Examples of How to Find Median of Data

Median is a key measure of central tendency in statistics, representing the middle value of a data set when arranged in ascending or descending order. Unlike the mean, the median is not affected by extreme values, making it a reliable measure for skewed distributions.

In this article, we will explore various examples that illustrate the steps to find the median in different scenarios, helping you grasp this important statistical concept with ease.

What is the Median?

Median is a measure of central tendency that represents the middle value in a data set when the values are arranged in ascending or descending order.

If the data set has an odd number of values, the median is the value exactly in the middle. If the data set has an even number of values, the median is the average of the two middle values.

Consider the following examples for better understanding.

• Odd Number of Data Points:
  • Data set: 3, 1, 9, 7, 5
  • Ordered: 1, 3, 5, 7, 9
  • Median: The middle value is 5.
• Even Number of Data Points:
  • Data set: 4, 1, 7, 3
  • Ordered: 1, 3, 4, 7
  • Median: The average of the two middle values 3 and 4, which is (3 + 4)/2 = 3.5

Read More about Median.

Methods to Find the Median

Finding the median of a data set involves several methods, depending on the nature of the data and the specific requirements of the analysis.

• Median for Ungrouped Data
  • Median for Odd Number of Data Points
  • Median for Even Number of Data Points
• Median for Grouped Data

Median for Odd Number of Data Points

To find the median of dataset having odd number of observations, we first arrange the data point in either ascendin or descending order and then Median is the data point on \left( \frac{n + 1}{2}\right)^{th} position.

Median for Even Number of Data Points

To find the median of dataset having even number of observations, we first arrange the data point in either ascendin or descending order and then Median is the average of data points on \left( \frac{n}{2}\right)^{th} and \left( \frac{n}{2}+ 1\right)^{th}position.

Median for Grouped Data

When data is grouped into classes, the median can be found using the median class and the formula:

Median = L + \left(\frac{\frac{n}{2} - C_F}{f}\right) \times h

Where:

• L is Lower boundary of the median class
• n is Total number of observations
• C_F is Cumulative frequency of the class preceding the median class
• f is Frequency of the median class
• h is Class width

Read More about Median of Grouped Data.

Examples of Finding the Median

Let's discuss some example for each case here.

Median of an Odd Set of Numbers

Example 1: Find the median of the following set of numbers: 12, 4, 8, 10, 14, 6, 2.

Solution:

Step 1: Arrange the numbers in ascending order:

2, 4, 6, 8, 10, 12, 14

Step 1: Count the numbers:

There are 7 numbers in the set.

Step 1: Find the middle position:

Since there are 7 numbers, the median is the number at the \left(\frac{7+1}{2}\right)^th position.

\text{Median position} = \frac{7+1}{2} = 4

Identify the median:

The number at the 4th position in the ordered set is 8.

Therefore, the median of 12, 4, 8, 10, 14, 6, 2 is 8.

Example 2: Find the median of the following set of numbers: 21, 13, 35, 46, 29, 55, 39, 11, 47.

Solution:

Step 1: Arrange the numbers in ascending order:

11, 13, 21, 29, 35, 39, 46, 47, 55

Step 2: Count the numbers:

There are 9 numbers in the set.

Step 3: Find the middle position:

Since there are 9 numbers, the median is the number at the \left(\frac{9+1}{2}\right)^th position.

\text{Median position} = \frac{9+1}{2} = 5

Step 4: Identify the median:

The number at the 5th position in the ordered set is 35.

Therefore, the median of 21, 13, 35, 46, 29, 55, 39, 11, 47 is 35.

Median of an Even Set of Numbers

Example 1: Find the median of the following set of numbers: 8, 3, 5, 7, 2, 4.

Solution:

Step 1: Arrange the numbers in ascending order:

2, 3, 4, 5, 7, 8

Step 2: Count the numbers:

There are 6 numbers in the set.

Step 3: Find the middle positions:

Since there are 6 numbers, the median is the average of the numbers at the 3rd and 4th positions.

