Generally, the word Cumulative means "how much so far". In statistics, it is the running total of all frequencies. Cumulative frequency corresponding to a particular value is the sum of all the frequencies up to and including that value.

For example, the below cumulative frequency table displays the valconic eruption between 1991 to 2000.

Years           Frequency            Cumulative Frequency       
1991 - 92   10  10
 1992 - 94  15 10 + 15 = 25
 1994 - 96  925 + 9 = 34
 1996 - 98  13 34 + 13 = 47
 1998 - 2000  7 47 + 7 = 54

From the table, the cumulative frequency for the total number of valconic eruption that took place between the years 1994 to 1998 is 34 + 13 = 47. The cumulative frequency is mostly used while analyzing the data, where the value of the cumulative frequency represents the number of samples in the data, that lie below the current value. It is also useful while displaying the data using the histograms.

The Cumulative Frequency can be clearly understood when displayed in a table. A table displaying the cumulative frequencies is called as cumulative frequency distribution and this is one of important type of frequency distribution.

There are main two types of cumulative frequency distribution as follows:

Less than cumulative frequency distribution:
Here, the Cumulative total of the frequencies are obtained by adding frequencies of lowest size to the highest size.

For example:
          
Marks of students             
Less than Cumulative frequency       
Less than 20 7
 Less than 30 
8
Less than 40 12
Less than 50  16
Less than 60  24
Less than 70  37
Less than 80  45
Less than 90  58
 Less than 100  65

From the table, we get to know that student scoring less than 50 is 16.  

More than cumulative frequency distribution:

Here, the Cumulative total of the frequencies are obtained by adding frequencies of the highest size to lowest size.

Marks of students             
  More than Cumulative frequency      
 More than 10  88
 More than 20  74
 More than 30  65
 More than 40  60
 More than 50  58
 More than 60  50
  More than  70  47
 More than 80  29
 More than 90  11

From the table, we can say that the student scoring marks between 40 and 50 is 2
ie, 60 - 58 = 2.
The quotient between the cumulative frequency of a particular value and the total number of data is called as relative cumulative frequency. It is calculated by dividing the cumulative frequency in a frequency distribution by the total number of data points. It can be expressed as percentage.

Relative Cumulative frequency can be expressed as follows:

Relative Cumulative FrequencyHere, f = Cumulative frequency
n = Total number of frequency.

Example:

    Upper Limit         Frequency       Less than Cumulative frequency         Relative cumulative frequency          
 20  11  11 11/56 = 0.196 = 19.6 %
 30  8 11 + 8 = 19 19/56 = 0.339 = 33.9 %
 40  10 19 + 10 = 29 29/56 = 0.517 = 51.7 %
 50  3 29 + 3 = 32 32/56 = 0.571 = 57.1 %
 60  7 32 + 7 = 39 39/56 = 0.696 = 69.6 %
 70  12 39 + 12 = 51 51/56 = 0.910 = 91 %
 80  5 51 + 5 = 56 56/56 = 1 = 100 %
 Total  56    
The cumulative frequency is the running total of the frequencies. As mentioned before, cumulative frequency is mainly used to construct a histogram. The graph is displayed like a bar graph that shows the data after it has been added from the smallest interval to the largest interval.

The shape of the histogram always have the rectangular bars getting bigger as you move to the right.

Solved Example

Question: Draw a cumulative frequency histogram for the following cumulative frequency distribution.

Score           Frequency          Cumulative frequency        
 10  2  2
 12  7  9
 14  8  17
 16  10  27
 18  5  32
 20  13  45
 22  17  62
 24  6  68

Solution:
Cumulative Frequency Histogram

In simple words, the word cumulative means increasing by progressive addition. In statistics, it is the running total of the frequencies. 

A cumulative frequency polygon is a line graph obtained by plotting the cumulative frequency on the vertical axis and the upper limit of each class interval along with horizontal axis. It is a way to display cumulative information graphically. The shape of a cumulative frequency will always be increasing order as we move to the right.
A cumulative frequency polygon is also called as Ogive. It is a variation on the frequency polygon. The cumulative frequencies are useful for knowing the number of the values that fall above or below a given value.

