Here we will learn cumulative frequency.
The cumulative frequency of a value of a variable is the number of values in the collection of data less than or equal to the value of the variable.
For example: Let the raw data be 2, 10, 18, 25, 15, 16, 15, 3, 27, 17, 15, 16. The cumulative frequency of 15 = 6 (Since, values ≤ 15 are 2, 10, 15, 15, 3, 15).
The cumulative frequency of a class interval (overlapping or nonoverlapping) is the sum of the frequencies of earlier class intervals and the concerned class interval.
For example: Consider the frequency distribution below.
Class Interval
0 - 20
20 - 40
40 - 60
60 - 80
80 - 100
Total
Frequency
4
7
2
5
3
21
The cumulative frequency of 0 – 20 is 4, of 20 – 40 is 11 (i.e., 4 + 7), of 40 – 60 is 13(i.e., 4 + 7 + 2), etc.
The following cumulative frequency table can be constructed from the above frequency table.
Class Interval 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100 |
Frequency 4 7 2 5 3 |
Cumulative Frequency 4 11 13 18 21 |
(= 4 + 7) (= 4 + 7 + 2) (= 4 + 7 + 2 + 5) (= 4 + 7 + 2 + 5 + 3) |
Another way of representing the same table is shown below.
Value of the Variable
Under 20
Under 40
Under 60
Under 80
Under 100
Frequency
4
11
13
18
21
Here the class interval 0 – 20 has the frequency 4. The class interval 20 – 40 includes those values of the variable which are under 40 but not under 20. So, the frequency of the class interval 20 – 40 is 11 – 4, that is 7. Clearly, such a table is in fact a cumulative frequency table for overlapping class intervals.
Solved Examples on Cumulative Frequency Table:
1. For the collection of numbers 12, 15, 8, 13, 12, 15, 9, 16, 24, 20, 20, 16 and 10, answer the following:
(i) What is the cumulative frequency of 16?
(ii) If 15 - 20 be an overlapping class interval when the numbers are grouped, find the cumulative frequency of the class interval.
(iii) If 15 - 20 be a nonoverlapping class interval when the numbers are grouped, find the cumulative frequency of the class interval.
Solution:
(i) 10 (Since values ≤ 16 are 12, 15, 8, 13, 12, 15, 9, 16, 16, 10)
(ii) 10 (Since values < 20 are 12, 15, 8, 13, 12, 15, 9, 16, 16, 10)
(iii) 12 (Since values ≤ 20 are 12, 15, 8, 13, 12, 15, 9, 16, 16, 10)
2. The marks of 200 students in a test were recorded and shown by the following frequency distribution.
Marks % 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69 70 - 79 80 - 89 |
Number of Students 7 11 20 46 57 37 15 7 |
Construct the cumulative frequency table.
Also answer the following.
(i) How many students obtained less than 50 marks?
(ii) How many students obtained at least 60 marks?
Solution:
The cumulative frequency table is as given below.
Class Interval
10 - 19
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
70 - 79
80 - 89
Frequency
7
11
20
46
57
37
15
7
Cumulative Frequency
7
18
38
84
141
178
193
200
(i) The number of students obtaining less than 50 marks
= the cumulative frequency of the class interval 40 - 49 = 84.
(ii) The number of students obtaining at least 60 marks
= total number of students - the number of students getting less than or equal to 59
= 200 - 141
= 59.
Alternative method
The number of students obtaining at least 60 marks
= Sum of the frequencies of the class intervals 60 - 69, 70 - 79 and 80 - 89
= 37 + 15 + 7
= 59.
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