Cumulative Frequency
2026-02-28 11:08 Diff
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Last updated on November 24, 2025

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 its class intervals. In this article, we will learn more about cumulative frequency and its types.

What is Cumulative Frequency?

In statistics, cumulative frequency (c.f.) is found by adding each class frequency to the total of all previous frequencies.
You begin with the first class, then add the second, then add the third, and continue this process.
The resulting values are called cumulative frequencies.
A table that shows these cumulative totals for different classes is known as a cumulative frequency distribution or a cumulative frequency table.

There are two types of cumulative frequency:

  • Less-than cumulative frequency
  • Greater-than cumulative frequency

Cumulative frequency helps us understand how many observations fall below or above a specific value in a dataset. It is especially useful in data analysis, surveys, and business decision-making.

For example,
Emma works at a bakery and tracks how many cupcakes were sold each day over a week. She wants to know the total number of cupcakes sold up to each day.

Day Number of cupcakes sold (Frequency) Cumulative Frequency Monday 25 25 Tuesday 18 25 + 18 = 43 Wednesday 22 43 + 22 = 65 Thursday 30 65 + 30 = 95 Friday 20 95 + 20 = 115

Here, the final cumulative frequency 115 represents the total number of cupcakes sold during the week.

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:

\(CFi = \sum (f_j) \) from j = 1 to i

Where: 

  • \(CF_i \)is the cumulative frequency up to the ith class interval
     
  • \(f_j\) 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: 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.
 

What are the Types of Cumulative Frequency?

Cumulative frequencies are categorized into two types:

  • Less than Cumulative Frequency
  • More than Cumulative Frequency
     

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Less than Cumulative Frequency

Less-than cumulative frequency, also known as a 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.

For example, 
In a test, if the mark intervals \(0–10, 10–20, 20–30, 30–40, \)and \(40–50\) have frequencies 4, 6, 10, 8, and 12, then their less-than cumulative frequencies are 4, 10, 20, 28, and 40, which form a rising curve called a less-than ogive.

More than Cumulative Frequency

More-than cumulative frequency is calculated by adding the frequencies of the last class to those of 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.

For example,
If the mark intervals have frequencies 4, 6, 10, 8, and 12, then the more-than cumulative frequencies from the highest class downward are 12, 20, 30, 36, and 40, which form a downward curve called a more-than ogive.

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 Frequency 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.

Tips and Tricks to Master Cumulative Frequency

These simple tips help students grasp cumulative frequency better by showing them where to begin. Let us look at a few of them.

  • Always start from the correct end: use the lowest class for less-than cumulative frequency and the highest class for more-than cumulative frequency.
  • Add each frequency to the previous total step by step to avoid mistakes.
  • Use an extra column to clearly keep track of cumulative totals.
  • Check that the final cumulative frequency equals the total number of observations.
  • Visualize the data using ogive graphs, which rise for less-than and fall for more-than cumulative frequency.
  • Teachers can guide children to start adding frequencies from the correct side, the lowest class for less-than type, and the highest class for more-than type.
  • Parents can encourage children to practice using simple real-life examples, like daily expenses or marks, to understand cumulative totals easily.
  • Children can double-check their work by verifying that the final cumulative frequency matches the total number of observations.

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.

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 institutions use cumulative frequency to analyze students' performance. This can help teachers understand how many students scored over a particular mark.
  • Businesses and analysts: Many companies use cumulative frequency to track the total sales over time. This helps firms analyze trends and predict the product demand for a particular product.
  • Traffic management: Traffic engineers use cumulative frequency to analyze vehicle speeds, accidents, and commute times to determine how often a particular road is used and whether the area is accident-prone.
  • Weather forecasting: Meteorologists use cumulative frequency to study rainfall patterns, temperature ranges, and storm occurrences. This helps them understand how often certain weather conditions occur.
  • Healthcare analysis: Hospitals and researchers use cumulative frequency to track patient data such as recovery time, age groups, or disease cases. This helps identify patterns and improve treatment strategies.

