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 |