Class 11 Maths Ncert Solutions Ex 15.1

Class 11 Maths Ncert Solutions Chapter 15 Ex 15.1

Q1. Calculate the mean deviation about the mean for the given data

5, 8, 9, 10, 11, 13, 14, 18

 

Sol:

The given data is,

5, 8, 9, 10, 11, 13, 14, 18

 

Mean,

x¯=5+8+9+10+11+13+14+188 x¯=888 x¯=11

 

The deviations of the respective observations from the mean x¯,

xix¯ are -6, -3, -2, -1, 0, 2, 3, 7

 

The absolute values of the deviations,

|xix¯| are 6, 3, 2, 1, 0 , 2, 3, 7

 

The required mean deviation about the mean is,

M.D.(x¯)=8i=1|xix¯|8 M.D.(x¯)=6+3+2+1+0+2+3+78

M.D.(x¯)=248 = 3

 

 

Q2. Calculate the mean deviation about the mean for the given data

39, 71, 49, 41, 43, 56, 64, 47, 55, 45

 

Sol:

The given data is,

39, 71, 49, 41, 43, 56, 64, 47, 55, 45

 

Mean,

x¯=39+71+49+41+43+56+64+47+55+4510 x¯=51010 x¯=51

 

The deviations of the respective observations from the mean x¯,

xix¯ are -12, 20, -2, -10, -8, 5, 13, -4, 4, -6.

 

The absolute values of the deviations,

|xix¯| are 12, 20, 2, 10, 8, 5, 13, 4, 4, 6.

 

The required mean deviation about the mean is,

M.D.(x¯)=10i=1|xix¯|10 M.D.(x¯)=12+20+2+10+8+5+13+4+4+610

M.D.(x¯)=8410 = 8.4

 

Q3. Calculate the mean deviation about the median for the given data:

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

 

Sol:

The given data is,

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

There are 12 observations which is even number.

 

Arrange the data in ascending order:

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

 

Median,

M=(122)thobservation+(122+1)thobservation2

= 6thobservation+7thobservation2

= 13+142

= 272

= 13.5

 

The deviations of the respective observations from the median x¯,

xiM 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

 

The absolute values of the deviations,

|xiM| 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

 

The required mean deviation about the median is,

M.D.(M)=12i=1|xiM|12

= 3.5+2.5+2.5+1.5+0.5+0.5+0.5+2.5+2.5+3.5+3.5+4.512

= 2812

= 2.33

 

Q4. Calculate the mean deviation about the median for the given data:

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

 

Sol:

The given data is,

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

There are 10 observations which is even number.

 

Arrange the data in ascending order:

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

 

Median,

M=(102)thobservation+(102+1)thobservation2

= 5thobservation+6thobservation2

= 46+492

= 952

= 47.5

 

The deviations of the respective observations from the median x¯,

xiM are –11.5, –5.5, –2.5, –1.5, –1.5, 1.5, 3.5, 5.5, 12.5, 24.5

 

The absolute values of the deviations,

|xiM| are 11.5, 5.5, 2.5, 1.5, 1.5, 1.5, 3.5, 5.5, 12.5, 24.5

 

The required mean deviation about the median is,

M.D.(M)=10i=1|xiM|10

= 11.5+5.5+2.5+1.5+1.5+1.5+3.5+5.5+12.5+24.510

= 7010

= 7

 

Q5. Calculate the mean deviation about the mean for the given data

 

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

 

Sol:

xi xi xi |xix¯| fi|xix¯|
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

 

N=5i=1fi=25

 

5i=1fixi=350

 

Therefore,

x¯=1N5i=1fixi

= 125×350

= 14

 

MD(x¯)=1N5i=1fi|xix¯|

= 125×158

= 6.32

 

Q6. Calculate the mean deviation about the mean for the given data

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

 

Sol:

xi xi xi |xix¯| fi|xix¯|
10 4 35 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

 

N=5i=1fi=80

 

5i=1fixi=4000

 

Therefore,

x¯=1N5i=1fixi

= 180×4000

= 50

 

MD(x¯)=1N5i=1fi|xix¯|

= 180×1280

= 16

 

Q7. Calculate the mean deviation about the median for the given data

xi 10 10 15 20 25
fi 7 4 6 3 5
xi 5 7 9 10 12 15
fi 8 6 2 2 2 6

 

 Sol:

The observations given are in ascending order.

