Methods for Finding the Quartiles
Methods for Finding the Quartiles
There are three quartiles in a set of numbers:

The lower quartile Q_{1}.

The middle quartile (called the median) Q_{2}.

The upper quartile Q_{3}.
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.

The Tukey method.
(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 Q_{2} 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 Q_{1} is the middle number (the median) of the lower half:

The upper quartile Q_{3} 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 Q_{2} 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 Q_{1} is the middle number (the median) of the lower half:

The upper quartile Q_{3} 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 Q_{2} 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 Q_{1} is the middle number (the median) of the lower half (including the middle quartile):

The upper quartile Q_{3} 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 Q_{2} 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 Q_{1} is the middle number (the median) of the lower half:

The upper quartile Q_{3} 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 Q_{2} is the middle number (the median).

To find which number is the lower quartile Q_{1}, 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 3^{rd} number in the set:

To find which number is the upper quartile Q_{3}, 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 9^{th} 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 Q_{2} 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 Q_{1}, 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 3^{rd} number in the set:

To find which number is the upper quartile Q_{3}, 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 8^{th} 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