Median of Grouped Data

The meaning of median is the middle value of a given distribution, and the median of grouped data remains the same as the meaning of median. In grouped data, we have data that is in the form of intervals or classes. We also have a median class to find the value of the median.

Let us see how to calculate the media of grouped data using an example.

Examples: Student Test Scores

How to find the median of grouped data for a class of 50 students.

Calculation

Step A: Locate the Median Class
First, calculate \(\frac{N}{2}\) where N is the total frequency.
N = 50

\(\frac{N}{2} = 25\)

Look for the class interval where the Cumulative Frequency (CF) is greater than or equal to 25.
The 20 - 30 group has a CF of 27. This is our Median Class.

Step B: Apply the Median of Grouped Data Formula

\(Median = L + \left( \frac{\frac{N}{2} - CF_{pre}}{f} \right) \times h\)

Where:

• L (Lower limit of median class) = 20
• N/2 = 251
• \(CF_{pre}\) (Cumulative frequency of the class preceding the median class) = 12
• f (Frequency of the median class) = 15
• h (Class height/width) = 10

Step C: Solve

• \(Median = 20 + \left( \frac{25 - 12}{15} \right) \times 10\)

   

• \(Median = 20 + \left( \frac{13}{15} \right) \times 10\)

   

• \(Median = 20 + (0.866...) \times 10\)

   

• \(Median = 20 + 8.67\)

   

• \(Median \approx 28.67\)

To calculate the median of grouped data, we follow the following steps:

Step 1: We have to find the total number of observations by summing up the frequencies that are given.
 

Step 2: After that, we need to find the median class, which is the class having cumulative frequency just greater than half of the total number of observations.

Step 3: Now we note the values of lower limit of median class (L), frequency of the median class (f), cumulative frequency of the class preceding median class (cf), and the class width (w).

Step 4: We then substitute the values in the given formula and calculate the median of grouped data.

            \(Median = l + ((n/2 - cf)/f) x w\)
 

To compare between mean, median, and mode of grouped data let us see the following table:

Mastering the median of grouped data helps summarize large datasets effectively. These tips guide you to identify the median class and calculate it accurately.

• Understand the concept of median and how it divides data into two equal halves.
   
• Learn to identify the median class using cumulative frequency.
   
• Memorize the formula for calculating the median of grouped data.
   
• Practice with different datasets to become comfortable with class intervals and frequency tables.
   
• Use step-by step methods to avoid mistakes in calculations, especially with large datasets.
   

• The "Middle Student" Analogy: Start without numbers. Ask the student to imagine a school assembly where students are lined up by height in groups (e.g., 140-150cm group, 150-160cm group). Explain that finding the median is like walking down the line to find the exact student standing in the middle position. Since we only know which group they are in, we have to use a formula to estimate exactly where they stand within that group.
   

• Visualize the Area Split: Use a histogram to show that the median is the "Equal Area Line." Draw a histogram on graph paper and explain that the median vertical line cuts the total area of the bars exactly in half (50% area on the left, 50% on the right). This visual connection helps them understand why we calculate N/2.
   

• Explain Interpolation Simply: The most confusing part for students is often why the formula looks so complex. Explain that the formula is just "interpolation." It assumes the data points are spread out evenly inside the median class. We are simply calculating how far into that specific class we need to go to reach the middle count.
   

• The "Running Total" (Cumulative Frequency): Before jumping to the formula, practice calculating Cumulative Frequency (cf) as a "running total." Use the analogy of a video game score or a bank savings account that keeps adding up. Emphasize that without the cf column, we cannot locate which group the middle value lives in.
   

• Decode the Variables: Students typically get overwhelmed by the letters (L, f, cf, h). Create a "Cheat Sheet" where they label these variables in plain English alongside the math symbol:

  • L = The Starting Line (Lower limit of the median group)

  • h = The Step Size (Width of the class interval)

  • f = The Crowd Size (Frequency of the median group)
     

• Use Discrete Examples First: Before using continuous grouped data (0-10, 10-20), revisit a simple list of numbers (e.g., 2, 5, 9, 12, 14). Show that the median is 9. Then group them and show how the formula gets us a value close to 9. This bridges the gap between what they already know (ungrouped median) and the new concept (grouped median).

The median of grouped data is applied in income distribution, exam results, and healthcare analysis. Let us now see the various fields and applications we use in median of grouped data:

Income and wealth distribution analysis: We use the median of grouped data in income and wealth distribution, where Governments and economists use the median income to understand income inequality in a country.

Exam results: We use the median of grouped data in exam results, where schools and universities analyze students’ marks using the median to determine a typical student’s performance.

Healthcare and medical studies: We use the median of grouped data in medical studies and healthcare, where hospitals use the median length of hospital stays for specific treatments to measure healthcare efficiency.

Real estate pricing: Median property prices in a region are calculated using grouped data to understand typical housing costs and market trends.

Customer spending analysis: Retailers use the median of grouped data to determine the typical amount spent by customers, helping in pricing and marketing strategies.