Concept of cumulative frequency, the frequency distribution (cumulative) and drawing of cumulative frequency curves(Ogive) is discussed in this chapter.
Firstly, the topic of Mean of Grouped Data is explained. The first solved example is based on the concept of finding the mean of the given data (here marks). These types of questions must be solved in tabular form. After that, the upper limit and the lower limit are explained.
It is assumed that the frequency of each class is centered around its midpoint.
 Class mark = (Upper class limit + Lower class limit)/2
 The class with the maximum frequency is called the modal class.
 Mode can be equal, less or more than mean.
 Empirical relationship: 3Median = Mode + 2Mean
Page No 270:
Question 1:
A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
Number of plants 
0 − 2 
2 − 4 
4 − 6 
6 − 8 
8 − 10 
10 − 12 
12 − 14 
Number of houses 
1 
2 
1 
5 
6 
2 
3 
Which method did you use for finding the mean, and why?
Answer:
To find the class mark (x_{i}) for each interval, the following relation is used.
Class mark (x_{i}) =
x_{i }and f_{i}x_{i} can be calculated as follows.
Number of plants 
Number of houses (f_{i}) 
x_{i} 
f_{i}x_{i} 
0 − 2 
1 
1 
1 × 1 = 1 
2 − 4 
2 
3 
2 × 3 = 6 
4 − 6 
1 
5 
1 × 5 = 5 
6 − 8 
5 
7 
5 × 7 = 35 
8 − 10 
6 
9 
6 × 9 = 54 
10 − 12 
2 
11 
2 ×11 = 22 
12 − 14 
3 
13 
3 × 13 = 39 
Total 
20 
162 
From the table, it can be observed that
Mean,
Therefore, mean number of plants per house is 8.1.
Here, direct method has been used as the values of class marks (x_{i}) and f_{i} are small.
Video Solution for real numbers (Page: 270 , Q.No.: 1)
NCERT Solution for Class 10 math  real numbers 270 , Question 1
Page No 270:
Question 2:
Consider the following distribution of daily wages of 50 workers of a factory.
Daily wages (in Rs) 
500 − 520 
520 − 540 
540 − 560 
560 − 580 
580 − 600 
Number of workers 
12 
14 
8 
6 
10 
Find the mean daily wages of the workers of the factory by using an appropriate method.
Answer:
To find the class mark for each interval, the following relation is used.
Class size (h) of this data = 20
Taking 550 as assured mean (a), d_{i}, u_{i}, and f_{i}u_{i} can be calculated as follows.
Daily wages (in Rs) 
Number of workers (f_{i}) 
x_{i} 
d_{i} = x_{i }− 550 
f_{i}u_{i} 

500 − 520 
12 
510 
− 40 
− 2 
− 24 
520 − 540 
14 
530 
− 20 
− 1 
− 14 
540 − 560 
8 
550 
0 
0 
0 
560 − 580 
6 
570 
20 
1 
6 
580 − 600 
10 
590 
40 
2 
20 
Total 
50 
− 12 
From the table, it can be observed that
$=550+\left(\frac{12}{50}\right)20\phantom{\rule{0ex}{0ex}}=550\frac{24}{5}\phantom{\rule{0ex}{0ex}}=5504.8\phantom{\rule{0ex}{0ex}}=545.2$
Therefore, the mean daily wage of the workers of the factory is Rs 545.20.
Video Solution for real numbers (Page: 270 , Q.No.: 2)
NCERT Solution for Class 10 math  real numbers 270 , Question 2
Page No 270:
Question 3:
The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs.18. Find the missing frequency f.
Daily pocket allowance (in Rs) 
11 − 13 
13 − 15 
15 −17 
17 − 19 
19 − 21 
21 − 23 
23 − 25 
Number of workers 
7 
6 
9 
13 
f 
5 
4 
Answer:
To find the class mark (x_{i}) for each interval, the following relation is used.
Given that, mean pocket allowance,
Taking 18 as assured mean (a), d_{i} and f_{i}d_{i} are calculated as follows.
