# NCERT Solutions for Class 10 Maths Chapter 14 Statistics Ex 14.3

NCERT Solutions for Class 10 Maths Chapter 14 Statistics Ex 14.3 are part of NCERT Solutions for Class 10 Maths. Here are we have given Chapter 14 Statistics Class 10 NCERT Solutions Ex 14.3.

 Board CBSE Textbook NCERT Class Class 10 Subject Maths Chapter Chapter 14 Chapter Name Statistics Exercise Ex 14.3 Number of Questions Solved 7 Category NCERT Solutions

## NCERT Solutions for Class 10 Maths Chapter 14 Statistics Ex 14.3

Question 1.
The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

 Monthly consumption (in units) Number of consumers 65 – 85 4 85 – 105 5 105 – 125 13 125 – 145 20 145 – 165 14 165 – 185 8 185 – 205 4

Solution:
We may find class marks by using the relation Taking 135 as assumed mean (a) we may find di, ui, fiui, according to step deviation method as following

 Monthly consumption  (in units) Number of consumers (f i) xi class mark di = xi – 135 fiui 65 – 85 4 75 – 60 – 3 – 12 85 – 105 5 95 – 40 – 2 – 10 105 – 125 13 115 – 20 – 1 – 13 125 – 145 20 135 0 0 0 145 – 165 14 155 20 1 14 165 – 185 8 175 40 2 16 185 – 205 4 195 60 3 12 Total 68 7

From the table we may observe that Now from table it is clear that maximum class frequency is 20 belonging to class interval 125 – 145.
Modal class = 125 – 145
Lower limit (l) of modal class = 125
Class size (h) = 20
Frequency (f1) of modal class = 20
Frequency (f0) of class preceding modal class = 13
Frequency (f2) of class succeeding the modal class = 14 We know that
3 median = mode + 2 mean
= 135.76 + 2 (137.058)
= 135.76 + 274.116
= 409.876
Median = 136.625
So median, mode, mean of given data is 136.625, 135.76, 137.05 respectively.

Question 2.
If the median of the distribution is given below is 28.5, find the values of x and y.

 Class interval Frequency 0 – 10 5 10 – 20 x 20 – 30 20 30 – 40 15 40 – 50 y 50 – 60 5 Total 60

Solution:
We may find cumulative frequency for the given data as following

 Class interval Frequency Cumulative frequency 0 – 10 5 5 10 – 20 x 5 + x 20 – 30 20 25 + x 30 – 40 15 40 + x 40 – 5 y 40 + x + y 50 – 60 5 45 + x + y Total (n) 60

It is clear that n = 60

45 + x + y = 60

x + y = 15        (1)
Median of data is given as 28.5 which lies in interval 20 – 30.
So, median class = 20 – 30
Lower limit (l) of median class = 20
Cumulative frequency (cf) of class preceding the median class = 5 + x
Frequency (f) of median class = 20
Class size (h) = 10 From equation (1)
8 + y = 15
y = 7
Hence values of x and y are 8 and 7 respectively.

Question 3

A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 year.

 Age (in years) Number of policy holders Below 20 2 Below 25 6 Below 30 24 Below 35 45 Below 40 78 Below 45 89 Below 50 92 Below 55 98 Below 60 100

Solution:
Here class width is not same. There is no need to adjust the frequencies according to class intervals. Now given frequency table is of less than type represented with upper class limits. As policies were given only to persons having age 18 years onwards but less than 60 years, we can define class intervals with their respective cumulative frequency as below

 Age (in years) Number of policy holders (fi) Cumulative frequency (cf) 18 – 20 2 2 20 – 25 6 – 2 = 4 6 25 – 30 24 – 6 = 18 24 30 – 35 45 – 24 = 21 45 35 – 40 78 – 45 = 33 78 40 – 45 89 – 78 = 11 89
 45 – 50 92 – 89 = 3 92 50 – 55 98 – 92 = 6 98 55 – 60 100 – 98 = 2 100 Total (n)

Now from table we may observe that n = 100.

Cumulative frequency (cf) just greater than is 78 belonging to interval 35 – 40
So, median class = 35 – 40
Lower limit (l) of median class = 35
Class size (h) = 5
Frequency (f) of median class = 33
Cumulative frequency (cf) of class preceding median class = 45 So, median age is 35.76 years.

Question 4.