Adding a column corresponding to cumulative frequencies of the given data, we obtain the table given below:

xi fi c.f.
5 8 8
7 6 14
9 2 16
10 2 18
12 2 20
15 6 26

 

Here, N = 26, which is even.

 

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

 

Median=13thobservation+14thobservation2

= 7+72

= 7

 

The absolute values of the deviations from median,

|xiM| 2 0 2 3 5 8
fi 8 6 2 2 2 6
fi|xiM| 16 0 4 6 10 48

 

6i=1fi=26

 

6i=1fi|xiM|=84

 

Therefore,

MD(M)=1N6i=1fi|xiM|

= 126×84

= 3.23

 

Q8. Calculate the mean deviation about the median for the given data

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

 

Sol:

The observations given are in ascending order.

 

Adding a column corresponding to cumulative frequencies of the given data, we obtain the table given below:

 

xi fi c.f.
15 3 3
21 5 8
27 6 14
30 7 21
35 8 29

 

Here, N = 29, which is odd.

 

Median=(29+12)thobservation

= 15th observation

 

This observation lies in the cumulative frequency 21, for which the corresponding observation is 30.

 

Therefore,

Median = 30

 

The absolute values of the deviations from median,

|xiM| 15 9 3 0 5
fi 3 5 6 7 8
fi|xiM| 45 45 18 0 40

 

5i=1fi=29

 

5i=1fi|xiM|=148

 

Therefore,

MD(M)=1N5i=1fi|xiM|

= 129×148

= 5.1

 

Q9. Calculate the mean deviation about the mean for the given data:

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

 

 Sol:

The given table is formed.

Income per day Number of persons fi Mid – point xi fi xi |xix¯| fi|xix¯|
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 1168
700 – 800 3 750 2250 392 1176
50 17900 7896

 

N=8i=1fi=50

 

8i=1fixi=17900

 

Therefore,

x¯=1N8i=1fixi

= 150×17900

= 358

 

MD(x¯)=1N8i=1fi|xix¯|

= 150×7896

= 175.92

 

Q10. Calculate the mean deviation about the mean for the given data:

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

 

 

Sol:

The given table is obtained:

 

Height in cm Number of boys fi Mid – point xi fi xi |xix¯| fi|xix¯|
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

 

N=6i=1fi=100

 

6i=1fixi=12530

 

Therefore,

x¯=1N6i=1fixi

= 1100×12530

= 125.3

 

MD(x¯)=1N6i=1fi|xix¯|

= 1100×1128.8

= 11.28

 

Q11. Calculate the mean deviation about the median for the given data:

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

 

 

Sol:

The given table is obtained:

 

Marks Number of girls fi Cumulative frequency (c.f.) Mid – point xi |xiM| fi|xix¯|
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 (N2)th or 25th item is 20 – 30.

Therefore, 20 -30 is the median class.

 

Now,

Median=l+N2Cf×h

Where, l = 20

C = 14

f = 14

h = 10

N = 50

 

Median=20+251414×10

= 20+11014

= 20 + 7.85

= 27.85

 

Mean deviation about the median is,

MD(M)=1N6i=1fi|xiM|

= 150×517.1

= 10.34

 

Q12.  Calculate the mean deviation about the median for the given data:

            

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

 

Sol:

The given data is not continuous. Therefore, it has to be converted into continuous frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval.

 

The table is as given below:

 

Age Number fi Cumulative frequency (c.f.) Mid – point xi |xiM| fi|xix¯|
16 – 20 5 5 18 20 100
21 – 25 6 11 23 15 90
26 – 30 12 23 28 10 120
31 – 35 14 37 33 5 70
36 – 40 26 63 38 0 0
41 – 45 12 75 43 5 60
46 – 50 16 91 48 10 160
51 – 55 9 100 53 15 135
100

 

The class interval containing (N2)th or 50th item is 35.5 – 40.5. Therefore, 35.5 – 40.5 is the median class.

 

Now,

Median=l+N2Cf×h

Where, l = 35.5

C = 37

f = 26

h = 5

N = 100

 

Median=35.5+503726×5

= 35.5+13×526

= 35.5 + 2.5

= 38

 

Mean deviation about the median is,

MD(M)=1N8i=1fi|xiM|

= 1100×735

= 7.35