Interquartile range

The interquartile range of a set of numbers is the difference between the third quartile (sometimes called the upper quartile) and the first quartile (sometimes called the lower quartile).

But what is a quartile?

Quartiles

A quartile is, as it sounds, (a multiple of) a quarter of the way through a set of data. It’s the value that you get when you go to the number 1/4 of the way into your dataset (as well as 2/4 and 3/4 of the way in).

Usually, we just focus on:

• The first quartile (Q1), which is the value that is 1/4 of the way through the dataset. We can call this the lower quartile.
• The third quartile (Q3), which is the value that is 3/4 of the way through the dataset. We can call this the upper quartile.

Finding the first quartile

The first quartile is half way through the lower half of the dataset.

That means that we can find it like this:

1. Arrange the numbers in order from smallest to largest.
2. Find the median of the dataset.
3. Get rid of the median, and everything above it.
   • Doing that leaves us with the lower half of the dataset.
4. Find the median of the lower half of the dataset.
   • That median is the first quartile.

Finding the third quartile

The third quartile is half way through the upper half of the dataset.

So, similarly to finding the first quartile, we can find it like this:

1. Arrange the numbers in order from smallest to largest.
2. Find the median of the dataset.
3. Get rid of the median, and everything below it.
   • Doing that leaves us with the upper half of the dataset.
4. Find the median of the upper half of the dataset.
   • That median is the third quartile.

Interquartile range

The interquartile range is the difference between the third quartile and the first quartile.

interquartile range = third quartile - first quartile

Examples

Find the lower quartile of \{1, 4, 9, 16, 25\}

• The numbers are already in order: \{1, 4, 9, 16, 25\}
• The median is 9.
• The lower half of the dataset is \{1, 4\}.
• The median of the lower half is the value in-between 1 and 4, which is \frac{1 + 4}{2} = \frac{5}{2} = 2.5.
• Answer: \text{first quartile} = 2.5

Find the upper quartile of \{1, 4, 9, 16, 25\}

• The numbers are already in order: \{1, 4, 9, 16, 25\}
• The median is 9.
• The upper half of the dataset is \{16, 25\}.
• The median of the upper half is the value in-between 16 and 25, which is \frac{16 + 25}{2} = \frac{41}{2} = 20.5.
• Answer: \text{third quartile} = 20.5

Find the interquartile range of \{1, 4, 9, 16, 25\}

• The first quartile is 2.5.
• The third quartile is 20.5.
• The interquartile range is the difference between the third quartile and the first quartile, which is 20.5 - 2.5 = 18.
• Answer: \text{interquartile range} = 18

Find the interquartile range of \{3, 1, 4, 1, 5, 9, 2, 6, 5, 8, 11\}

• Arrange the numbers in order: \{1, 1, 2, 3, 4, 5, 5, 6, 8, 9, 11\}
• The median is 5.
• The lower half of the dataset is \{1, 1, 2, 3, 4\}.
• The median of the lower half is 2, so the first quartile is 2.
• The upper half of the dataset is \{5, 6, 8, 9, 11\}.
• The median of the upper half is 8, so the third quartile is 8.
• \text{interquartile range} = \text{third quartile} - \text{first quartile}
• \text{interquartile range} = 8 - 2
• Answer: \text{interquartile range} = 6