\text{Median position} = \frac{3^\text{rd} \, \text{value} + 4^\text{th} \, \text{value}}{2}

Step 4: Identify the median:

The numbers at the 3^rd and 4^th positions in the ordered set are 4 and 5, respectively.

Median = (4 + 5)/2 = 4.5

Therefore, the median of 8, 3, 5, 7, 2, 4 is \boxed{4.5}.

Example 2: Find the median of the following set of numbers: 45, 78, 22, 90, 67, 56, 89, 34, 12, 49, 23, 68.

Solution:

Step 1: Arrange the numbers in ascending order:

12, 22, 23, 34, 45, 49, 56, 67, 68, 78, 89, 90

Step 2: Count the numbers:

There are 12 numbers in the set.

Step 3: Find the middle positions:

Since there are 12 numbers, the median is the average of the numbers at the 6^th and 7^th positions.

\text{Median position} = \frac{6^\text{th} \, \text{value} + 7^\text{th} \, \text{value}}{2}

Step 4: Identify the median:

The numbers at the 6^th and 7^th positions in the ordered set are 49 and 56, respectively.

\text{Median} = \frac{49 + 56}{2} = \frac{105}{2} = 52.5

Therefore, the median of 45, 78, 22, 90, 67, 56, 89, 34, 12, 49, 23, 68 is 52.5.

Example 3: Median of Grouped Data

Example 1: The following table shows the distribution of the marks obtained by students in a test. Find the median.

Solution:

Step 1: Calculate the cumulative frequency (C.F.):

Step 2: Determine the total number of observations (N):

N = 30

Step 3: Find the median class:

The median class is the class where N/2 falls. Here, 30/2 = 15. The median class is 20 - 30 because the 15th observation lies in this class.

Step 4: Identify the following for the median formula:

• l = 20 (lower boundary of the median class)
• N = 30 (total number of observations)
• cf = 7 (cumulative frequency of the class before the median class)
• f = 8 (frequency of the median class)
• h = 10 (class interval)

Step 5: Apply the median formula:

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

⇒ Median = 20 + \left( \frac{15 - 7}{8} \right) \times 10

⇒ Median = 20 + \left( \frac{8}{8} \right) \times 10

⇒ Median = 20 + 10 = 30

Therefore, the median of the grouped data is 30.

Example 2: The following table shows the distribution of the monthly salaries of employees in a company. Find the median.

Solution:

Step 1: Calculate the cumulative frequency (C.F.):

Step 2: Determine the total number of observations (N):

N = 80

Step 3: Find the median class:

The median class is the class where \frac{N}{2} falls. Here, \frac{80}{2} = 40. The median class is 15-20 because the 40th observation lies in this class.

Step 4: Identify the following for the median formula:

• l = 15 (lower boundary of the median class)
• N = 80 (total number of observations)
• cf = 34 (cumulative frequency of the class before the median class)
• f = 30 (frequency of the median class)
• h = 5 (class interval)

Step 5: Apply the median formula:

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

⇒ Median = 15 + \left( \frac{40 - 34}{30} \right) \times 5

⇒ Median = 15 + \left( \frac{6}{30} \right) \times 5

⇒ Median = 15 + 0.2 × 5

⇒ Median = 15 + 1 = 16

Therefore, the median of the grouped data is \boxed{16}.

Read More,

• Statistics
• Mean, Median and Mode
• Difference Between Mean, Median, and Mode
• Relation Between Mean Median and Mode

Practice Problems on Median of Data

Problem 1: Find the median of the following set of numbers: 17, 23, 12, 14, 19, 21, 25

Problem 2: Find the median of the following set of numbers: 3, 8, 2, 10, 5, 6, 7

Problem 3: Find the median of the following data set: 2, 4, 6, 8

Problem 4: Find the median of the following data set: 7, 3, 1, 9, 5, 11

Problem 5: Find the median class for the following frequency distribution:

Problem 6: Find the median for the following frequency distribution: