# Methods for Finding the Quartiles (Mathematics Lesson)

# 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}

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
**Q**is the middle number (the median)._{2}

**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**is the middle number (the median) of the lower half:_{1}

- The upper quartile
**Q**is the middle number (the median) of the upper half:_{3}

**Even Numbered Set**

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

- The middle quartile
**Q**is the median. Because it is an even numbered set, the median is halfway between the middle two numbers._{2}

**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**is the middle number (the median) of the lower half:_{1}

- The upper quartile
**Q**is the middle number (the median) of the upper half:_{3}

# Method 2: Tukey

**Odd Numbered Set**

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

- The middle quartile
**Q**is the middle number (the median)._{2}

**include**the median in both halves.

- The lower quartile
**Q**is the middle number (the median) of the lower half (including the middle quartile):_{1}

- The upper quartile
**Q**is the middle number (the median) of the upper half (including the middle quartile):_{3}

**Even Numbered Set**

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

- The middle quartile
**Q**is the median. Because it is an even numbered set, the median is halfway between the middle two numbers._{2}

**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**is the middle number (the median) of the lower half:_{1}

- The upper quartile
**Q**is the middle number (the median) of the upper half:_{3}

# 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**is the middle number (the median)._{2}

- To find which number is the lower quartile
**Q**, use the formula below:_{1}

In this formula,**n**is how many numbers there are in the set. In our example,**n**= 11.

(n + 1) ÷ 4 = (11 + 1) ÷ 4The lower quartile is the

(n + 1) ÷ 4 = 12 ÷ 4

(n + 1) ÷ 4 = 3**3**number in the set:^{rd}

- To find which number is the upper quartile
**Q**, use the formula below:_{3}

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) ÷ 4The upper quartile is the

3(n + 1) ÷ 4 = 3 × 12 ÷ 4

3(n + 1) ÷ 4 = 36 ÷ 4

3(n + 1) ÷ 4 = 9

**9**number in the set:^{th}

**Even Numbered Set**

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

- The middle quartile
**Q**is the median. Because it is an even numbered set, the median is halfway between the middle two numbers._{2}

- To find which number is the lower quartile
**Q**, use the formula below:_{1}

In this formula,**n**is how many numbers there are in the set. In our example,**n**= 10.

(n + 1) ÷ 4 = (10 + 1) ÷ 4The lower quartile is the

(n + 1) ÷ 4 = 11 ÷ 4

(n + 1) ÷ 4 = 2.75

(n + 1) ÷ 4 = 3 rounded up to the nearest integer**3**number in the set:^{rd}

- To find which number is the upper quartile
**Q**, use the formula below:_{3}

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) ÷ 4The upper quartile is the

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**8**number in the set:^{th}

# 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.