Question 1: Define cumulative frequency distribution.

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.

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.

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.

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.

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.

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

 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.

 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.