Median of Grouped Data

In statistics, the three most important measures of central tendency are mean, median, and mode. The mean represents the average of a dataset, the mode represents the most frequent value, and the median represents the middle value.

When data is organized into class intervals, we use the median of grouped data approach to calculate the middle value. This involves a specific median formula, which we'll explore below along with examples and practice problems.

 

Table of Contents

• Finding Median of Ungrouped Data
• How to Find the Median of Grouped Data
• Median of Grouped Data Example
• Practice Problems
• Conclusion
• FAQs on Median of Grouped Data

 

Finding Median of Ungrouped Data 

To find the median of ungrouped data:

 1. Arrange the data in ascending order.

 2. If the number of values (n) is odd, the median is the (n + 1)/2th observation.

 3. If the number of values is even, the median is the average of the (n/2)th and (n/2 + 1)th observations. 

Example: Given data: 6, 4, 7, 3, 2 Ascending order: 2, 3, 4, 6, 7 n = 5 -> Median = 3rd observation = 4

 

How to Find the Median of Grouped Data

To find the median of grouped data, we follow these steps:

1. Determine the total number of observations (n).

2. Calculate n/2.

3. Find the cumulative frequency for each class interval.

4. Identify the median class - the class interval whose cumulative frequency is greater than or equal to n/2.

5. Apply the median formula.
   
   

Median Formula:
Median = l + [(n/2 − cf) / f] × h

Where:
l = lower limit of the median class
n = total number of observations
cf = cumulative frequency of the class before the median class
f = frequency of the median class
h = class width

 

Median of Grouped Data Example

Question:
The table below shows the heights (in cm) of 51 girls. Find the median height.

Height (in cm) -Cumulative Frequency
Less than 140 -4
Less than 145 -11
Less than 150 -29
Less than 155 -40
Less than 160 -46
Less than 165 -51

Step 1: Convert into class intervals and calculate frequencies.

Class Interval -Frequency -Cumulative Frequency
Below 140 -4 -4
140 -145 -7 -11
145 -150 -18 -29
150 -155 -11 -40
155 -160 -6 -46
160 -165 -5 -51

n = 51 → n/2 = 25.5
The cumulative frequency just greater than 25.5 is 29.
So, median class = 145 -150

Using the formula:
l = 145
cf = 11
f = 18
h = 5

Median = 145 + [(25.5 -11)/18] × 5
= 145 + (14.5 / 18) × 5
= 145 + 4.03
= 149.03 cm

Answer: The median of grouped data (height) = 149.03 cm

 

Practice Problems

1. The median of the following data set is 525. Total frequency = 100. Find x and y.
   
   

Class Interval -Frequency
0 -100 -2
100 -200 -5
200 -300 -x
300 -400 -12
400 -500 -17
500 -600 -20
600 -700 -y
700 -800 -9
800 -900 -7
900 -1000 -4

2. The following table shows monthly electricity usage for 68 households. Find the median of grouped data.
   
   

Class Interval (in units) -Number of Consumers
65 -85 -4
85 -105 -5
105 -125 -13
125 -145 -20
145 -165 -14
165 -185 -8
185 -205 -4

Conclusion  

The median of grouped data is an essential statistical tool that helps identify the central value of large datasets organized into class intervals. Unlike ungrouped data, where the middle value can be directly observed, grouped data requires the use of a specific median formula to estimate the middle point accurately. By understanding how to identify the median class, apply the formula, and work through cumulative frequencies, students can solve real-world problems with confidence.

 

Frequently Asked Questions on Median of Grouped Data

Q1. What is the median in statistics?

 Answer.The median is the middle value of a dataset when arranged in order.

Q2. What is the formula to find the median of grouped data? 

 Answer.Median = l + [(n/2 − cf) / f] × h

Q3. What is the median class?

  Answer.The median class is the class interval whose cumulative frequency is just greater than or equal to n/2.

Q4. Why do we need a formula to calculate the median of grouped data?

 Answer.Because exact data points are grouped in intervals, the median formula estimates the middle value using cumulative frequency and class widths.

Q5. Where is the concept of median of grouped data applied?

 Answer. In economics, education, surveys, and research studies - especially when analyzing large datasets divided into class intervals.

 

 Learn how to calculate the median of grouped data step by step. Practice applying the median formula with real examples at Orchids The International School and strengthen your foundation in statistics.