Problem 1

Create a cumulative frequency table for the number of pages Emma reads each day. Monday: 10 pages, Tuesday: 12 pages, Wednesday: 8 pages, Thursday: 15 pages, Friday: 20 pages, Saturday: 5 pages, Sunday: 10 pages.

Okay, lets begin

Day Frequency (Pages) Cumulative Frequency (Pages) Monday 10 10 Tuesday 12 10 + 12 = 22 Wednesday 8 22 + 8 = 30 Thursday 15 30 + 15 = 45 Friday 20 45 + 20 = 65 Saturday 5 65 + 5 = 70 Sunday 10 70 + 10 = 80

Explanation

The cumulative frequency table is created by starting with Monday’s pages and continuously adding each day’s reading to the previous total, helping us track how many pages Emma has read up to any point in the week, ending with a total of 80 pages.

Well explained 👍

Problem 2

Create a cumulative frequency table showing the number of glasses of water David drinks each day. Monday: 4 glasses, Tuesday: 5 glasses, Wednesday: 3 glasses, Thursday: 6 glasses, Friday: 4 glasses, Saturday: 7 glasses, Sunday: 5 glasses.

Okay, lets begin

Day Frequency (Glasses) Cumulative Frequency (Glasses) Monday 4 4 Tuesday 5 4 + 5 = 9 Wednesday 3 9 + 3 = 12 Thursday 6 12 + 6 = 18 Friday 4 18 + 4 = 22 Saturday 7  22 + 7 = 29 Sunday 5 29 + 5 = 34

Explanation

To find the cumulative frequency, we start from the first day and keep adding each day's water intake to the total of the previous days. This helps us know how many glasses of water David has consumed up to any day of the week.

Well explained 👍

Problem 3

A fitness coach records the number of minutes spent exercising by 12 participants in a week. The data (in minutes) is as follows: 32, 45, 28, 52, 47, 39, 41, 55, 48, 33, 60, 52. Create a frequency table for the data.

Okay, lets begin

Frequency represents how many times a value falls within a particular interval.
We will choose suitable class intervals and count how many data points fall into each one.

Interval (Minutes) Frequency 25-34 3 35-44 3 45-54 5 55-64 1

Explanation

Here, we create the frequency table by selecting suitable class intervals and then counting how many exercise-time values fall into each interval, which gives us the frequency for every group.

Well explained 👍

Problem 4

A teacher recorded the number of hours 12 students studied in a day: 1, 2, 3, 2, 4, 3, 5, 1, 2, 4, 3, 2. Create a frequency table using class intervals.

Okay, lets begin

Choose the class intervals: 1–2, 3–4, 5–6.

Interval Frequency 1-2 5 3-4 5 5-6 1

Explanation

We group the study hours into intervals and count how many values fall in each interval to fill the frequency table.

Well explained 👍

Problem 5

The quiz scores of 15 students are: 12, 18, 15, 17, 20, 19, 13, 16, 14, 18, 20, 15, 17, 16, 14. Create a frequency table.

Okay, lets begin

Choose class intervals: 10–14, 15–17, 18–20.

Interval Frequency 10-14 4 15-17 6 18-20 5

Explanation

We divide the scores into intervals and count how many scores fall in each range to determine the frequencies.

Well explained 👍

FAQs on Cumulative Frequency

1. How many types of cumulative frequencies are there?

There are two types of cumulative frequency. The first type is called less than cumulative frequency and the second type is called more than cumulative frequency.
 

2.What is an ogive?

 An ogive is a graphical representation of cumulative frequency. It gives a quick glance of how concentrated or spread out the data is.
 

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

Frequency is the number of data points in a specific class or category. Cumulative frequency is the total count of data points accumulated up to that class or category.
 

4.When should we use cumulative frequency graphs?

We use cumulative frequency graphs for various reasons. They are used to compare data sets visually, find medians, quartiles, and percentiles, and see how data builds up over time.
 

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

Yes. For grouped data, cumulative frequency can be calculated for each class interval. For ungrouped data, the data needs to be arranged in ascending order. Then, calculate the cumulative frequency by determining how many values are equal to or less than each data point.