Daily pocket allowance (in Rs) 
Number of children f_{i} 
Class mark x_{i} 
d_{i} = x_{i }− 18 
f_{i}d_{i} 
11 −13 
7 
12 
− 6 
− 42 
13 − 15 
6 
14 
− 4 
− 24 
15 − 17 
9 
16 
− 2 
− 18 
17 −19 
13 
18 
0 
0 
19 − 21 
f 
20 
2 
2 f 
21 − 23 
5 
22 
4 
20 
23 − 25 
4 
24 
6 
24 
Total 
2f − 40 
From the table, we obtain
Hence, the missing frequency, f, is 20.
Video Solution for real numbers (Page: 270 , Q.No.: 3)
NCERT Solution for Class 10 math  real numbers 270 , Question 3
Page No 271:
Question 4:
Thirty women were examined in a hospital by a doctor and the number of heart beats per minute were recorded and summarized as follows. Fine the mean heart beats per minute for these women, choosing a suitable method.
Number of heart beats per minute 
65 − 68 
68 − 71 
71 −74 
74 − 77 
77 − 80 
80 − 83 
83 − 86 
Number of women 
2 
4 
3 
8 
7 
4 
2 
Answer:
To find the class mark of each interval (x_{i}), the following relation is used.
Class size, h, of this data = 3
Taking 75.5 as assumed mean (a), di, u_{i}, f_{i}u_{i} are calculated as follows.
Number of heart beats per minute 
Number of women f_{i} 
x_{i} 
d_{i} = x_{i }− 75.5 
f_{i}u_{i} 

65 − 68 
2 
66.5 
− 9 
− 3 
− 6 
68 − 71 
4 
69.5 
− 6 
− 2 
− 8 
71 − 74 
3 
72.5 
− 3 
− 1 
− 3 
74 − 77 
8 
75.5 
0 
0 
0 
77 − 80 
7 
78.5 
3 
1 
7 
80 − 83 
4 
81.5 
6 
2 
8 
83 − 86 
2 
84.5 
9 
3 
6 
Total 
30 
4 
From the table, we obtain
Therefore, mean hear beats per minute for these women are 75.9 beats per minute.
Video Solution for real numbers (Page: 271 , Q.No.: 4)
NCERT Solution for Class 10 math  real numbers 271 , Question 4
Page No 271:
Question 5:
In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.
Number of mangoes 
50 − 52 
53 − 55 
56 − 58 
59 − 61 
62 − 64 
Number of boxes 
15 
110 
135 
115 
25 
Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?
Answer:
Number of mangoes 
Number of boxes f_{i} 
50 − 52 
15 
53 − 55 
110 
56 − 58 
135 
59 − 61 
115 
62 − 64 
25 
It can be observed that class intervals are not continuous. There is a gap of 1 between two class intervals. Therefore, has to be added to the upper class limit and has to be subtracted from the lower class limit of each interval.
Class mark (x_{i}) can be obtained by using the following relation.
Class size (h) of this data = 3
Taking 57 as assumed mean (a), d_{i}, u_{i}, f_{i}u_{i} are calculated as follows.
Class interval 
f_{i} 
x_{i} 
d_{i}_{ }= x_{i} − 57 
f_{i}u_{i} 

49.5 − 52.5 
15 
51 
− 6 
− 2 
− 30 
52.5 − 55.5 
110 
54 
− 3 
− 1 
− 110 
55.5 − 58.5 
135 
57 
0 
0 
0 
58.5 − 61.5 
115 
60 
3 
1 
115 
61.5 − 64.5 
25 
63 
6 
2 
50 
Total 
400 
25 
It can be observed that
Mean number of mangoes kept in a packing box is 57.19.
Step deviation method is used here as the values of f_{i, }d_{i} are big and also, there is a common multiple between all d_{i}.
Video Solution for real numbers (Page: 271 , Q.No.: 5)
NCERT Solution for Class 10 math  real numbers 271 , Question 5
Page No 271:
Question 6:
The table below shows the daily expenditure on food of 25 households in a locality.
