Question 1: Define cumulative frequency distribution.

Answer:

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 table.

Cumulative frequency distributions are of two types viz. less than and greater than or more than. For less than cumulative frequency distributions we add up the frequencies from the above and for greater than cumulative frequencies we add up the frequencies from below.

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Question 2: Explain the difference between a frequency distribution and a cumulative frequency distribution.

Answer:

Frequency Distribution: Frequency distribution in statistics provides the information of the number of occurrences (frequency) of distinct values distributed within a given period of time or interval, in a list, table, or graphical representation.

Frequency distributions are of two types:

(i) Discrete frequency distribution     (ii) Continuous or grouped frequency distribution

Cumulative frequency distribution: Refer to the above question.

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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.

Answer:

Less than cumulative frequency

Marks Cumulative Frequency
Less than 5 2
Less than 10 8
Less than 15 21
Less than 20 38
Less than 25 49
Less than 30 53
Less than 35 55

More than cumulative frequency table

Marks Cumulative Frequency
More than 0 55
More than 4 53
More than 9 47
More than 14 34
More than 19 17
More than 24 6
More than 29 2

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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 40-40 50-60 60-70
No. of Patients 90 50 60 80 50 30

Construct a cumulative frequency distribution.

Answer:

Less than cumulative frequency

Marks Cumulative Frequency
Less than 20 90
Less than 30 140
Less than 40 200
Less than 50 280
Less than 60 330
Less than 70 360

More than cumulative frequency table

Marks Cumulative Frequency
More than 10 360
More than 19 270
More than 29 220
More than 39 160
More than 49 80
More than 59 30

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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.

Answer:

First understand the frequency distribution. Arrange the given data in ascending order. Which is:

24,27,30,32,35,35,35,38,43,43,44,45,47,52,54,55,56,56,58,59,63,65, \\ 65,67,68,69,72,74,75,84,85,95

Hence, we get

Class Interval Frequency
20-29 2
30-39 6
40-49 5
50-59 7
60-69 6
70-79 3
80-89 2
90-99 1

Less than cumulative frequency

Water Bill Cumulative Frequency
Less than 30 2
Less than 40 8
Less than 50 13
Less than 60 20
Less than 70 26
Less than 80 29
Less than 90 31
Less than 100 32

More than cumulative frequency table

Marks Cumulative Frequency
More than 19 32
More than 29 30
More than 39 24
More than 49 19
More than 59 12
More than 69 6
More than 79 3
More than 89 1

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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.

Answer:

First arrange the data in ascending order:

16,18,19,19,19,20,20,21,22,23,23,23,24,24,24,24,24,24,25,25,25,26,27,27,27,27,27,28, \\ 28,28,28,28,29,29,29,30,30,31,31,31,32,32,32,32,32,34,34,34,35,36,37,37,38,38,39, \\ 40,42,42,43,45

Hence, we get

Class Interval Frequency
15-19 5
20-24 13
25-29 17
30-34 13
35-39 7
40-44 4
45-49 1

Less than cumulative frequency

Water Bill Cumulative Frequency
Less than 20 5
Less than 25 18
Less than 30 35
Less than 35 48
Less than 40 55
Less than 45 59
Less than 50 60

More than cumulative frequency table

Marks Cumulative Frequency
More than 14 60
More than 19 55
More than 24 42
More than 29 25
More than 34 12
More than 39 5
More than 44 1

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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

Answer:

Weight No. of Students
25-30 24
30-35 54
35-40 105
40-45 111
45-50 114
50-55 135
55-60 78
60-65 53
65-70 11

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Question 8: The following cumulative frequency distribution table shows the daily electricity consumption (in kVV) 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.

Answer:

Consumption in kW No. of families
0-240 1
240-270 3
270-300 4
300-330 16
330-360 9
360-390 5
390-420 2

 

Consumption in kW No. of families
More than 0 40
More than 240 39
More than 270 36
More than 300 32
More than 330 16
More than 360 7
More than 390 2
More than 420 0

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Question 9: Given below is a cumulative frequency distribution table showing the 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.

Answer:

Age in years No of persons
0-12 98
12-24 217
24-36 382
36-48 269
48-60 138
60-72 15
72-84 2
84-96 2
96-108 1

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