Median of Grouped Data 

The median of grouped data is the middle value of a dataset organized into class intervals. Unlike ungrouped data, where the median is a specific number, grouped data requires estimation using a formula. It helps identify the central tendency when exact values aren't available. By locating the median class and applying the median formula, we can effectively determine the approximate center of the data distribution. This method is widely used in statistics to analyze large, categorized datasets.

1.0What is the Median of Grouped Data?

The median is the middle value that separates the higher half from the lower half of a data set. For ungrouped data, it's relatively easy to calculate. However, when the data is grouped into classes (like in frequency tables), we need to use a different approach.

Grouped data doesn’t show individual values but rather the range in which values fall. Hence, finding the median of grouped data involves estimating the value that lies at the midpoint of the cumulative frequency distribution.

2.0Formula for Calculating Median of Grouped Data

To find the median of grouped data, use the following formula:

$\text{ Median }=L+\left(\frac{\frac{N}{2}-CF}{f}\right)\times h$

Where:

• L = Lower boundary of the median class
• N = Total frequency
• CF = Cumulative frequency of the class before the median class
• f = Frequency of the median class
• h = Class width (upper boundary - lower boundary)

3.0Steps to Find the Median of Grouped Data

1. Create a cumulative frequency column from the frequency distribution.
2. Calculate N/2, where N is the total frequency.
3. Identify the median class (the class interval whose cumulative frequency is just greater than or equal to N/2).
4. Apply the median formula to get the result.

4.0Solved Examples of Median of Grouped Data

Example 1: Find the median of the following frequency distribution:

Solution: 

Step 1: Calculate cumulative frequency

Step 2: Find N and N/2

Total frequency, N = 45

N/2 = 22.5

Step 3: Locate the Median Class

The cumulative frequency just greater than 22.5 is 28, which corresponds to the class 20 - 30. So, this is our median class.

Step 4: Apply the Formula

• L = 20
• CF = 13
• f = 15
• h = 10

$\begin{array}{rl}& \text{ Median }=20+\left(\frac{22.5-13}{15}\right)\times 10\\ & \text{ Median }=20+\left(\frac{9.5}{15}\right)\times 10\\ & \text{ Median }=20+6.33\\ & \text{ Median }=26.33\end{array}$

Final Answer: Median = 26.33

Example 2: Find the median of the following frequency distribution:

Solution:

1. Calculate cumulative frequency (CF):

2. Total frequency (N) = 35

N/2 = 35/2 = 17.5 

3. Median class = 20 – 30 (CF just greater than 17.5 is 18)
4. Apply the formula:

• L = 20
• CF = 10
• f = 8
• h = 10

$\begin{array}{rl}& \text{ Median }=20+\left(\frac{17.5-10}{8}\right)\times 10\\ & =20+\left(\frac{7.5}{8}\right)\times 10\\ & =20+9.375=29.375\end{array}$

Example 3: Find the median of the following frequency distribution:

Solution:

• N = 50, N/2 = 25
• Median class = 60 – 80
• L = 60, CF = 22, f = 17, h = 20

$\begin{array}{rl}& \text{ Median }=60+\left(\frac{25-22}{17}\right)\times 20\\ & =60+\left(\frac{3}{17}\right)\times 20\\ & =60+3.53=63.53\end{array}$

Example 4: The median of the following data is 47.5. If the total frequency is 150, find the missing frequencies x and y.

Solution: 

Step 1: Write total frequency equation

Sum of known frequencies:

5 + 9 + x + 25 + 30 + y + 20 + 10 = 99 + x + y 

Set total equal to 150:

x + y + 99 = 150 

$\Rightarrow x+y=51$ (Eq ①)

Step 2: Find N/2 = 150/2 = 75

We need cumulative frequency to find median class

Build CF up to median class:

We’re told median is 47.5 → So median class is 40 – 50

So:

• L = 40
• f = 30
• CF before median = x + 39
• h = 10
• N/2 = 75

Step 3: Apply Median Formula

Step 4: Use Eq ① → x + y = 51

$14+y=51\Rightarrow y=37$

Final Answer: x = 14, y = 37

Example 5: In the table below, the median is 67, and the total frequency is 120. Find the values of x and y.

Solution: 

Step 1: Total Frequency Equation

7 + 13 + x + 30 + y + 10 + 8 

$\Rightarrow 78+x+y=120\Rightarrow x+y=42(Eq①)$

Step 2: N/2 = 60

Build CF to find median class:

Since median = 67 → Median class is 60 – 80

So:

• L = 60
• CF before median = 20 + x
• f = 30
• h = 20
• N/2 = 60

Step 3: Apply Formula

$\begin{array}{rl}& 67=60+\left(\frac{60-(20+x)}{30}\right)\cdot 20\\ & \Rightarrow 7=\left(\frac{40-x}{30}\right)\cdot 20\\ & \Rightarrow \frac{40-x}{30}=\frac{7}{20}\\ & \Rightarrow 40-x=\frac{7}{20}\cdot 30=10.5\\ & \Rightarrow x=29.5\end{array}$

Step 4: Find y from Eq ①

$x+y=42\Rightarrow 29.5+y=42\Rightarrow y=12.5$

Final Answer: x = 29.5, y = 12.5

Example 6: The median of the following grouped data is 62.5, and the total number of observations is 180. Find x and y.

Solution: 

Step 1: Total Frequency

12 + x + 30 + 50 + y + 28

 = 120 + x + y = 180 

$\Rightarrow x+y=60\text{(Eq ①)}$

Step 2: N/2 = 90

Build CF:

Median = 62.5 → So median class = 60 – 80

So:

• L = 60
• f = 50
• CF before median class = x + 42
• h = 20

Step 3: Apply Formula

$\begin{array}{rl}& 62.5=60+\left(\frac{90-(x+42)}{50}\right)\cdot 20\\ & \Rightarrow 2.5=\left(\frac{48-x}{50}\right)\cdot 20\\ & \Rightarrow \frac{48-x}{50}=\frac{2.5}{20}=0.125\\ & \Rightarrow 48-x=6.25\\ & \Rightarrow x=41.75\end{array}$

Step 4: Use Eq ①:

$x+y=60\Rightarrow 41.75+y=60\Rightarrow y=18.25$ 

Final Answer: x = 41.75, y = 18.25

5.0Practice Questions on Median of Grouped Data 

Question 1: Find the median of the following frequency distribution:

Question 2: Find the median of the following frequency distribution:

Question 3: The median of the following data is 60, and the total frequency is 150. Find the missing frequencies x and y.

Question 4: The median of the data below is 45, and the total frequency is 120. Find the values of x and y.