Daily expenditure (in Rs) 
100 − 150 
150 − 200 
200 − 250 
250 − 300 
300 − 350 
Number of households 
4 
5 
12 
2 
2 
Find the mean daily expenditure on food by a suitable method.
Answer:
To find the class mark (x_{i}) for each interval, the following relation is used.
Class size = 50
Taking 225 as assumed mean (a), d_{i}, u_{i}, f_{i}u_{i} are calculated as follows.
Daily expenditure (in Rs) 
f_{i} 
x_{i} 
d_{i}_{ }= x_{i} − 225 
f_{i}u_{i} 

100 − 150 
4 
125 
− 100 
− 2 
− 8 
150 − 200 
5 
175 
− 50 
− 1 
− 5 
200 − 250 
12 
225 
0 
0 
0 
250 − 300 
2 
275 
50 
1 
2 
300 − 350 
2 
325 
100 
2 
4 
Total 
25 
− 7 
From the table, we obtain
Therefore, mean daily expenditure on food is Rs 211.
Video Solution for real numbers (Page: 271 , Q.No.: 6)
NCERT Solution for Class 10 math  real numbers 271 , Question 6
Page No 271:
Question 7:
To find out the concentration of SO_{2} in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below:
concentration of SO_{2} (in ppm) 
Frequency 
0.00 − 0.04 
4 
0.04 − 0.08 
9 
0.08 − 0.12 
9 
0.12 − 0.16 
2 
0.16 − 0.20 
4 
0.20 − 0.24 
2 
Find the mean concentration of SO_{2} in the air.
Answer:
To find the class marks for each interval, the following relation is used.
Class size of this data = 0.04
Taking 0.14 as assumed mean (a), d_{i}, u_{i},_{ }f_{i}u_{i} are calculated as follows.
Concentration of SO_{2} (in ppm) 
Frequency f_{i} 
Class mark x_{i} 
d_{i}_{ }= x_{i }− 0.14 
f_{i}u_{i} 

0.00 − 0.04 
4 
0.02 
− 0.12 
− 3 
− 12 
0.04 − 0.08 
9 
0.06 
− 0.08 
− 2 
− 18 
0.08 − 0.12 
9 
0.10 
− 0.04 
− 1 
− 9 
0.12 − 0.16 
2 
0.14 
0 
0 
0 
0.16 − 0.20 
4 
0.18 
0.04 
1 
4 
0.20 − 0.24 
2 
0.22 
0.08 
2 
4 
Total 
30 
− 31 
From the table, we obtain
Therefore, mean concentration of SO_{2} in the air is 0.099 ppm.
Video Solution for real numbers (Page: 271 , Q.No.: 7)
NCERT Solution for Class 10 math  real numbers 271 , Question 7
Page No 272:
Question 8:
A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.
Number of days 
0 − 6 
6 − 10 
10 − 14 
14 − 20 
20 − 28 
28 − 38 
38 − 40 
Number of students 
11 
10 
7 
4 
4 
3 
1 
Answer:
To find the class mark of each interval, the following relation is used.
Taking 17 as assumed mean (a), d_{i} and f_{i}d_{i} are calculated as follows.
Number of days 
Number of students f_{i} 
x_{i} 
d_{i}_{ }= x_{i} − 17 
f_{i}d_{i} 
0 − 6 
11 
3 
− 14 
− 154 
6 − 10 
10 
8 
− 9 
− 90 
10 − 14 
7 
12 
− 5 
− 35 
14 − 20 
4 
17 
0 
0 
20 − 28 
4 
24 
7 
28 
28 − 38 
3 
33 
16 
48 
38 − 40 
1 
39 
22 
22 
Total 
40 
− 181 
From the table, we obtain
Therefore, the mean number of days is 12.48 days for which a student was absent.
