Cumulative Frequency Distribution

What Is Cumulative Frequency Distribution?

Distribution of any data using a table or graph makes the data comprehensive. There are different classes and subclasses, which indicates the frequency,  and this is known as the frequency distribution of data. The frequency is actually the number of times an observation occurs for a particular time period.

Cumulative frequency helps to determine the number of operations, lying above specific observation. For calculating the increasing frequency of any data, the frequency of the observation is added to the sum of frequencies of the predecessors. This previous sum is called the cumulative frequency distribution of the entire set of data.

How to Find Cumulative Frequency and Why to Use Graphs to Represent it

While graphical representation makes the data simpler to understand, one can even see the fluctuations and ups and downs easily by drawing graphs. Usually, bar graphs and frequency polygons are used for graphical representation of the frequencies. The graphical representation of cumulative incidence is called ogive. 

The Graph for the Frequencies Can Be Plotted in two Different Ways, Like:

  • Cumulative frequency distribution curve, less than type

  • Cumulative frequency distribution curve, more than type

The below-given solution will provide a stepwise understanding of how to find more than the cumulative distribution frequency and how to calculate less than type cumulative distribution frequency.

How to Calculate Cumulative Frequency Distribution

How to Calculate More Than Cumulative Frequency

The Daily Income of Workers (in rupee)

No.of Workers 

(F)

Lower limits

Cumulative Frequency (cf)

100-120

10

100

50

120-140

12

120

50-10= 40

140-160

15

140

40-12= 28

160-180

13

160

28-15= 13


In the above-given table, four different groups are representing the daily wage of workers, along with the number of workers in the company. For example, when looking at the first group in the table, there are ten workers with regular income between 100-120 rupees.

Step 1: The number of worker columns is the frequency column. Frequency is nothing but the rate at which activity has occurred, which is further denoted by F. The sum of all the frequency is 50. Cumulative frequency is the running frequency of every group present in the table. 

Step 2: As we are solving more than type, it's essential to take the lower limits from the table, which is 100,120,140 and 160. If we calculate the lower bound from the table, it will give a result like more than 100, more than 120, and so on. 

Step 3: As the total frequency, we have obtained 50; the very first cumulative frequency will be written as 50. The cumulative frequency of the second row will be more than 120 and will be under 180. Hence, we have to subtract ten from 50 and progressively subtract the frequency from the resultant cumulative frequency. In more than cumulative type frequency, the last frequency should match with the cumulative frequency. 

Plotting the frequency distribution graph 

To draw the cumulative frequency distribution graph, the following steps have to be taken care of:

  • As the cumulative frequency is a dependent variable, it will come on the Y-axis, and the daily income will be shown on the X-axis. If there's a discontinuity of the numbers on the x-axis, then a key should be made before all the numbers. Don't forget to set the scale before jotting the graph. Start plotting the points with the help of numbers given on the table. 

  • After joining all the points through the freehand, the frequency distribution curve will be a decreasing curve, because in more than type the curve which will be obtained will always be a decreasing curve.

Graph for Cumulative Frequency Distribution More than Type

How to Calculate Less than Cumulative Frequency?

The Daily Income of Workers (in rupee)

No.of Workers 

(F)

Upper Limits

Cumulative Frequency (cf)

100-120

10

120

10

120-140

12

140

10+12= 22

140-160

15

160

22+15= 37

160-180

13

180

37+13= 50


The above-given example is similar to the more than type, but there are some changes made. The class size of every group is 20. Over here, we will be using the upper limits and not the lower limits. 

Step 1 and Step 2 are the same as the more than type frequency. Hence we will straight away move to step 3.

Step 3:  The frequency and the cumulative frequency for the first group will be the same, which is 12.  For the second group, one has to add the frequencies between 100-140, that's 10+12, 22. In the end, the cumulative incidence obtained will be 50. 

The total frequency will always be equal to the last cumulative frequency. 

Plotting the frequency distribution graph

To draw a cumulative frequency distribution graph, The X-axis will be of daily income, in rupees and the Y-axis will be of cumulative frequency. After setting the scale, start to plot the points on the graph. After making a freehand curve, the frequency distribution curve will be an increasing curve, unlike the more than type graphs. 

Calculating the median is also of less than cumulative type frequency is as same as more than type frequency graph. 

Note: A sigma (Σ) denotes frequency. 

FAQ (Frequently Asked Questions)

1. The Cumulative Frequency Distribution is Shown By

Graphs can represent the cumulative frequency in statistics. All the numbers have a specific rate, and with this data, a chart is made. A frequency polygon does the graphical representation of the data. Usually, a histogram is provided, and one has to find the cumulative frequency of that histogram. Technically, the aggregate amount obtained by you at the end is summarised by all the other numbers provided by you. Although there are two ways to find the cumulative frequency, there is just the slightest difference in the calculation part, like in less than type the first frequency and cf will be the same, but in more than type, the cf will be the sum of all the frequencies.

2. How to Construct Cumulative Frequency Distribution

There will be a table and a frequency column provided to you already. Calculate the cf, if it is more than type, then the first cf will be the sums of all the frequencies, and all the further will be subtracted progressively by previous frequencies. And for less than type, the first frequency and cf will be the same, and you have to add all the further frequencies progressively. It's not mandatory to draw a graph, but a graphical representation of the data not only helps to show the trend of that particular question but also it helps to find the mean and the median in a distinctive way.