Class 10 - Median of Grouped Data: Formula and Steps

The median of grouped data is a statistical measure that helps identify the central value in a dataset that is divided into class intervals. Unlike ungrouped data, finding the median of grouped data involves using the formula and cumulative frequencies. This concept is widely used in data analysis. In this guide, you'll learn simple steps, key formulas, and examples to easily calculate the median of grouped data.

Table of Contents

• What is Median of a Grouped Data
• Median of a Grouped Data Formula
• How to Calculate the Median of a Grouped Data
• Solved Examples on Median of a Grouped Data
• Practice Questions on Median of a Grouped Data
• FAQS

What is Median of a Grouped Data

The median of grouped data represents the central value of large datasets when data is presented in class intervals (grouped form). The median of grouped data divides the entire distribution into two equal parts, where half of the observations lie below it, and the other half above it. Since the data is grouped, the exact middle value is not directly visible, like in ungrouped data, so it is estimated based on the cumulative distribution of the data.

Median of a Grouped Data Formula

$Median = l + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$

Where:

• l = lower limit of median class
• N = total frequency
• cf = cumulative frequency before median class
• f = frequency of median class
• h = class width
• The median class is the class interval where $\frac{N}{2}$ lies.

How to Calculate the Median of a Grouped Data

The following steps to compute the median of grouped data:

• Step 1: Prepare the frequency distribution table such that its first column consists of the observations and the second column consists of the frequency.
• Step 2: Find the cumulative frequency and add it as the third column.
• Step 3: Obtain N and find the value of $\frac{N}{2}$.
• Step 4: Find the class whose cumulative frequency is just greater than the value $\frac{N}{2}$. This class is known as the median class.
• Step 5: Find l, cf, f, and h
• Step 6: Calculate the median using the formula

Solved Examples on Median of a Grouped Data

Example 1: Find the median of the following data set

Class Interval	Frequency
0–10	5
10–20	9
20–30	14
30–40	8
40–50	4

Solution: 

Class Interval	Frequency	Cumulative Frequency
0–10	5	5
10–20	9	14
20–30	14	28
30–40	8	36
40–50	4	40

$Median = l + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$

$= 20 + \left(\frac{\frac{40}{2} - 14}{14}\right) \times 10 = 20 + \frac{6}{14} \times 10 = 20 + 4.28 = 24.28$

Example 2: The median of the following data set is 42. Find the values of x and y if the total frequency is 60.

Class Interval	Frequency
0–10	5
10–20	7
20–30	x
30–40	12
40–50	15
50–60	y
60–70	6

Solution:
$Median = 42 = l + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h$

$42 = 40 + \left(\frac{\frac{60}{2} - (24 + x)}{15}\right) \times 10$

$42 = 40 + \frac{6 - x}{15} \times 10 \quad 2 = \frac{60 - 10x}{15}$

$2 \times 15 = 60 - 10x \quad x = 3 \quad x + y = 15$

⇒ y = 12

∴ x = 3 and y = 12

Practice Questions on Median of a Grouped Data

If the median of the following frequency distribution is 28.5, find the missing frequencies. The total frequency is 60.

Class	0–10	10–20	20–30	30–40	40–50	50–60	60–70
Frequency	x	8	20	15	7	y	60

Find the median of the following frequency distribution:

Age (years)	Frequency
0–10	5
10–20	5
20–30	12
30–40	6
40–50	4