Video Solution for real numbers (Page: 272 , Q.No.: 8)
NCERT Solution for Class 10 math  real numbers 272 , Question 8
Page No 272:
Question 9:
The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
Literacy rate (in %) 
45 − 55 
55 − 65 
65 − 75 
75 − 85 
85 − 95 
Number of cities 
3 
10 
11 
8 
3 
Answer:
To find the class marks, the following relation is used.
Class size (h) for this data = 10
Taking 70 as assumed mean (a), d_{i}, u_{i}, and f_{i}u_{i} are calculated as follows.
Literacy rate (in %) 
Number of cities f_{i} 
x_{i} 
d_{i}_{ }= x_{i} − 70 
f_{i}u_{i} 

45 − 55 
3 
50 
− 20 
− 2 
− 6 
55 − 65 
10 
60 
− 10 
− 1 
− 10 
65 − 75 
11 
70 
0 
0 
0 
75 − 85 
8 
80 
10 
1 
8 
85 − 95 
3 
90 
20 
2 
6 
Total 
35 
− 2 
From the table, we obtain
Therefore, mean literacy rate is 69.43%.
Video Solution for real numbers (Page: 272 , Q.No.: 9)
NCERT Solution for Class 10 math  real numbers 272 , Question 9
Page No 275:
Question 1:
The following table shows the ages of the patients admitted in a hospital during a year:
age (in years) 
5 − 15 
15 − 25 
25 − 35 
35 − 45 
45 − 55 
55 − 65 
Number of patients 
6 
11 
21 
23 
14 
5 
Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.
Answer:
To find the class marks (x_{i}), the following relation is used.
Taking 30 as assumed mean (a), d_{i} and f_{i}d_{i}are calculated as follows.
Age (in years) 
Number of patients f_{i} 
Class mark x_{i} 
d_{i}_{ }= x_{i} − 30 
f_{i}d_{i} 
5 − 15 
6 
10 
− 20 
− 120 
15 − 25 
11 
20 
− 10 
− 110 
25 − 35 
21 
30 
0 
0 
35 − 45 
23 
40 
10 
230 
45 − 55 
14 
50 
20 
280 
55 − 65 
5 
60 
30 
150 
Total 
80 
430 
From the table, we obtain
Mean of this data is 35.38. It represents that on an average, the age of a patient admitted to hospital was 35.38 years.
It can be observed that the maximum class frequency is 23 belonging to class interval 35 − 45.
Modal class = 35 − 45
Lower limit (l) of modal class = 35
Frequency (f_{1}) of modal class = 23
Class size (h) = 10
Frequency (f_{0}) of class preceding the modal class = 21
Frequency (f_{2}) of class succeeding the modal class = 14
Mode =
Mode is 36.8. It represents that the age of maximum number of patients admitted in hospital was 36.8 years.
Video Solution for real numbers (Page: 275 , Q.No.: 1)
NCERT Solution for Class 10 math  real numbers 275 , Question 1
Page No 275:
Question 2:
The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:
Lifetimes (in hours) 
0 − 20 
20 − 40 
40 − 60 
60 − 80 
80 − 100 
100 − 120 
Frequency 
10 
35 
52 
61 
38 
29 
Determine the modal lifetimes of the components.
Answer:
From the data given above, it can be observed that the maximum class frequency is 61, belonging to class interval 60 − 80.
Therefore, modal class = 60 − 80
Lower class limit (l) of modal class = 60
Frequency (f_{1}) of modal class = 61
Frequency (f_{0}) of class preceding the modal class = 52
Frequency (f_{2}) of class succeeding the modal class = 38
Class size (h) = 20
Therefore, modal lifetime of electrical components is 65.625 hours.
Video Solution for real numbers (Page: 275 , Q.No.: 2)
NCERT Solution for Class 10 math  real numbers 275 , Question 2
Page No 275:
Question 3:
The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure.
Expenditure (in Rs) 
Number of families 
1000 − 1500 
24 
1500 − 2000 
40 
2000 − 2500 
33 
2500 − 3000 
28 
3000 − 3500 
30 
3500 − 4000 
22 
4000 − 4500 
16 
4500 − 5000 
7 
Answer:
It can be observed from the given data that the maximum class frequency is 40, belonging to 1500 − 2000 intervals.
