# Quartiles

## Quartiles Revision

**Quartiles**

Where the **median** is the value that splits the data set evenly in two, **the quartiles **can be used to further separate the data into four equally sized sets.

**The lower quartile** can be thought of as the “median” of the lower half of the data.

Similarly **the upper quartile** can be thought of as the “median” of the higher half of the data.

**Upper and Lower Quartiles**

The **lower quartile (Q_1)** is the value in the ordered data such that 75\% of the data is** larger** than it and 25\% of the data is** smaller**.

In a data set size n, Q_1 is in position \dfrac{n+1}{4}

The **upper quartile (Q_3)** is the value in the ordered data such that 75\% of the data is** smaller **than it and 25\% of the data is** larger**.

In a data set size n, Q_3 is in position \dfrac{3(n+1)}{4}

**Q_2 **denotes the **median**, found at \dfrac{n+1}{2}

If the position of the upper or lower quartile is not a whole number, we must take an average of the data in the next position up, and the data in the next position down. For example if n = 26, then

\dfrac{26+1}{4} = 6.75 and \dfrac{3(25+1)}{4} = 20.25

In this case Q_1 will be the mean of the data in position 6 and 7, added together and divided by two. Q_3 will be the mean of the data in position 20 and 21.

**Interquartile Range**

**Interquartile range (IQR) **is a measure of the distribution of the data.

IQR = Q_3 - Q_1

IQR can be more useful than range as extreme outliers will lie outside it and do not affect the IQR.

**Example 1: Finding the UQ **

Considering the following data set

2, 36, 48, 49, 58, 62, 92

find the upper quartile.

**[1 mark]**

There are 7 numbers in this data set. We can find the position of Q_3.

Q_3 \text{ position } = \dfrac{3(7+ 1)}{4} = 6th,

The sixth value is 62, so Q_3 = 62.

**Example 2: Finding the LQ **

Considering the integers from 1 to 159, find their lower quartile.

**[1 mark]**

There are 159 numbers in this data set. We can find the position of Q_1.

Q_1 \text{ position } = \dfrac{159+ 1}{4} = 40th,

As the first value is 1 and the second value is 2 and so on, the 40^{\text{th}} value is 40.

Hence, Q_1 = 40

**Example 3: Finding the IQR **

Considering the following data set

-45, 3, 5, 5, 5, 6, 7, 8, 10, 10, 118

find the interquartile range.

**[3 marks]**

As the data set has 11 elements, we can find the position of Q_1.

Q_1 \text{ position } = \dfrac{11+ 1}{4} = 3 rd,

The third number is 5, so Q_1 = 5. Now for Q3,

Q_3 \text{ position } = \dfrac{3(11+ 1)}{4} = 9th,

The ninth number is 10, so Q_3 = 10. Now for the IQR.

IQR = Q_3 - Q_1 = 10 - 5 = 5

The IQR = 5.

Notice how this differs from the range of 163 as outliers are ignored.

## Quartiles Example Questions

**Question 1:** Consider the following data set

10, 14, 23, 35, 43, 60, 70, 78, 86, 100, 118, 131, 160, 164, 168, 169, 171, 173, 198

a) Find the lower quartile.

**[1 mark]**

b) Find the upper quartile.

**[1 mark]**

a) We can find the position of the lower quartile in the data set with the formula

Q_1 Position = \dfrac{n+1}{4}

There are 19 entries in the data set and so n = 19

Q_1 Position = \dfrac{19+1}{4}=\dfrac{20}{4}=5,

The 5^{th} value in the data set is the lower quartile, which is 43.

b) We can find the position of the upper quartile in the data set with the formula

Q_3 Position = \dfrac{3(n+1)}{4}

There are 19 entries in the data set and so n = 19

Q_3 Position = \dfrac{3(19+1)}{4}

Q_3 Position = \dfrac{3 \times 20}{4} = \dfrac{60}{4} = 15

The 15^{th} value in the data set is the upper quartile, which is 168

**Question 2: **Consider the following data set

33, 97, 37, 71, 13, 77, 84, 55, 57, 27, 94

Find the interquartile range.

**[2 marks]**

Firstly we must order this data. Which when done gives us

13, 27, 33, 37, 55, 57, 71, 77, 84, 94, 97

The data set has 11 elements. So we can find the position of Q_1 and Q_3.

Q_1 Position = \dfrac{11+1}{4} = 3

The third element is 33 and so it is the lower quartile.

Q_3 Position = \dfrac{3(11+1)}{4} = 9

The ninth element is 84 and so it is the upper quartile.

Then the interquartile range

IQR = Q_3 -Q_1

IQR = 84 - 33 = 51

**Question 3: **Two maths classes sat a test. Below are there scores in percentages.

**Class 1:** 36, 65, 72, 88, 89, 89, 90, 92, 99, 99, 99

**Class 2: **9, 42, 71, 71, 72, 75, 75, 77, 79, 81, 100

By comparing their interquartile ranges, find which class has more consistent grades.

**[3 marks]**

Both classes 1 and 2 have 11 students. So for both, the lower quartile is in the 3^{\text{rd}} position and the upper quartile is in the 9^{\text{th}} position.

**Class 1:
**Q_3 =99

Q_1 = 72

IQR = 99 - 72 = 27

**Class 2:
**Q_3 =79

Q_1 = 71

IQR = 79 - 71 = 8

While class 2 had the lower median grade, it did have the more consistent grades, despite outliers.

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