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How to Find the Median (Mathematics Lesson)

Finding the Median

We can find the median of a set of numbers.

Imagine a teacher wanted to find the class's average test score in mathematics.

How to Find the Median

The median is found by ordering the numbers and finding the middle number.

The middle number is slightly different depending on whether there are an odd or even number of numbers in the set.
  • If there are an odd number of numbers, the median is simply the middle number in the set.

  • If there are an even number of numbers, the median is halfway between the middle two numbers. (It is the mean of the middle two numbers).
Let's look at an example of an odd numbered set and an even numbered set separately.

Odd Numbered Set

An Example Question

What is the median of the test scores?

Step 1
List the numbers in numerical order (going from the smallest to the largest number).
7 10 8 6 4 → 4 6 7 8 10
Step 2
Find the middle number. In our example, the middle number is the 3rd number of 5.
4 6 7 8 10

Median = 7
The median of the test scores is 7.

Even Numbered Set

This time, 6 students take a mathematics test. There test scores are given below:



An Example Question

What is the median of the test scores?

Step 1
List the numbers in numerical order (going from the smallest to the largest number).
8 9 5 6 5 10 → 5 5 6 8 9 10
Step 2
Find the middle two numbers. In our example, the middle two numbers are the 3rd 4th numbers of 6.
5 5 6 8 9 10

Middle two numbers = 6 8
Step 3
Find the mean of the middle two numbers (6 and 8) by adding them together and then dividing by 2.
(6 + 8) ÷ 2 = 14 ÷ 2 = 7
The median of the test scores is 7.

A Formula to Find the Middle Number

In the two examples above, we have quoted that the 3rd number is the middle of 5 and that the 3rd and 4th are the middle two of 6.

It is relatively easy to see by eye where the middle number is when there are relatively few of them and they are written out in front of us. It may not always be this easy.

Luckily, there is a formula for finding which is the middle number.



In this formula, n is how many numbers there are in the set.

Let's apply this formula to the two examples above.
  • In the first example, we are asked to find the median of 5 numbers (7, 10, 8, 6 and 4). This means that n = 5.

    Using the formula to find the middle number gives:
    Middle number = (n + 1) ÷ 2 = (5 + 1) ÷ 2 = 6 ÷ 2 = 3
    The median of 7 10 8 6 4 is the 3rd number when they are ordered:
    7 10 8 6 4 → 4 6 7 8 10

    Median = 7

  • In the second example, we are asked to find the median of 6 numbers (8, 9, 5, 6, 5 and 10). This means that n = 6.

    Using the formula to find the middle number gives:
    Middle number = (n + 1) ÷ 2 = (6 + 1) ÷ 2 = 7 ÷ 2 = 3.5
    The median of 8 9 5 6 5 10 is the 3.5th number when they are ordered, which means halfway between the 3rd and 4th numbers:
    8 9 5 6 5 10 → 5 5 6 8 9 10

    Halfway between 3rd and 4th numbers = (6 + 8) ÷ 2 = 7

    Median = 7

A Real Example of How to Find the Median

The slider below gives another example of finding the median.

How to Find the Median from a Frequency Table

Sometimes data is presented in frequency tables.

A frequency table representing the test scores is shown below:



It is possible to find that the median test score is 7.

Read more about finding the median from a frequency table.

Interactive Widget
Here is an interactive widget to help you learn about the median.
Interactive Test
  show
 


Here's a second test on finding the median.
Here's a third test on finding the median.
Top Tip

The Median is the Middle

The median gives the number exactly in the middle of a set of numbers.

Half of the numbers in the set are less than the median.

Half of the numbers in the set are greater than the median.

Note

What Is the Median?

The median is an average of a set of numbers.

The median is the middle number in a set of numbers that has been arranged in order. (If there are an even number of numbers in a set, the median is the mean of the middle two numbers).

What's in a Name?

"Median" comes from the Latin "medianus", meaning "middle".

87% of People Think They Are Above Average

In a survey of MBA students at Stanford University, 87% of the students believed themselves to be smarter than the median.

This is impossible! By definition, only 50% of students will be smarter then the average.

Similar surveys and experiments have shown that people often make the same mistake, and overestimate how rich, clever, good looking and popular they are relative to others.

Other Types of Average

The most commonly commonly used types of average, other than the median, are:

Outliers

Sometimes in a group of numbers, a few of them are much larger...

3, 5, 2, 7, 150

...or much smaller than the rest:

1, 502, 847, 564, 980

These relatively large (150) or small (1) numbers are called outliers.

The median is not affected by outliers, unlike the mean. For example, consider the numbers:
1, 2, 3
The mean is 2 and the median is 2.

If instead the numbers have a large outlier:
1, 2, 300
The median is still 2 but the mean is now 101.

If you read about the average salary in a country, ask if it is the mean or median average?

The median average won't be dragged up by some very rich people, while the mean average may be.