Therefore, modal class = 1500 − 2000
Lower limit (l) of modal class = 1500
Frequency (f_{1}) of modal class = 40
Frequency (f_{0}) of class preceding modal class = 24
Frequency (f_{2}) of class succeeding modal class = 33
Class size (h) = 500
Therefore, modal monthly expenditure was Rs 1847.83.
To find the class mark, the following relation is used.
Class size (h) of the given data = 500
Taking 2750 as assumed mean (a), d_{i}, u_{i}, and f_{i}u_{i}are calculated as follows.
Expenditure (in Rs) 
Number of families f_{i} 
x_{i} 
d_{i} = x_{i} − 2750 
f_{i}u_{i} 

1000 − 1500 
24 
1250 
− 1500 
− 3 
− 72 
1500 − 2000 
40 
1750 
− 1000 
− 2 
− 80 
2000 − 2500 
33 
2250 
− 500 
− 1 
− 33 
2500 − 3000 
28 
2750 
0 
0 
0 
3000 − 3500 
30 
3250 
500 
1 
30 
3500 − 4000 
22 
3750 
1000 
2 
44 
4000 − 4500 
16 
4250 
1500 
3 
48 
4500 − 5000 
7 
4750 
2000 
4 
28 
Total 
200 
− 35 
From the table, we obtain
Therefore, mean monthly expenditure was Rs 2662.50.
Video Solution for real numbers (Page: 275 , Q.No.: 3)
NCERT Solution for Class 10 math  real numbers 275 , Question 3
Page No 276:
Question 4:
The following distribution gives the statewise teacherstudent ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures.
Number of students per teacher 
Number of states/U.T 
15 − 20 
3 
20 − 25 
8 
25 − 30 
9 
30 − 35 
10 
35 − 40 
3 
40 − 45 
0 
45 − 50 
0 
50 − 55 
2 
Answer:
It can be observed from the given data that the maximum class frequency is 10 belonging to class interval 30 − 35.
Therefore, modal class = 30 − 35
Class size (h) = 5
Lower limit (l) of modal class = 30
Frequency (f_{1}) of modal class = 10
Frequency (f_{0}) of class preceding modal class = 9
Frequency (f_{2}) of class succeeding modal class = 3
It represents that most of the states/U.T have a teacherstudent ratio as 30.6.
To find the class marks, the following relation is used.
Taking 32.5 as assumed mean (a), d_{i}, u_{i}, and f_{i}u_{i} are calculated as follows.
Number of students per teacher 
Number of states/U.T (f_{i}) 
x_{i} 
d_{i} = x_{i} − 32.5 
f_{i}u_{i} 

15 − 20 
3 
17.5 
− 15 
− 3 
− 9 
20 − 25 
8 
22.5 
− 10 
− 2 
− 16 
25 − 30 
9 
27.5 
− 5 
− 1 
− 9 
30 − 35 
10 
32.5 
0 
0 
0 
35 − 40 
3 
37.5 
5 
1 
3 
40 − 45 
0 
42.5 
10 
2 
0 
45 − 50 
0 
47.5 
15 
3 
0 
50 − 55 
2 
52.5 
20 
4 
8 
Total 
35 
− 23 
Therefore, mean of the data is 29.2.
It represents that on an average, teacher−student ratio was 29.2.
Video Solution for real numbers (Page: 276 , Q.No.: 4)
NCERT Solution for Class 10 math  real numbers 276 , Question 4
Page No 276:
Question 5:
The given distribution shows the number of runs scored by some top batsmen of the world in oneday international cricket matches.
Runs scored 
Number of batsmen 
3000 − 4000 
4 
4000 − 5000 
18 
5000 − 6000 
9 
6000 − 7000 
7 
7000 − 8000 
6 
8000 − 9000 
3 
9000 − 10000 
1 
10000 − 11000 
1 
Find the mode of the data.
