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.
 

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: 

• 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: 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.

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 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. 
 

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. 
 

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
 

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.