Solved Example

Question: Plot a cumulative frequency polygon to the following data of cumulative frequency distribution.

 Limit      Frequency        Cumulative frequency       
20   8  8
 25  10  18
 30  12  30
 35  5  35
 40  7  42
 45  9  51

Solution:
For the above data, the Ogive will look as shown below:

Cumulative Frequency Graph

Cumulative percentage is just another way of presenting the frequency distribution. It helps us to know the percentage of the cumulative frequency within each interval. The important purpose of cumulative percentage on cumulative frequency is to provide an easier way to compare various sets of data.

Cumulative frequency and cumulative percentage graphs are one and the same and it is calculated by dividing the cumulative frequency by the total number observation, and then multiplying it by 100.

It is expressed as follows:

Cumulative Percentage Frequency

Solved Example

Question: From the given data, construct a cumulative percentage frequency distribution and plot a graph.
5, 4, 2, 4, 7, 4, 5, 4, 5, 2, 6, 7, 3, 4, 6.
Solution:
Now, let us construct the frequency distribution table, by finding the frequency, cumulative frequency and the cumulative percentage frequency.


Classes      Class Interval        Frequency       Cumulative frequency          Cumulative percentage frequency       
 2  1.5 - 2.5  2  2  2/15 * 100 = 13
 3  2.5 - 3.5  1  3  3/15 * 100 = 20
 4  3.5 - 4.5  5  8  8/15 * 100 = 53
 5  4.5 - 5.5  3  11  11/15 * 100 = 73
 6  5.5 - 6.5  2  13  13/15 * 100 = 86
 7  6.5 - 7.5  2  15  15/15 * 100 = 100

From the above table, the cumulative percentage graph will displayed as follows:

Cumulative Percentage Frequency Example

Given below is a problem based on cumulative frequency.

Solved Example

Question: Calculate the "less than'' and "more than" cumulative frequency for the following data of 70 students and plot a graph for both the cumulative frequency.

   Class Score               Frequency          
 0 - 10
 10 - 20  7
 20 - 30  11
 30 - 40 16 
 40 - 50 10 
 50 - 60 15 
 60 - 70
 70 - 80

Solution:
First, let us construct a frequency distribution table for the marks obtained by 70 students.

   Class Score            Frequency            Cumulative frequency          
 0 - 10  3
 10 - 20 10 
 20 - 30 11   21
 30 - 40 16  37 
 40 - 50 10  47 
50 - 60  15   62
60 - 70   5 67 
70 - 80  70 

The above distribution is a ''less than'' cumulative frequency distribution. Here, you can see that 3 students have scored ''less than 10'' and 10 students got '' less than 20 '' and so on.

Therefore, the above cumulative frequency distribution can be rewritten as:

Class                        Cumulative frequency              
 Less than 10
 Less than 20 10 
 Less than 30 21 
 Less than 40 37 
 Less than 50 47 
 Less than 60 62 
 Less than 70 67 
 Less than 80 70 

When these data are plotted on graph, then the "less than" cumulative frequency polygon will look as follows:

Cumulative Frequency Example Problems

Now, let us calculate the "more than" cumulative frequency by adding the frequencies in a reverse order.

 Class Marks     Frequency          Cumulative frequency                 
 0 - 10 3 + 7 + 11 + 16 + 10 + 15 + 5 + 3 = 70
10 - 20  7 + 11 + 16 + 10 + 15 + 5 + 3 = 67 
 20 - 30  11  11 + 16 + 10 + 15 + 5 + 3 = 60
 30 - 40  16  16 + 10 + 15 + 5 + 3 = 49
 40 - 50  10  10 + 15 + 5 + 3 = 33
 50 - 60  15  15 + 5 + 3 = 23
 60 - 70  5  5 + 3 = 8
 70 - 80  3  3

The above table can be rewritten as follows:

Class                     Cumulative frequency   
 More than 10 70 
 More than 20 67 
 More than 30 60 
 More than 40 49 
 More than 50 33 
 More than 60 23 
 More than 70
 More than 80

And, the cumulative frequency polygon will look as follows:

Cumulative Frequency Example Problem