Answer:
From the given data, it can be observed that the maximum class frequency is 18, belonging to class interval 4000 − 5000.
Therefore, modal class = 4000 − 5000
Lower limit (l) of modal class = 4000
Frequency (f_{1}) of modal class = 18
Frequency (f_{0}) of class preceding modal class = 4
Frequency (f_{2}) of class succeeding modal class = 9
Class size (h) = 1000
Therefore, mode of the given data is 4608.7 runs.
Video Solution for real numbers (Page: 276 , Q.No.: 5)
NCERT Solution for Class 10 math  real numbers 276 , Question 5
Page No 276:
Question 6:
A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data:
Number of cars 
0 − 10 
10 − 20 
20 − 30 
30 − 40 
40 − 50 
50 − 60 
60 − 70 
70 − 80 
Frequency 
7 
14 
13 
12 
20 
11 
15 
8 
Answer:
From the given data, it can be observed that the maximum class frequency is 20, belonging to 40 − 50 class intervals.
Therefore, modal class = 40 − 50
Lower limit (l) of modal class = 40
Frequency (f_{1}) of modal class = 20
Frequency (f_{0}) of class preceding modal class = 12
Frequency (f_{2}) of class succeeding modal class = 11
Class size = 10
Therefore, mode of this data is 44.7 cars.
Video Solution for real numbers (Page: 276 , Q.No.: 6)
NCERT Solution for Class 10 math  real numbers 276 , Question 6
Page No 287:
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 
Answer:
To find the class marks, the following relation is used.
Taking 135 as assumed mean (a), d_{i}, u_{i}, f_{i}u_{i} are calculated according to step deviation method as follows.
Monthly consumption (in units) 
Number of consumers (f _{i}) 
x_{i} class mark 
d_{i}= x_{i}− 135 

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 obtain
From the table, it can be observed that the 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 (f_{1}) of modal class = 20
Frequency (f_{0}) of class preceding modal class = 13
Frequency (f_{2}) of class succeeding the modal class = 14
To find the median of the given data, cumulative frequency is calculated as follows.
Monthly consumption (in units) 
Number of consumers 
Cumulative frequency 
65 − 85 
4 
4 
85 − 105 
5 
4 + 5 = 9 
105 − 125 
13 
9 + 13 = 22 
125 − 145 
20 
22 + 20 = 42 
145 − 165 
14 
42 + 14 = 56 
165 − 185 
8 
56 + 8 = 64 
185 − 205 
4 
64 + 4 = 68 
From the table, we obtain
n = 68
Cumulative frequency (cf) just greater than is 42, belonging to interval 125 − 145.
Therefore, median class = 125 − 145
Lower limit (l) of median class = 125
Class size (h) = 20
Frequency (f) of median class = 20
Cumulative frequency (cf) of class preceding median class = 22
Therefore, median, mode, mean of the given data is 137, 135.76, and 137.05 respectively.
The three measures are approximately the same in this case.
Video Solution for real numbers (Page: 287 , Q.No.: 1)
NCERT Solution for Class 10 math  real numbers 287 , Question 1
Page No 287:
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 
Answer:
The cumulative frequency for the given data is calculated as follows.
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 − 50 
y 
40+ x + y 
50 − 60 
5 
45 + x + y 
Total (n) 
60 
From the table, it can be observed that n = 60
45 + x + y = 60
x + y = 15 (1)
Median of the data is given as 28.5 which lies in interval 20 − 30.
Therefore, 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, the values of x and y are 8 and 7 respectively.
Video Solution for real numbers (Page: 287 , Q.No.: 2)
NCERT Solution for Class 10 math  real numbers 287 , Question 2
Page No 287:
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 
Answer:
Here, class width is not the same. There is no requirement of adjusting the frequencies according to class intervals. The given frequency table is of less than type represented with upper class limits. The policies were given only to persons with age 18 years onwards but less than 60 years. Therefore, class intervals with their respective cumulative frequency can be defined as below.
