Cumulative frequency distribution with example

Cumulative frequency distribution introduction:

From a grouped frequency distribution, we can simply read how many observations fall within different classes. E.g. consider the following frequency distance.

Mark:                      0 – 20             20 – 40          40 – 60          60 – 80

No. of students:         5                      25                  45                   19

The number of students who secured marks 40 or more but less than 60 are 45, those who secured marks 20 or more but less than 40 are 25. But if one wants to know the number of students who obtained 40 or more marks or those who obtained less than 60 marks, then the given frequency distribution does not serve our purpose. To answer such type of questions we construct a frequency distribution known as cumulative frequency distribution.

Definition:

The total frequency of all the classes less than the upper class boundary or move than or equal to the lower class boundary of a particular class is called the cumulative frequency of that class. A table showing the cumulative frequencies of different classes is called the cumulative frequency distribution.

There are two methods of constructing the cumulative frequency distributions.

i.                        Less than type cumulative frequency distance:

A table showing total frequencies of all the classes less than the upper class boundaries is called less than type cumulative frequency distribution. In this type the frequencies are serially added from top to bottom. The cumulative frequency of last class should equal the total frequency.

ii.                        More than type cumulative frequency distance:

A table showing total frequencies of all the classes more than equal to the lower class boundaries is called the more than type cumulative frequency distribution. In this type the frequencies are serially added from bottom to top. The cumulative frequency of first class should equal the total frequency.

Question:

From the following frequency distribution construct less than type and more than type cumulative frequency distribution.

Marks:    0 – 10     10 – 20       20 – 30         30 – 40    40 – 50      50 - 60

Frequency:   5              25                37                  43            13                7

Sol:

a)                 Less type cumulative frequency distribution

Marks	No. of students	Cumulative frequency
Below 10	5	5
Below 20	5 + 25	30
Below 30	5 + 25 +37	67
Below 40	5 + 25 + 37 + 43	110
Below 50	5 + 25 + 37 + 43 + 13	123
Below 60	5 + 25 + 37 + 43 + 13 +7	130

b)                More than type cumulative frequency distribution

Marks	No. of students	Cumulative frequency
Above 0	7 + 13 + 43 + 37 + 25 + 5	130
Above 10	7 + 13 + 43 + 37 + 25	125
Above 20	7 + 13 + 43 + 37	100
Above 30	7 + 13 + 43	63
Above 40	7 + 13	20
Above 50	7	7