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Methods for Finding the Quartiles (Mathematics Lesson)

Methods for Finding the Quartiles

There are three quartiles in a set of numbers: The quartiles divide the set, when they are in numerical order, into four equal groups.

It is possible to find the quartiles. However, there are different methods for finding the quartiles, which give different values for them. (Note: Within each method, the method is slightly different dependent on whether there are an odd or even number of numbers in the set).

Method 1: Moore and McCabe (M & M)

Odd Numbered Set
Imagine you wanted to find the quartiles of the set of numbers shown below:

  • The middle quartile Q2 is the middle number (the median).

If we exclude the middle quartile, it divides the set into two equal groups either side of it: a lower half and an upper half.
  • The lower quartile Q1 is the middle number (the median) of the lower half:


  • The upper quartile Q3 is the middle number (the median) of the upper half:

The table below summarizes the quartiles:



Even Numbered Set
Imagine you wanted to find the quartiles of the set of numbers shown below:

  • The middle quartile Q2 is the median. Because it is an even numbered set, the median is halfway between the middle two numbers.



    Note: The median of an even numbered set is the mean of the middle two numbers, 5 and 6.
    (5 + 6) ÷ 2 = 5.5
The middle quartile divides the set into two equal groups either side of it: a lower half and an upper half.
  • The lower quartile Q1 is the middle number (the median) of the lower half:


  • The upper quartile Q3 is the middle number (the median) of the upper half:

The table below summarizes the quartiles:

Method 2: Tukey

Odd Numbered Set
Imagine you wanted to find the quartiles of the set of numbers shown below:

  • The middle quartile Q2 is the middle number (the median).

The middle quartile divides the set into two equal groups: a lower half and an upper half. We include the median in both halves.
  • The lower quartile Q1 is the middle number (the median) of the lower half (including the middle quartile):


  • The upper quartile Q3 is the middle number (the median) of the upper half (including the middle quartile):

The table below summarizes the quartiles:



Even Numbered Set
Imagine you wanted to find the quartiles of the set of numbers shown below:

  • The middle quartile Q2 is the median. Because it is an even numbered set, the median is halfway between the middle two numbers.



    Note: The median of an even numbered set is the mean of the middle two numbers, 5 and 6.
    (5 + 6) ÷ 2 = 5.5
The middle quartile divides the set into two equal groups either side of it: a lower half and an upper half.
  • The lower quartile Q1 is the middle number (the median) of the lower half:


  • The upper quartile Q3 is the middle number (the median) of the upper half:

The table below summarizes the quartiles:

Method 3: Mendenhall and Sincich (M & S)

Odd Numbered Set
Imagine you wanted to find the quartiles of the set of numbers shown below:

  • The middle quartile Q2 is the middle number (the median).


  • To find which number is the lower quartile Q1, use the formula below:



    In this formula, n is how many numbers there are in the set. In our example, n = 11.
    (n + 1) ÷ 4 = (11 + 1) ÷ 4
    (n + 1) ÷ 4 = 12 ÷ 4
    (n + 1) ÷ 4 = 3
    The lower quartile is the 3rd number in the set:


  • To find which number is the upper quartile Q3, use the formula below:



    In this formula, n is how many numbers there are in the set. In our example, n = 11.
    3(n + 1) ÷ 4 = 3 × (11 + 1) ÷ 4
    3(n + 1) ÷ 4 = 3 × 12 ÷ 4
    3(n + 1) ÷ 4 = 36 ÷ 4
    3(n + 1) ÷ 4 = 9
    The upper quartile is the 9th number in the set:

The table below summarizes the quartiles:



Even Numbered Set
Imagine you wanted to find the quartiles of the set of numbers shown below:

  • The middle quartile Q2 is the median. Because it is an even numbered set, the median is halfway between the middle two numbers.


  • To find which number is the lower quartile Q1, use the formula below:



    In this formula, n is how many numbers there are in the set. In our example, n = 10.
    (n + 1) ÷ 4 = (10 + 1) ÷ 4
    (n + 1) ÷ 4 = 11 ÷ 4
    (n + 1) ÷ 4 = 2.75

    (n + 1) ÷ 4 = 3 rounded up to the nearest integer
    The lower quartile is the 3rd number in the set:


  • To find which number is the upper quartile Q3, use the formula below:



    In this formula, n is how many numbers there are in the set. In our example, n = 10.
    3(n + 1) ÷ 4 = 3 × (10 + 1) ÷ 4
    3(n + 1) ÷ 4 = 3 × 11 ÷ 4
    3(n + 1) ÷ 4 = 33 ÷ 4
    3(n + 1) ÷ 4 = 8.25

    3(n + 1) ÷ 4 = 8 rounded down to the nearest integer
    The upper quartile is the 8th number in the set:

The table below summarizes the quartiles:

Comparison of Methods

The table below compares the quartiles found from the different methods.

It finds the quartiles for the odd and even numbered sets of numbers below:
Set A: 1 2 3 4 5 6 7 8 9 10 11
Set B: 1 2 3 4 5 6 7 8 9 10

A Real Example of the Methods for Finding the Quartiles

The slider below gives an example of finding the quartiles of a set of numbers using different methods.
Interactive Test
  show
 


Here's a second test on finding the quartiles.
Here's a third test on 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 quartiles of the numbers below. Don't be tempted to jump right in.
3 2 4 5 1
Put the numbers in order and then find the quartile:
1 2 3 4 5
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'.