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.

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

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.

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

Cumulative frequencies are categorized into two types:

• Less than Cumulative Frequency
• More 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 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.

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

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.

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.

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.