Age (in years) 
Number of policy holders (f_{i}) 
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) 
From the table, it can be observed that n = 100.
Cumulative frequency (cf) just greater than is 78, belonging to interval 35 − 40.
Therefore, 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
Therefore, median age is 35.76 years.
Video Solution for real numbers (Page: 287 , Q.No.: 3)
NCERT Solution for Class 10 math  real numbers 287 , Question 3
Page No 288:
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 f_{i} 
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)
Answer:
The given data does not have continuous class intervals. It can be observed that the difference between two class intervals is 1. Therefore, has to be added and subtracted to upper class limits and lower class limits respectively.
Continuous class intervals with respective cumulative frequencies can be represented as follows.
Length (in mm) 
Number or leaves f_{i} 
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, it can be observed that the cumulative frequency just greater than 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
Median
Therefore, median length of leaves is 146.75 mm.
Video Solution for real numbers (Page: 288 , Q.No.: 4)
NCERT Solution for Class 10 math  real numbers 288 , Question 4
Page No 289:
Question 5:
Find the following table 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.
Answer:
Thecumulative frequencies with their respective class intervals are as follows.
Life time 
Number of lamps (f_{i}) 
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 
It can be observed that the 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
Median
= 3406.976
Therefore, median life time of lamps is 3406.98 hours.
Video Solution for real numbers (Page: 289 , Q.No.: 5)
NCERT Solution for Class 10 math  real numbers 289 , Question 5
Page No 289:
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.
Answer:
The cumulative frequencies with their respective class intervals are as follows.
Number of letters 
Frequency (f_{i}) 
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 
It can be observed that the 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
Median
= 8.05
To find the class marks of the given class intervals, the following relation is used.
Taking 11.5 as assumed mean (a), d_{i}, u_{i}, and f_{i}u_{i} are calculated according to step deviation method as follows.
Number of letters 
Number of surnames f_{i} 
x_{i} 
d_{i} = x_{i}− 11.5 
f_{i}u_{i} 

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 
From the table, we obtain
∑f_{i}u_{i} = −106
∑f_{i} = 100
Mean,
= 11.5 − 3.18 = 8.32
The data in the given table can be written as
Number of letters 
Frequency (f_{i}) 
1 − 4 
6 
4 − 7 
30 
7 − 10 
40 
10 − 13 
16 
13 − 16 
4 
16 − 19 
4 
Total (n) 
100 
From the table, it can be observed that the maximum class frequency is 40 belonging to class interval 7 − 10.
Modal class = 7 − 10
Lower limit (l) of modal class = 7
Class size (h) = 3
Frequency (f_{1}) of modal class = 40
Frequency (f_{0}) of class preceding the modal class = 30
Frequency (f_{2}) of class succeeding the modal class = 16
Therefore, median number and mean number of letters in surnames is 8.05 and 8.32 respectively while modal size of surnames is 7.88.
Video Solution for real numbers (Page: 289 , Q.No.: 6)
NCERT Solution for Class 10 math  real numbers 289 , Question 6
Page No 289:
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 
Answer:
The cumulative frequencies with their respective class intervals are as follows.
Weight (in kg) 
Frequency (fi) 
Cumulative frequency 
40 − 45 
2 
2 
45 − 50 
3 
2 + 3 = 5 
50 − 55 
8 
5 + 8 = 13 
55 − 60 
6 
13 + 6 = 19 
60 − 65 
6 
19 + 6 = 25 
65 − 70 
3 
25 + 3 = 28 
70 − 75 
2 
28 + 2 = 30 
Total (n) 
30 
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
Median
= 56.67
Therefore, median weight is 56.67 kg.
Video Solution for real numbers (Page: 289 , Q.No.: 7)
NCERT Solution for Class 10 math  real numbers 289 , Question 7
Page No 293:
Question 1:
The following distribution gives the daily income of 50 workers of a factory.
Daily income (in Rs) 
100 − 120 
120 − 140 
140 − 160 
160 − 180 
180 − 200 
Number of workers 
12 
14 
8 
6 
10 
Convert the distribution above to a less than type cumulative frequency distribution, and draw its ogive.
Answer:
The frequency distribution table of less than type is as follows.
Daily income (in Rs) (upper class limits) 
Cumulative frequency 
Less than 120 
12 
Less than 140 
12 + 14 = 26 
Less than 160 
26 + 8 = 34 
Less than 180 
34 + 6 = 40 
Less than 200 
40 + 10 = 50 
Taking upper class limits of class intervals on xaxis and their respective frequencies on yaxis, its ogive can be drawn as follows.
Video Solution for real numbers (Page: 293 , Q.No.: 1)
NCERT Solution for Class 10 math  real numbers 293 , Question 1
Page No 293:
Question 2:
During the medical checkup of 35 students of a class, their weights were recorded as follows:
Weight (in kg) 
Number of students 
Less than 38 
0 
Less than 40 
3 
Less than 42 
5 
Less than 44 
9 
Less than 46 
14 
Less than 48 
28 
Less than 50 
32 
Less than 52 
35 
Draw a less than type ogive for the given data. Hence obtain the median weight from the graph verify the result by using the formula.
Answer:
The given cumulative frequency distributions of less than type are
Weight (in kg) upper class limits 
Number of students (cumulative frequency) 
Less than 38 
0 
Less than 40 
3 
Less than 42 
5 
Less than 44 
9 
Less than 46 
14 
Less than 48 
28 
Less than 50 
32 
Less than 52 
35 
Taking upper class limits on xaxis and their respective cumulative frequencies on yaxis, its ogive can be drawn as follows.
Here, n = 35
So, = 17.5
Mark the point A whose ordinate is 17.5 and its xcoordinate is 46.5. Therefore, median of this data is 46.5.
It can be observed that the difference between two consecutive upper class limits is 2. The class marks with their respective frequencies are obtained as below.
Weight (in kg) 
Frequency (f) 
Cumulative frequency 
Less than 38 
0 
0 
38 − 40 
3 − 0 = 3 
3 
40 − 42 
5 − 3 = 2 
5 
42 − 44 
9 − 5 = 4 
9 
44 − 46 
14 − 9 = 5 
14 
46 − 48 
28 − 14 = 14 
28 
48 − 50 
32 − 28 = 4 
32 
50 − 52 
35 − 32 = 3 
35 
Total (n) 
35 
The cumulative frequency just greater thanis 28, belonging to class interval 46 − 48.
Median class = 46 − 48
Lower class limit (l) of median class = 46
Frequency (f) of median class = 14
Cumulative frequency (cf) of class preceding median class = 14
Class size (h) = 2
Therefore, median of this data is 46.5.
Hence, the value of median is verified.
Video Solution for real numbers (Page: 293 , Q.No.: 2)
NCERT Solution for Class 10 math  real numbers 293 , Question 2
Page No 293:
Question 3:
The following table gives production yield per hectare of wheat of 100 farms of a village.
Production yield (in kg/ha) 
50 − 55 
55 − 60 
60 − 65 
65 − 70 
70 − 75 
75 − 80 
Number of farms 
2 
8 
12 
24 
38 
16 
Change the distribution to a more than type distribution and draw ogive.
Answer:
The cumulative frequency distribution of more than type can be obtained as follows.
Production yield (lower class limits) 
Cumulative frequency 
more than or equal to 50 
100 
more than or equal to 55 
100 − 2 = 98 
more than or equal to 60 
98 − 8 = 90 
more than or equal to 65 
90 − 12 = 78 
more than or equal to 70 
78 − 24 = 54 
more than or equal to 75 
54 − 38 = 16 
Taking the lower class limits on xaxis and their respective cumulative frequencies on yaxis, its ogive can be obtained as follows.
Video Solution for real numbers (Page: 293 , Q.No.: 3)
NCERT Solution for Class 10 math  real numbers 293 , Question 3
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