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Upper Quartiles (Mathematics Glossary)

What Is the Upper Quartile?

The upper quartile is the middle number between the median and the highest number.



It is the middle number of the upper half of a set of numbers.

The upper quartile is denoted Q3.

Understanding the Upper Quartile and the Quartiles

The upper quartile is the highest of the three quartiles.

The three quartiles divide a set of numbers, that are in numerical order, into four equal groups:

  • The middle quartile is also known as the median. It is the middle number in the set. It divides the set in two halves.

  • The lower quartile is the middle number of the lower half. It divides the lower half in two.

  • The upper quartile is the middle number of the upper half. It divides the upper half in two.

Finding the Upper Quartile

There are different methods for finding the upper quartile, which give different values.
  • The Moore and McCabe method excludes the median from the upper half. The upper quartile is the middle number of the upper half:


  • The Tukey method includes the median in the upper half. The upper quarter is the middle number of the upper half:



    Note: When the upper half has an even number of numbers, the middle number is halfway between the middle two numbers. It is the mean of the middle two numbers:
    Q3 = (8 + 9) ÷ 2 = 8.5

  • The Mendenhall and Sincich method uses a formula to find the position of the upper quartile. The formula is shown below:



    In this formula, n is how many numbers there are in the set. In our example, n is 11.
    Position of Q3 = 3(n + 1) ÷ 4 = 3 × (11 + 1) ÷ 4
    Position of Q3 = 36 ÷ 4
    Position of Q3 = 9th
    The upper quartile is the 9th number in the set:

Interactive Test
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Here's a second test on the upper quartile.
Here's a third test on the upper quartile.
Note

What Is a Quartile?

A quartile is one of three numbers that divide a set into four equal groups.

A quartile can also describe each of the four groups.

What's In a Name?

Quartile comes from the Latin word 'quartus', meaning 'a fourth'. It comes from the same root as 'quarter'.

The Other Quartiles

The upper quartile is just one of the three quartiles. The other quartiles are:
  • The lower quartile. It is the middle number between the smallest number and the median.

  • The middle quartile (also called the median). It is the middle number of the whole set of numbers.
There are different methods for finding the quartiles, which give different values.

Read more about the methods for finding the quartiles

Beware

Put Your Numbers in Order

The quartiles of a set of numbers divide the numbers into four equal groups when the numbers are in order.

Imagine you were asked to find the upper quartile of the numbers below. Don't be tempted to jump right in.
3 1 2 5 4
Put the numbers in order and then find the upper quartile:
1 2 3 4 5