5.91/51) × 100

= 11.58

Now, we have to calculate Mean for y,

Mean ȳ = ∑yi/n

Where, n = number of terms

= 1050/10

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 42= 105

Then, Variance for y =

= (1/102)[(10 × 110290) – 10502]

= (1/100) (1102900 – 1102500)

= 400/100

= 4

WKT Standard deviation = √variance

= √4

= 2

So, co-efficient of variation = (σ/ x̅) × 100

= (2/105) × 100

= 1.904

By comparing C.V. of X and Y.

C.V of X > C.V. of Y

So, Y is more stable than X.

3. An analysis of monthly 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 wages earners 586 648
Mean of monthly wages Rs 5253 Rs 5253
Variance of the distribution of wages 100 121

(i) Which firm A or B pays larger amount as monthly wages?

(ii) Which firm, A or B, shows greater variability in individual wages?

Solution:-

From the given table,

Mean monthly wages of firm A = Rs 5253

and Number of wage earners = 586

Then,

Total amount paid = 586 × 5253

= Rs 3078258

Mean monthly wages of firm B = Rs 5253

Number of wage earners = 648

Then,

Total amount paid = 648 × 5253

= Rs 34,03,944

(i) So, firm B pays larger amount as monthly wages.

(ii) Variance of firm A = 100

We know that, standard deviation (σ)= √100

=10

Variance of firm B = 121

Then,

Standard deviation (σ)=√(121 )

=11

Hence the standard deviation is more in case of Firm B that means in firm B there is greater variability in individual wages.

4. The following is the record of goals scored by team A in a football session:

No. of goals scored 0 1 2 3 4
No. of matches 1 9 7 5 3

For the team B, mean number of goals scored per match was 2 with a standard deviation 1.25 goals. Find which team may be considered more consistent?

Solution:-

From the given data,

Let us make the table of the given data and append other columns after calculations.

Number of goals scored xi Number of matches fi fixi Xi2 fixi2
0 1 0 0 0
1 9 9 1 9
2 7 14 4 28
3 5 15 9 45
4 3 12 16 48
Total 25 50 130

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 43

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 44

Since C.V. of firm B is greater

∴ Team A is more consistent.

5. The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are given below:

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 45

Which is more varying, the length or weight?

Solution:-

First we have to calculate Mean for Length x,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 46

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 47

 

Miscellaneous Exercise Page: 380

1. The mean and variance of eight 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:-

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 48

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 49

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 50

2. The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12, 14. Find the remaining two observations.

Solution:-

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 51

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 52

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 53

Hence, from equation (i) and (v) we have:

When x – y = 2 then x = 8 and y = 6

And, when x – y = – 2 then x = 6 and y = 8

∴ the remaining observations are 6 and 8

3. The mean and standard deviation of six 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:-

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 54

We know that,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 55

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 56

4. Given that x̅ is the mean and σ2 is the variance of n observations x1, x2, …,xn . Prove that the mean and variance of the observations ax1, ax2, ax3, …., axn are ax̅ and a2σ2, respectively, (a ≠ 0).

Solution:-

From the question it is given that, n observations are x1, x2,…..xn

Mean of the n observation = x̅

Variance of the n observation = σ2

As we know that,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 57

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 58

5. The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases: (i) If wrong item is omitted. (ii) If it is replaced by 12

Solution:-

(i) If wrong item is omitted,

From the question it is given that,

The number of observations i.e. n = 20

The incorrect mean = 20

The incorrect standard deviation = 2

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 59

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 61
NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 60

(ii) If it is replaced by 12,

From the question it is given that,

The number of incorrect sum observations i.e. n = 200

The correct sum of observations n = 200 – 8 + 12

n = 204

Then, correct mean = correct sum/20

= 204/20

= 10.2

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 62

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 63

6. The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below:

Subject Mathematics Physics Chemistry
Mean 42 32 40.9
Standard deviation 12 15 20

Which of the three subjects shows the highest variability in marks and which shows the lowest?

