How to Find the Median

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

Question

What is the median of the test scores below?

Step-by-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

2

Find the middle number. In our example, the middle number is the 3rd number of 5.

4 6 7 8 10

Median = 7

Answer:

The median of the test scores is 7.

Even Numbered Set

This time, 6 students take a mathematics test.

Question

What is the median of the test scores below?

Step-by-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

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

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

Answer:

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

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See Also

What is a frequency table? Finding the median from a frequency table Learn more about the mean ( interactive widget)