Median of Ungrouped Data Formula in Statistics | Definition, Solved Examples, Important Questions, FAQs

Median in Statistics | Class 11 Notes

Written by Neeraj Anand | Published by ANAND TECHNICAL PUBLISHERS

Introduction:

The Median is the middle value of a dataset when arranged in ascending or descending order. It divides the data into two equal halves, making it an important measure of central tendency. In Class 11 Statistics, the concept of the Median is essential for Board Exams, JEE Mains, and Advanced preparation.

For ungrouped data, the median is simply the middle observation, while for grouped data, it is calculated using a specific formula based on cumulative frequencies.

Median Definition in Maths

In Mathematics, the median is defined as the middle value of a sorted list of numbers.

Median is the middle value of the dataset when arranged in ascending or descending order. If the dataset has an odd number of values, the median is the middle value. If the dataset has an even number of values, the median is the average of the two middle values. A median divides the data into two halves.

Median of Ungrouped Data for Odd number of Observations

It is easy to find the median for the dataset, that has an odd number of observations.

Eg. Median of 2, 5, 8 is 5

Median of Ungrouped Data for Even number of Observations

If the dataset is even, then the mean value or average for the middle two numbers is called the median of the given data set.

Eg. Median of 4, 5, 6 and 7 is the mean of 5 and 6, i.e.,5.5.

Median Formula for Ungrouped Data

Based on the definition, the formula to find the median of the dataset is depends upon the number of observation as follows :

If n is Odd

If the given number of sorted observations/data (n) is odd, then the formula to calculate the median is:

Median = {(n+1)/2}^th term

If n is Even

If the given number of sorted observations/data (n) is even, then the formula to find the median is given by:

Median = [(n/2)^th term + {(n/2)+1}^th term]/2

Where,

“n” is the number of observations.

How to Find Median for Ungrouped Data?

To find the median of the data we can use the steps discussed below,

Step 1: Arrange the given data in ascending or descending order.

Step 2: Count the number of data values(n)

Step 3: Use the formula to find the median if n is even, or the median formula when n is odd, accordingly based on the value of n from step 2.

Step 4: Simplify to get the required median.

Solved Examples on Median for Ungrouped Data

Example 1: Determine the median for the given dataset:

5, 7, 4, 8, 6

Solution:

Given dataset: 5, 7, 4, 8, 6

Here, the number of observations is odd, i.e., 5 observations are given.

n = 5

Now, arrange the numbers in ascending order

4, 5, 6, 7, 8

The formula to calculate the median for odd observations is:

Median = {(n+1)/2}^th term

Median = {(5+1)/2}^th term

Median = 3rd term

Here, the 3rd term is 6.

Therefore, the median for the given dataset is 6.

Example 2: Determine the median for the given dataset:

4, 7, 3, 8, 6, 2

Solution:

Given dataset: 4, 7, 3, 8, 6, 2

Here, the number of observations is even, i.e., 6 observations are given.

n = 6

Now, arrange the numbers in ascending order

2, 3, 4, 6, 7, 8

The formula to calculate the median for odd observations is:

Median = [(n/2)^th term + {(n/2)+1}^th term]/2

Median = [(6/2)^th term + {(6/2)+1}^th term]/2

Median = (3^rd term + 4^th term)/2

Here, the 3^rd term is 4 and the 4^th term is 6

Therefore, median = (4+6)/2

= 10/2 = 5

Therefore, the median for the given dataset is 5.

Example 3: Find the median of given data set 30, 40, 10, 20, and 50

Solution:

Median of the data 30, 40, 10, 20, and 50 is,

Step 1: Order the given data in ascending order as:

10, 20, 30, 40, 50

Step 2: Check if n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.

Step 3: Here, n = 5 (odd)

Median = [(n + 1)/2]^th term

Median = [(5 + 1)/2]^th term = 3^3r term = 30

Thus, the median is 30.

Example 4: Find the median of the given data set 60, 70, 10, 30, and 50

Solution:

Median of the data 60, 70, 10, 30, and 50 is,

Step 1: Order the given data in ascending order as:

10, 30, 50, 60, 70

Step 2: Check if n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.

Step 3: Here, n = 5 (odd)

Median = [(n + 1)/2]^th term

Median = [(5 + 1)/2]^th term = 3^rd term

            = 50

Example 5: Find the median of the given data set 13, 47, 19, 25, 75, 66, and 50

Solution:

Median of the data 13, 47, 19, 25, 75, 66, and 50 is,

Step 1: Order the given data in ascending order as:

13, 19, 25, 47, 50, 66, 75

Step 2: Check if n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.

Step 3: Here, n = 7 (odd)

Median = [(n + 1)/2]^th term

Median = [(7 + 1)/2]^th term = 4^th term

            = 47

Practice Questions on Median for Ungrouped Data

1. Find the median of 2, 8, 3, 7, 5.
2. What is the median of 65, 76, 4, 17, 68, 12, 54, 68?
3. Evaluate the median of the numbers: 6, -4, 41, 85, 50.
4. The median of 87, 56, 99, 43, and 67 is?

Frequently Asked Questions – FAQs of Median for Ungrouped Data

Q1

What do you mean by median?

Median is the middle value of set of numbers, when arranged in an order.

Q2

Is median also an average?

Median is a type of average, where we find the middle value of a sequence of numbers, when arranged in order.

Q3

How do you calculate the median?

If the given set of data is in odd number, then arrange the numbers in an order, then find the middle value to get the median.
If the given set of data is an even number, then arrange the numbers in order, then find the average of the two central values. This will be the required median.

Q4

What is the median of 7 and 7?

The median of 7 and 7 is 7.

Q5

What is the median 8 5 7 9 11 6 10?

8, 5, 7, 9, 11, 6, 10 arranged in ascending order is 5, 6, 7, 8, 9, 10, 11 and thus, the median of given data is, 8.

Q6

What is the median of 7 6 4 8 2 5 and 11?

7 6 4 8 2 5 and 11 arranged in ascending order is 2, 4, 5, 6, 7, 8, 11 and thus, the median of given data is, 6.

Why is the Median Important?

✔️ Less affected by extreme values (outliers).
✔️ Provides a better central value for skewed distributions.
✔️ Useful in real-life applications like income distribution and percentile ranking.

---

📥 Download PDF Notes:

Get Class 11 Statistics Notes on Median Calculation from ANAND CLASSES for Board and JEE exam preparation.