Solution:-

From the question it is given that,

Mean of Mathematics = 42

Standard deviation of Mathematics = 12

Mean of Physics = 32

Standard deviation of physics = 15

Mean of Chemistry = 40.9

Standard deviation of chemistry = 20

As we know that,

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 64

7. The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.

Solution:-

From the question it is given that,

The total number of observations (n) = 100

Incorrect mean, (x̅) = 20

And, Incorrect standard deviation (σ) = 3

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 65

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 66

NCERT Soluitons for Class 11 Maths Chapter 15 Statistics Image 67


NCERT Solutions for Class 11 Maths Chapter 15 – Statistics

The topics of Class 11 Chapter 15 – Statistics of NCERT Solution are as follows:

15.1 Introduction

This section talks about the concept of central tendency, mean and median [during even and the odd number of observations] with examples. It introduces the concept of measure of dispersion.

The values which cluster around the middle or centre of the distribution are measures of central tendency. They are mean, median and mode.

In a class, mean can be used to find the average marks scored by the students.

When calculating the height of students, the median can be used to find the middlemost value.

15.2 Measures of Dispersion

This section defines measures of dispersion, different measures of dispersion [range, quartile deviation, mean deviation, standard deviation]

Measures of dispersion explain the relationship with measures of central tendency. For example, the spread of data tells how well the mean represents the data. If the spread is large, then mean is not representative of data.

15.3 Range

This section defines the range, its formula and an example.

The range gives the variability of scores using the maximum and minimum values in the set.

In a cricket match, Batsman A range = 121 – 0 = 121 and Batsman B range = 52 – 22 = 30

Range A > Range B. So, the scores are spread in case of A whereas for B they are close to each other.

15.4 Mean Deviation

This section defines mean deviation, the formula for mean deviation.

The concept of mean deviation can be used by biologists in the comparison of different animal weights and decide what would be a healthy weight.

15.4.1Mean deviation for ungrouped data

The process of obtaining the mean deviation for ungrouped data is elaborated in this section.

Find the mean, deviations from the mean, absolute deviations and substitute the values in the mean deviation formula and arrive at the answer.

15.4.2 Mean deviation for grouped data

The process of obtaining mean deviation for discrete and continuous frequency distributions are elaborated in this section with solved examples.

15.4.3 Limitations of mean deviation

  • If in a series the degree of variability is very high, then median will not be a representative of data. Hence the mean deviation calculated about such median cannot be relied on.
  • If the sum of deviations from the mean is greater than the sum of deviations from the median, then the mean deviation about mean is not very specific.
  • The absolute mean deviation calculated can’t be subjected to further algebraic treatment. It can’t be used as an appropriate measure of dispersion.

15.5 Variance and standard deviation

15.5.1 Standard Deviation

This section involves the variance and standard deviation definition, formula and solved examples on the discrete and continuous frequency distribution.

A science test was taken by a class of students. The mean score of the test was 85% on calculation. The teacher found the standard deviation of other scores and noticed that a very small standard deviation exists which suggests that most of the students scored very close to 85%.

15.5.2 Shortcut method to find Variance and standard deviation

This section deals with the simpler way of calculating the standard deviation with a few illustrations.

15.6 Analysis of Frequency Distributions

This section deals with the process of comparing the variability of two series having the same mean, coefficient of variation with few solved problems.

A few points on Chapter 15 Statistics

  • Range is defined as the difference between the maximum and minimum value of the given data.
  • If there exists series with equal means, then the series with lesser standard deviation is more consistent or less scattered.
  • The addition or subtraction of a positive number to each data point of the data set will not affect the variance.

The solutions give substitute strategies and clarifications to take care of problems which makes the student feel sure while taking the test. Additionally, taking care of many muddled issues upgrades the information on Mathematical abilities in the students. The solutions cater to all the vital inquiries which a student should and ought to have aced to show up for the test. The BYJU’S subject specialists who have written these solutions have complete knowledge about the question paper setting and the marks distributed across the chapters.

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