Median: Definition, Formula, How to Find Median with Examples

In statistics, the median is a measure of central tendency that represents the middle value in a data set when the numbers are arranged in order. Understanding the median is essential for analyzing data and summarizing large sets of information. In this guide, we will discuss the median formula, how to find the median, and the importance of medians in data analysis.

Table of Contents

• What is the Median?
• Median Formula
• How to Find the Median
• Importance of the Median
• Applications of Median in Real Life
• Solved Examples on Median
• Practice Questions on Median
• Conclusion
• Frequently Asked Questions on Median
  
  

What is Median?

The median is the middle number in a set of numbers arranged in ascending or descending order. If the data set has an odd number of values, the median is the number in the middle. If the data set has an even number of values, the median is the average of the two middle numbers.

For example, in the data set 3, 5, 7, the median is 5, since it is the middle value. However, in the data set 3, 5, 7, 9, the median is the average of the two middle numbers, 5 and 7, which is 6.

Key Points:

• The median divides the data into two equal halves.
• It is less affected by outliers and skewed data than the mean, making it a better measure of central tendency when data contains extreme values.
  
  

Median Formula

The median is calculated differently depending on whether the number of observations is odd or even.

For Odd Number of Data Points:

When the total number of observations (n) is odd, the median is the middle value.

Median = Value of (n + 1) / 2 th observation

For Even Number of Data Points:

When the total number of observations (n) is even, the median is the average of the two middle values.

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

Note: Before applying the formula, the data must always be arranged in either ascending or descending order.

Read more:

• Mode
• Arithmetic Mean
• Empirical Relation between Mean, Median and Mode
• Median for Class 7
  
  

How to Find Median

To calculate the median, follow these steps:

1. Arrange the data: Order the numbers in either ascending or descending order.
2. Identify the middle number:
   • If there is an odd number of values, the median is the middle number.
   • If there is an even number of values, the median is the average of the two middle numbers.

Example 1: Odd Number of Values

Data set: 7, 3, 9, 5, 11
Step 1: Arrange the data in ascending order: 3, 5, 7, 9, 11
Step 2: The median is 7, as it is the middle value.

Example 2: Even Number of Values

Data set: 8, 3, 9, 5
Step 1: Arrange the data in ascending order: 3, 5, 8, 9
Step 2: The median is (5 + 8) / 2 = 6.5, as it is the average of the two middle values.

Importance of the Median

The median is one of the key measures of central tendency, alongside the mean and mode. It is particularly useful in scenarios where:

1. Outliers are present: In a data set with extreme values, the median is less sensitive to outliers than the mean.
2. Data is skewed: The median provides a better representation of the typical value when the data is not symmetrically distributed.
3. Ordinal data: When dealing with ordered categories, the median can help determine the middle category.

In these cases, the median offers a more accurate representation of the central value of the data than other measures such as the mean.

Applications of Median in Real Life

The median is not just useful in mathematics; it also has several practical applications:

• Income distribution: In economics, the median income is often used to measure the central tendency of income distribution, as it is less affected by extremely high or low incomes.
• Real estate: In real estate, the median price of homes is often quoted as a better indicator of typical home prices in a given area than the average price.
• Sports statistics: In sports, the median can be used to determine the middle performance level of players, eliminating outliers that may distort the average.
• Medical data: The median is used in healthcare to find the middle value in a set of data like patient ages or recovery times, especially when the data is not normally distributed.
  
  

Learn more about mean, median and mode
Also, check for mean, median and mode questions

Solved Examples on Median

Example 1: Finding the Median in a Group of Numbers

Given data set: 7, 1, 4, 9, 6
Step 1: Arrange the data in ascending order: 1, 4, 6, 7, 9
Step 2: The median is the middle number.

Median = 6

Example 2: Using the Median Formula

Given data set: 4, 7, 1, 9, 5, 2
Step 1: Arrange the data in ascending order: 1, 2, 4, 5, 7, 9
Step 2: Since there are 6 numbers (even), the median is the average of the two middle values.

Median = (4 + 5) / 2 = 4.5

Example 3: Finding the Median (Odd Number of Values)

Given data set: 11, 3, 8, 6, 10
Step 1: Arrange the data in ascending order: 3, 6, 8, 10, 11
Step 2: Since there are 5 numbers (odd), the median is the middle value.

Median = 8

Example 4: Finding the Median (Even Number of Values)

Given data set: 14, 9, 5, 12, 7, 10
Step 1: Arrange the data in ascending order: 5, 7, 9, 10, 12, 14
Step 2: Since there are 6 numbers (even), the median is the average of the two middle values.

Median = (9 + 10) / 2 = 9.5

Example 5: Median with Larger Numbers

Given data set: 25, 18, 32, 40, 21
Step 1: Arrange the data in ascending order: 18, 21, 25, 32, 40
Step 2: Since there are 5 numbers (odd), the median is the middle value.

Median = 25

Practice Questions on Median

1. Find the median of: 9, 3, 7, 1, 5
2. The ages of 5 children are 12, 10, 14, 9, and 11. Find the median age.
3. Find the median of the first 7 natural numbers.
4. The marks scored by 6 students are 45, 38, 50, 42, 47, 40. Find the median.
5. A data set contains 8 values: 6, 12, 8, 10, 14, 16, 18, 20. Find the median.
6. The heights (in cm) of 5 students are 150, 145, 160, 155, 148. Find the median height.
7. Find the median of: 25, 30, 20, 35, 40, 45, 50, 55, 60.
8. The scores in a game are: 18, 22, 15, 20, 25, 17, 19. Find the median score.
9. A teacher recorded the number of books read by 6 students: 3, 5, 2, 7, 4, 6. Find the median.
10. The daily temperatures (°C) for a week are: 30, 32, 31, 29, 28, 33, 34. Find the median temperature.
    
    

Conclusion

The median is a fundamental statistical measure that provides valuable insights into the central tendency of data. Whether working with small or large data sets, the median helps to find the middle value and is especially useful when dealing with outliers or skewed data. By understanding the median formula and learning how to find the median, you can improve your data analysis skills.

Frequently Asked Questions on Medians

1. What is the median?

Answer: The median is the middle value of a data set when the numbers are arranged in order.

2. How do you find the median?

Answer: To find the median, arrange the data in order and identify the middle value (for odd numbers) or average the two middle values (for even numbers).

3. Why is the median useful?

Answer: The median is useful when the data contains outliers or is skewed, as it provides a better representation of the central tendency than the mean.

4. What is the difference between the mean and median?

Answer: The mean is the average of all the numbers, while the median is the middle value when the data is arranged in order.

5. Can the median be used in all data sets?

Answer: The median is particularly useful for ordinal and skewed data sets, where the mean may not accurately represent the data.

Learn more about the median and other statistical measures at Orchids The International School. Start mastering medians today!