Question 1. Define cumulative frequency distribution.
Solution:
The structure showing the distribution of cumulative frequencies over various classes is known as cumulative frequency distribution.
Question 2. Explain the difference between a frequency distribution and a cumulative frequency distribution.
Solution:
The method representing raw data which is easily understandable is called frequency distribution. The structure showing the distribution of cumulative frequencies over various classes is known as cumulative frequency distribution.
Question 3. The marks scored by 55 students in a test are given below:
Marks | 0-5 | 5-10 | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 |
No. of students | 2 | 6 | 13 | 17 | 11 | 4 | 2 |
Prepare a cumulative frequency table.
Solution:
Marks
No. of students
Marks
No. of students
0-5
2
Less than 5
2
5-10
6
Less than 10
8
10-15
13
Less than 15
21
15-20
17
Less than 20
38
20-25
11
Less than 25
49
25-30
4
Less than 30
53
30-35
2
Less than 35
55
N = 55
Question 4. Following are the ages of 360 patients getting medical treatment in a hospital on a day:
Age (in years) | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 |
No. of Patients | 90 | 50 | 60 | 80 | 50 | 30 |
Construct a cumulative frequency distribution.
Solution:
Age (in years)
No. of patients
Age (in years)
No. of patients
10-20
90
Less than 20
90
20-30
50
Less than 30
140
30-40
60
Less than 40
200
40-50
80
Less than 50
280
50-60
50
Less than 60
330
60-70
30
Less than 70
360
N = 360
Question 5. The water bills (in rupees) of 32 houses in a certain street for the period 1.1.98 to 31.3.98 are given below :
56, 43, 32, 38, 56, 24, 68, 85, 52, 47, 35, 58, 63, 74, 27, 84, 69, 35, 44, 75, 55, 30, 54, 65, 45, 67, 95, 72, 43, 65, 35, 59.
Tabulate the data and present the data as a cumulative frequency table using 70-79 as one of the class intervals.
Solution:
The minimum and maximum bills are Rs 24 and Rs 95 respectively.
Range of the class interval is 95-24 = 71.
Class interval given = 70-79, with the class size being 9.
number of classes = range/class-size = 71/9 = 7.80 which is approximately 8.
Cumulative frequency distribution is as follows :
Bills (Rs.)
No. of houses(Frequency)
16-25
1
25-34
3
34-43
5
43-52
4
52-61
7
61-70
6
70-79
3
79-88
2
88-97
1
Question 6. The number of books in different shelves of a library are as follows :
30, 32, 28, 24, 20, 25, 38, 37, 40, 45, 16, 20, 19, 24, 27, 30, 32, 34, 35, 42, 27, 28, 19, 34, 38, 39, 42, 29, 24, 27, 22, 29, 31, 19, 27, 25, 28, 23, 24, 32, 34, 18, 27, 25, 37, 31, 24, 23, 43, 32, 28, 31, 24, 23, 26, 36, 32, 29, 28, 21.
Prepare a cumulative frequency distribution table using 45-49 as the last class interval.
Solution:
The minimum and maximum book shelves are 16 and 45 respectively.
The range of the book shelves 45-16 = 29
We have the following class interval , 45 – 49, with the class size =4.
Number of classes = Range/Class size = 29/4 = 7.25 , which approximates to 8 nearly.
No. of books
No. of shelves (Frequency)
Cumulative Frequency
13-17
1
1
17-21
6
7
21-25
11
18
25-29
15
33
29-33
12
45
33-37
5
50
37-41
6
56
41-45
3
59
45-49
1
50
Question 7. Given below are the cumulative frequencies showing the weights of 685 students of a school. Prepare a frequency distribution table.
Weight (in kg) | No. of students |
Below 25 | 0 |
Below 30 | 24 |
Below 35 | 78 |
Below 40 | 183 |
Below 45 | 294 |
Below 50 | 408 |
Below 55 | 543 |
Below 60 | 621 |
Below 65 | 674 |
Below 70 | 685 |
Solution:
Weight (in kg)
No. of students
Class interval
Frequency
Below 30
24
25-30
24 – 0 =24
Below 35
78
30-35
78- 24 = 54
Below 40
183
35-40
183 – 78 = 105
Below 45
294
40-45
294 – 183 = 111
Below 50
408
45-50
408 – 294 = 114
Below 55
543
50-55
543 – 408 = 135
Below 60
621
55-60
621 – 543 = 78
Below 65
674
60-65
674 – 621 = 53
Below 70
685
65-70
685 – 674 = 11
Question 8. The following cumulative frequency distribution table shows the daily electricity consumption (in kW) of 40 factories in an industrial state:
Consumption (in kW) | No. of factories |
Below 240 | 1 |
Below 270 | 4 |
Below 300 | 8 |
Below 330 | 24 |
Below 360 | 33 |
Below 390 | 38 |
Below 420 | 40 |
(i) Represent this as a frequency distribution table.
(ii) Prepare a cumulative frequency table.
Solution:
(i)
Consumption (in kW)
No. of factories
Class interval
Frequency
Below 240
1
0-240
1
Below 270
4
240-270
4-1 = 3
Below 300
8
270-300
8-4 = 4
Below 330
24
300-330
24-8 = 16
Below 360
33
330-360
33 – 24 = 9
Below 390
38
360-390
38-33 = 5
Below 420
40
390-420
40-38 = 2
(ii)
Class interval
Frequency
Consumption (in kW)
No. of factories
0-240
1
More than 0
40
240-270
3
More than 270
40-1 = 39
270-300
4
More than 270
39-3 = 36
300-330
16
More than 300
36-4 = 32
330-360
9
More than 330
32-16= 16
360-390
5
More than 360
16-9 = 7
390-420
2
More than 390
7-5=2
More than 420
2-2 = 0
N = 40
Question 9. Given below is a cumulative frequency distribution table showing ages of people living in a locality:
Age in years | No. of persons |
Above 108 | 0 |
Above 96 | 1 |
Above 84 | 3 |
Above 72 | 5 |
Above 60 | 20 |
Above 48 | 158 |
Above 36 | 427 |
Above 24 | 809 |
Above 12 | 1026 |
Above 0 | 1124 |
Prepare a frequency distribution table.
Solution:
Age (in years)
No. of persons
Class interval
Frequency
Above 0
1124
0-12
1124-1026 = 98
Above 12
1026
12-24
1026-809 = 217
Above 24
809
24-36
809-427 = 382
Above 36
427
36-48
427-158= 269
Above 48
158
48-60
158-20 = 138
Above 60
20
60-72
20 -5 = 15
Above 72
5
72-84
5-3 = 2
Above 84
3
84-96
3-1 = 2
Above 96
1
96-108
1-0 = 1