The lengths of 40 leaves of a plant are measured correct to the nearest millimeter, and the data obtained is represented in the following table:

 Length (in mm) Number or leaves fi 118 – 126 3 127 – 135 5 136 – 144 9 145 – 153 12 154 – 162 5 163 – 171 4 172 – 180 2

Find the median length of the leaves.
(Hint: The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to 117.5 – 126.5, 126.5 – 135.5 – 171.5 – 180.5)
Solution:

The given data is not having continuous class intervals. We can observe that difference between two class intervals is 1. So, we have to add and subtract 1/2=0.5  to upper class limits and lower class limits.
Now continuous class intervals with respective cumulative frequencies can be represented as below

 Length (in mm) Number or leaves fi Cumulative frequency 117.5 – 126.5 3 3 126.5 – 135.5 5 3 + 5 = 8 135.5 – 144.5 9 8 + 9 = 17 144.5 – 153.5 12 17 + 12 = 29 153.5 – 162.5 5 29 + 5 = 34 162.5 – 171.5 4 34 + 4 = 38 171.5 – 180.5 2 38 + 2 = 40

From the table we may observe that cumulative frequency just greater then is 29, belonging to class interval 144.5 – 153.5.
Median class = 144.5 – 153.5
Lower limit (l) of median class = 144.5
Class size (h) = 9
Frequency (f) of median class = 12
Cumulative frequency (cf) of class preceding median class = 17 So, median length of leaves is 146.75 mm.

Question 5.
Find the following tables gives the distribution of the life time of 400 neon lamps:

 Life time (in hours) Number of lamps 1500 – 2000 14 2000 – 2500 56 2500 – 3000 60 3000 – 3500 86 3500 – 4000 74 4000 – 4500 62 4500 – 5000 48

Find the median life time of a lamp.
Solution:
We can find cumulative frequencies with their respective class intervals as below –

 Life time Number of lamps (fi) Cumulative frequency 1500 – 2000 14 14 2000 – 2500 56 14 + 56 = 70 2500 – 3000 60 70 + 60 = 130 3000 – 3500 86 130 + 86 = 216 3500 – 4000 74 216 + 74 = 290 4000 – 4500 62 290 + 62 = 352 4500 – 5000 48 352 + 48 = 400 Total (n) 400

Now we may observe that cumulative frequency just greater than is 216 belonging to class interval 3000 – 3500.
Median class = 3000 – 3500
Lower limit (l) of median class = 3000
Frequency (f) of median class = 86
Cumulative frequency (cf) of class preceding median class = 130
Class size (h) = 500 So, median life time of lamps is 3406.98 hours.

Question 6.
100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

 Number of letters 1 – 4 4 – 7 7 – 10 10 – 13 13 – 16 16 – 19 Number of surnames 6 30 40 6 4 4

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.
Solution:
We can find cumulative frequencies with their respective class intervals as below

 Number of letters Frequency (fi) Cumulative frequency 1 – 4 6 6 4 – 7 30 30 + 6 = 36 7 – 10 40 36 + 40 = 76 10 – 13 16 76 + 16 = 92 13 – 16 4 92 + 4 = 96 16 – 19 4 96 + 4 = 100 Total (n) 100

Now we may observe that cumulative frequency just greater than  is 76 belonging to class interval 7 – 10.
Median class = 7 – 10
Lower limit (l) of median class = 7
Cumulative frequency (cf) of class preceding median class = 36
Frequency (f) of median class = 40
Class size (h) = 3 Now we can find class marks of given class intervals by using relation Taking 11.5 as assumed mean (a) we can find di, ui and fiui according to step deviation method as below.

 Number of letters Number of surnames xi xi – a fiui 1 – 4 6 2.5 -9 -3 -18 4 – 7 30 5.5 -6 -2 -60 7 – 10 40 8.5 -3 -1 -40 10 – 13 16 11.5 0 0 0 13 – 16 4 14.5 3 1 4 16 – 19 4 17.5 6 2 8 Total 100 -106 We know that
3 median = mode + 2 mean
3(8.05) = mode + 2(8.32)
24.15 – 16.64 = mode
7.51 = mode
So, median number and mean number of letters in surnames is 8.05 and 8.32 respectively while modal size of surnames is 7.51.

Question 7

The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

 Weight  (in kg) 40 – 45 45 – 50 50 – 55 55 – 60 60 – 65 65 – 70 70 – 75 Number of students 2 3 8 6 6 3 2

Solution:
We may find cumulative frequencies with their respective class intervals as below
Cumulative frequency just greater than is 19, belonging to class interval 55 – 60.
Median class = 55 – 60
Lower limit (l) of median class = 55
Frequency (f) of median class = 6
Cumulative frequency (cf) of median class = 13
Class size (h) = 5 So, median weight is 56.67 kg.

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