Follow Us on Twitter

Median (Mathematics Glossary)

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).

Dictionary Definition

The Oxford English Dictionary defines the median as "the middle term of a discrete series arranged in order of magnitude (or, if there is no middle term, the mean of the middle two terms)."

Understanding the Median

The median is the middle number in a set. Half of the numbers are less than the median. Half of the numbers are more than the median.



Imagine you wanted to find the median height of a group of friends.

You would line the friends up in height order, from shortest to tallest. The friend in the middle would have the median height.



There are an equal number of friends shorter than the median and friends taller than the median.

Finding the Median

Imagine we wanted to find the median of the numbers given below:



To find the median, order the numbers in numerical order and find the middle number:
3 2 4 5 1 → 1 2 3 4 5

Median = 3
In the example above, there were an odd number of numbers in the set.

Imagine we wanted to find the median of the even number of numbers given below:



To find the median, order the numbers in numerical order, find the middle two numbers and take the mean of those numbers:
3 2 4 5 1 6 → 1 2 3 4 5 6

Middle two numbers = 3 4

Mean of middle two numbers = (3 + 4) ÷ 2 = 3.5

Median = 3.5
Don't forget: The mean of two numbers is found by adding them together and dividing by 2.

Read more about how to find the median

A Formula for Finding the Middle Number

The median is the middle number in an ordered set. If you know how many numbers there are in a set, which is the middle number?

The formula for finding the middle number is:



In this formula, n is how many numbers there are in the set.
  • If there are 5 numbers in a set (n = 5), the middle number is the 3rd number:
    Middle number = (n + 1) ÷ 2 = (5 + 1) ÷ 2 = 6 ÷ 2 = 3
    The median of 3 2 4 5 1 is the 3rd number when they are ordered:
    3 2 4 5 1 → 1 2 3 4 5

    Median = 3

  • If there are 6 numbers in a set (n = 6), the middle number is the 3.5th number:
    Middle number = (n + 1) ÷ 2 = (6 + 1) ÷ 2 = 7 ÷ 2 = 3.5
    The median of 3 2 4 5 1 6 is the 3.5th number when they are ordered, which means halfway between the 3rd and 4th numbers:
    3 2 4 5 1 6 → 1 2 3 4 5 6

    Halfway between 3rd and 4th numbers = (3 + 4) ÷ 2 = 3.5

    Median = 3.5

Other Types of Average

Along with the median, there are other types of average. The most commonly used are: Read more about the mean
Read more about the mode

Interactive Widget
Here is an interactive widget to help you learn about averages.

Interactive Test
  show
 


Here's a second test on the median.
Here's a third test on the median.
Beware

Put Your Numbers in Order

The median of a set of numbers is the middle number when the numbers are in order.

Imagine you were asked to find the median of the numbers below. Don't be tempted to jump right in and select 4 as the middle number.
3 2 4 5 1
Put the numbers in order and see that 3 is the middle number:
1 2 3 4 5
Note

What's in a Name?

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

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.

Finding the Median from a Frequency Table

A frequency table shows how often (how frequently) each number appears in a list of numbers.

We could write the following numbers in a frequency table:
1, 2, 3, 4, 5


With a little further calculation, the median can be found from the frequency table.

Read more about finding the median from a frequency table