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Chapter 22: Tabular Representation of Statistical Data Exercise – 22.2

Question: 1

Define cumulative frequency distribution.

Solution:

Cumulative frequency distribution:

A table which displays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequency distribution or cumulative frequency distribution table.

 

Question: 2

Explain the difference between a frequency distribution and a cumulative frequency distribution.

Solution:

Frequency table or frequency distribution is a method to represent raw data in the form from which one can easily understand the information contained in a raw data, where as a table which plays the manner in which cumulative frequencies are distributed over various classes is called a 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 Cumulative Frequency
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 table.

Solution:

Age (in years) No. of students Marks Cumulative Frequency
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 1.198 to 31.398 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, 69.

Tabulate the data and present the data as a cumulative frequency table using 70-79 as one of the class intervals.

Solution:

The minimum bill is Rs 24

The maximum bill is Rs 95

Range = Maximum bill-Minimum bill = 95 - 24 = 71

Given class interval is 70-79. So, Class size = 79 – 70 = 9

Therefore number of classes = Range/class size = 71/9 = 7.80

Number of classes = 8

The cumulative frequency distribution is as follows:

Bills No. of Houses (Frequency) Cumulative Frequency
16 – 25 1 1
25 – 34 3 4
34 – 43 5 9
43 – 52 4 13
52 – 61 7 20
61 – 70 6 26
70 – 79 3 29
79 – 88 2 31
88 – 97 1 32


Question: 6

The number of books in different shelves of a library is 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 number of bookshelves is 16

and maximum number of bookshelves is 45

Range = Maximum book shelves-Minimum book shelves = 45 - 16 = 29

Given class interval is 45 – 49.So, class size = 49 - 45 = 4

Therefore number of classes = Range/class size = 29/4 = 7.25

Number of classes = 8

The cumulative frequency distribution is as follows:

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 60


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 30 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 – 65 671 – 621 = 53
Below 70 685 65 – 70 685 – 671 = 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

(1) Represent this as a frequency distribution table.

(2) Prepare a cumulative frequency table.

Solution:

(1)

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

(2)

Class interval Frequency Consumption(in KW) No. of factories
0 – 240 1 More than 0 40
240 – 270 3 More than 240 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 the people living in a locality:

Age in years No. of years
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 – 24 217
Above 24 809 24 – 36 382
Above  36 427 36 – 48 269
Above 38 158 48 – 60 138
Above 60 20 60 – 72 15
Above 72 5 72 – 84 5 – 3 = 2
Above 84 3 84 – 96 3 – 1 = 2
Above 96 3 96 – 108 1 – 0 = 1

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