The Lesson
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 Moore and McCabe method.
 The Tukey method.
 The Mendenhall and Sincich method.
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).

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:
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 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:
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 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):
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 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:
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 = 3The 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 = 9The upper quartile is the 9^{th} number in the set:
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 integerThe 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 integerThe upper quartile is the 8^{th} number in the set:
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
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