BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon104 Learners

Last updated on July 4th, 2025

Math Whiteboard Illustration

Cumulative Frequency

Professor Greenline Explaining Math Concepts

Cumulative frequency is the progressive total of frequencies in a data set. The total data is arranged in a table where the frequency is divided according to their class intervals. In this article, we will learn more about cumulative frequency and its types.

Cumulative Frequency for Indonesian Students
Professor Greenline from BrightChamps

What is Cumulative Frequency?

When we get the running sum of frequencies in a given dataset, we call it cumulative frequency. The frequency of the first class interval is added to the second frequency of the second class interval and so on, until the last frequency. 

The cumulative frequency shows how many values in a dataset fall at or are below a particular value. It accumulates the frequency of data as you move through the dataset.
 

Professor Greenline from BrightChamps

How to Calculate Cumulative Frequency?

To calculate cumulative frequency, we take the frequency of the first class interval, then we add it to the frequency of the second class interval, and so on. The cumulative frequency is calculated using the formula:

 

Where: 

  • CFi is the cumulative frequency up to the ith class interval
     
  • fj is the frequency of the jth class interval 
     
  • i is the index of the class interval up to which the cumulative frequency is calculated 
     
  • j = it is the index used, which indicates that you need to start from the 1st frequency.
     

Below are the steps to calculate the cumulative frequency:
 

Step 1: First we sort the data and arrange it in a table

Step 2: Then we calculate the frequency of each value in the dataset 

Step 3: The next step is to calculate the cumulative frequency

Step 4: The cumulative frequency of the first class interval is the same as the frequency of the first class interval

Step 5: Find the cumulative frequency of the next class interval

Step 6: Repeat for the remaining class intervals

Step 7: Double-check your answers to avoid careless errors
 

Below is a table explaining how cumulative frequency is calculated.

 

 

Month Frequency (number of toys sold)  Cumulative frequency (total number of toys sold)
January 50 50
February 60 50 + 60 = 110
March 70 110 + 70 = 180
April 80 180 + 80 = 260


You can see that the last cumulative frequency is equal to the total of all observations. This is true for the final cumulative frequency.
 

Professor Greenline from BrightChamps

What are the Types of Cumulative Frequency?

Cumulative frequencies are categorized into two types:

  • Less than Cumulative Frequency
  • More than Cumulative Frequency
     
Professor Greenline from BrightChamps

Less than Cumulative Frequency

Less than cumulative frequency, also known as less than ogive. It is obtained by adding the frequency of the first class interval to the frequency of the second-class interval and so on. Here, the cumulative frequency begins from the lowest class to the highest class. In a graph, less than cumulative frequency is shown as a rising curve. 
 

Professor Greenline from BrightChamps

More than Cumulative Frequency

More than cumulative frequency is calculating by the cumulative frequency from the last class to the first class. In this method, we start the cumulative frequency from the highest to the lowest class. In a graph, more than cumulative frequency is drawn as a downward curve. 
 

Professor Greenline from BrightChamps

How to Draw Cumulative Frequency?

To plot the points in a graph we use the cumulative frequency. To draw a cumulative graph (also called ogive, follow these steps:

Step 1: Create a cumulative frequency table

 

Score Frequeny Cumulative Frequency
0 - 10   2 2
10 - 20  5  2 + 5 = 7
20 - 30 8 7 + 8 = 15
30 - 40 6 15 + 6 = 21
40 - 50 4 21 + 4 = 25

 

Step 2: Identify the scales of the graph

Here, in the x-axis we represent the scores and the y-axis represents the cumulative frequency.

The x-axis would be 10, 20, 30, 40, 50 and the y-axis would be 0, 5, 10, 15, 20, 25.
 

Step 3: Plot the points in the graph.

Here the points are:

  • (10, 2)
  • (20, 7)
  • (30, 15)
  • (40, 21)
  • (50, 25)

Step 4: Connect the points in the graph to complete the ogive. 


We can use three methods to graphically represent cumulative frequency data:

Cumulative Frequency Curve - In this method, we will be creating the graph by plotting cumulative frequencies against the upper class boundaries of the dataset. We then use a smooth curve to connect the points.

Here is a cumulative frequency curve for better understanding: 

 

 

Cumulative Frequency Polygon: A line graph connecting cumulative frequencies at class midpoints.

Here is the cumulative frequency polygon using the example of score

 

 

Cumulative Frequency Graph: It can be represented as any kind of graph, even a bar graph showing the cumulative frequency.

Here is a cumulative frequency graph using the above example
 


 

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Cumulative Frequency

When calculating cumulative frequency and plotting graphs students may get confused and make mistakes. So here are a few mistakes that students make and ways to avoid them.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

 Not adding the frequencies

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students might not add the frequencies from the first interval to the next. Make sure you add the frequencies of each interval until the last interval in the dataset.
 

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Plotting the frequency instead of cumulative frequency

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

 Sometimes when plotting a graph, you would notice that the graph is in the shape of a bell curve or a mountain range. This indicates that the student might have plotted using the frequency data and not the cumulative frequency. A cumulative frequency data would usually look like an S-curve.
 

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Plotting the data using midpoints of the class interval
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Using the midpoint of the class interval to plot would not result in a curve. When plotting a cumulative frequency graph we use the upper-class interval. 

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Not starting from the first class interval
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Always start from the first class interval when calculating cumulative frequency. Its cumulative frequency should be the same as its own frequency. 

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Not arranging data before calculating cumulative frequency
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

 Class intervals must be arranged in ascending order before the calculation. Failing to do so will result in incorrect cumulative totals. 

arrow-right
Professor Greenline from BrightChamps

Real-life Applications of Cumulative Frequency

Cumulative frequency is widely used in the real world. It helps us understand how data is accumulated over a period of time. Here are a few real-world applications of cumulative frequency.
 

  • Exam results: Educational institutes use cumulative frequency to analyze the student's performance. This can help teachers understand how many students scored over a certain mark.
     
  • Businesses and analysts: Many businesses use cumulative frequency to track the total sales over time. This helps business analyze trends and predict the product demand for a particular product.
     
  • Traffic management: Traffic engineers use cumulative frequency to study vehicle speeds, accidents, or commute times to determine how often a particular road is used or whether the area is accident-prone.
Ray Thinking Deeply About Math Problems

FAQs on Cumulative Frequency

1. How many types of cumulative frequencies are there?

Math FAQ Answers Dropdown Arrow

2.What is an ogive?

Math FAQ Answers Dropdown Arrow

3.What is the difference between frequency and cumulative frequency?

Math FAQ Answers Dropdown Arrow

4.When should we use cumulative frequency graphs?

Math FAQ Answers Dropdown Arrow

5.Can grouped and ungrouped data be used with cumulative frequency?

Math FAQ Answers Dropdown Arrow
INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
Dubai - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom