Frequency Distribution

A frequency distribution is a technique used to organize collected data so that it becomes easy to understand and analyze. Whether the data represents a student’s marks, temperatures of towns, or points scored in a volleyball match, arranging it clearly helps reveal patterns and trends. When this data is summarized into rows and columns showing how often each value appears, we call it a frequency distribution table.

For example, consider the marks scored by 10 students:
Marks: \(5, 7, 5, 8, 9, 7, 6, 8, 5, 9\). A simple frequency distribution table example would show how many students scored each mark.

There are four types of frequency distribution, which are listed below:

Ungrouped Frequency Distribution

Data is presented in a list or a table without being grouped into intervals. For example, test scores of students:

 It is used for small datasets with distinct values that do not need grouping.

Grouped Frequency Distribution

Data is divided into intervals or class groups to make it easier to analyze. For example:

We use it when the data set is large and individual values can be grouped into meaningful ranges.

Cumulative Frequency Distribution

Shows the sum of frequencies up to a certain class interval. For example:

We use it when analyzing percentiles, medians, or data trends over time.

Relative Frequency Distribution

Expresses frequency as a percentage of the total number of observations. The formula used is:

        \(\ \text{Relative Frequency} = \frac{\text{Class Frequency}}{\text{Total Frequency}} \times 100 \ \)

For example:

We use it to compare distributions with total frequencies or for probability based studies.
 

There are two ways to make a frequency table, that is for an ungrouped data and for a grouped data. Let us see what steps are involved to make frequency tables for both types of data:

For Ungrouped Frequency:

To create an ungrouped frequency table, we have to follow the below-mentioned steps:

Step 1: Create a table with two columns and rows. Label the first column using the variable name and the second column named as frequency.

Step 2: Then we must count the frequencies. Frequencies are the number of times each value occurs. Enter the number of frequencies in the frequency column.

For Grouped Data:

The following steps must be followed in order to create a table for grouped data: 

Step 1: First we must divide the variable into class intervals. To do that, we need to calculate the range by subtracting the lowest value from the highest value. Then we need to find the class width. To calculate the width, we have to use the following formula:

       \(\ \text{Width} = \frac{\text{Range}}{\sqrt{\text{Sample Size}}} \ \)

Then we have to calculate the class intervals. The observations in a class interval are greater than or equal to the lower limit and less than the upper limit.

Step 2: We have to create a table with the class interval, and the frequency.

Step 3: Then we have to count the frequency. Frequencies are the number of times each value occurs. Enter the frequencies in the frequency column.
 

In frequency distribution, different formulas help us analyze and interpret data effectively. One important formula is the coefficient of variation, which helps compare the spread of two datasets.


Frequency (f) simply refers to how many times a particular value appears in the dataset.


Coefficient of variation
While the mean and standard deviation describe the average and the spread of a dataset, comparing two distributions can be difficult, especially when the datasets use different units or scales. To solve this, we use the coefficient of variation (CV).


The coefficient of variation is defined as:
Coefficient of Variation (CV) = \(σx × 100\)


Where,

σ = Standard deviation

x = mean of the observations

A frequency distribution table is a simple and effective way to organize and present data in a tabular format. It helps summarize a large dataset into a clear and concise form. In a frequency distribution table, one column represents the data values, either individual numbers or ranges. In contrast, the other column shows how often each value or interval occurs, known as the frequency.


For example,
Let’s say we have a dataset of weekly hours spent on social media by teenagers.

Another effective way to present data is through graphical methods, using a frequency distribution graph. Graphs make it easier to understand patterns, comparisons, and the overall structure of the collected data. A frequency distribution can be visually represented using the following types of graphs:
 

• Bar graphs: These display data using rectangular bars of equal width with uniform spacing between them. Each bar represents a category or value.
   

• Histograms: A histogram is similar to a bar graph but is used for continuous data. The bars are placed side by side with no gaps, and their heights indicate the frequency of each class interval.
   

• Pie chart: a circular graph that divides the dataset into sectors. Each sector represents a portion of the data, making it easy to compare parts of a whole.

• Frequency polygon: This graph is created by joining the midpoints of each class interval, usually taken from a histogram with straight lines, forming a polygonal shape that shows how frequencies change across intervals.

Learn easy methods to organize, understand, and analyze data using frequency tables and graphs. Let us explore a few simple tips and tricks to master frequency distribution.

• Understand what “frequency” is, which is just how many times something happens.
  
   
• Write the data in order, from smallest to largest. It makes everything easier.
  
   
• Tally marks help you count quickly without making mistakes.
  
   
• A small frequency error can change the whole table, so double-check it before writing.
  
   
• Practice simple frequency tables, grouped tables, relative frequency, and cumulative frequency to understand them better.
  
   
• Teachers can guide children to write the data in order from smallest to largest. 
  
   
• Parents and teachers should encourage children to recheck their counts to avoid small mistakes.
  
   
• Children should learn simple formulas, such as how to find the mean using a frequency table.

The frequency distribution tables have numerous applications across various fields. Let us explore how the frequency table is used in different areas:

• Education and academics - In schools and colleges, frequency distribution tables help track student’s marks, attendance, and performance trends. Teachers use them to understand how many students score within certain ranges and to identify areas where students need support.
  
   
• Business and sales - Businesses use frequency tables to analyze how often products are purchased. This helps companies understand customer demand, identify best-selling items, and plan inventory and sales strategies based on trends.
  
   
• Healthcare and medicine - In hospitals and medical research, frequency tables are used to record how frequently different diseases, symptoms, or conditions occur in patients. Pharmaceutical companies also use them to track prescription patterns and medicine usage.
  
   
• Weather and climate studies - Meteorologists use frequency distribution to track how often certain temperatures, rainfall levels, or weather events occur. This helps in predicting climate patterns, planning agricultural activities, and issuing weather alerts. 
  
   
• Transportation and traffic management - Traffic authorities use frequency tables to analyze how many vehicles pass through a road at different times. This helps in planning road improvements, setting signal timings, and reducing traffic congestion.