# Cumulative Frequency

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 Interval0 - 2020 - 4040 - 6060 - 8080 - 100 Frequency47253 Cumulative Frequency411131821 (= 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 - 1920 - 2930 - 3940 - 4950 - 5960 - 6970 - 7980 - 89 Number of Students71120465737157

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.