Median Of Grouped Data

The median of grouped data is a key concept in statistics that relates to the three most important measures of central tendency: 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 

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. To find the median of ungrouped data perform the following steps:

 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+[n2−cff]×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

 The following data represents the marks obtained by 60 students in a mathematics exam. Find the median marks.

Step 1 : Convert cumulative frequencies to class intervals.
The given data is in the 'less than' form, so we rewrite it into class intervals with frequencies.

Step 2: 

Identify median class.

Here, total students n = 60.
So, n/2 = 60/2 = 30.
The cumulative frequency just greater than 30 is 50, which corresponds to the class interval 60–80.
Thus, the median class = 60–80.

Step 3: Apply the formula for median of grouped data.

Median =L+(n2−cff)×h

Where:
- L = lower boundary of median class = 60
- n = total frequency = 60
- cf = cumulative frequency before median class = 30
- f = frequency of median class = 20
- h = class size = 20

Step 4: Substitution.

Median = 60 + ((30 – 30) / 20) × 20
Median = 60 + (0 / 20) × 20
Median = 60

 

Practice Problems

1. The median of the following data set is 350. Find the values of p and q, if the total frequency is 120.

   Class Interval	Frequency
   0 – 100	6
   100 – 200	10
   200 – 300	p
   300 – 400	20
   400 – 500	18
   500 – 600	q
   600 – 700	12
   700 – 800	8
   800 – 900	6
   900 – 1000	4

   Practice Problem 2

   The following frequency distribution table shows the weekly wages of 72 factory workers. Find the median.

   Weekly Wages (in ₹)	Number of Workers
   500 – 600	6
   600 – 700	10
   700 – 800	15
   800 – 900	20
   900 – 1000	12
   1000 – 1100	6
   1100 – 1200	3

   Practice Problem 3

   The following table shows the daily income of 60 shopkeepers in a market. Calculate the median income.

   Daily Income (₹)	Number of Shopkeepers
   100 – 200	5
   200 – 300	7
   300 – 400	12
   400 – 500	15
   500 – 600	9
   600 – 700	7
   700 – 800	5

    

   Practice Problem 4

   The marks obtained by 80 students in a test are shown below. Find the median marks.

   Marks Interval	Number of Students
   0 – 10	6
   10 – 20	10
   20 – 30	12
   30 – 40	20
   50 – 60	10
   60 – 70	8

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.Practicing mean median mode questions further strengthens problem-solving skills and deepens understanding of statistical concepts.

 

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+(n2−cff)×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.