## Quartiles

$Q_1$ represents the first quartile, and is a value that borders the first 25% of the data values.
$Q_3$ represents the third quartile, that is, the value that cuts off the highest 25% of the data values.

## IQR

The Interquartile range (IQR) is defined to be the spread of the middle 50% of data values. So the   range of the third quartile and the first quartile.

To calculate the Interquartile range you have to first find the third quartile $Q_3$ and first quartile $Q_1$ then find the difference of the two numbers.

$IQR = Q_3 - Q_1$

### Application

Given the following data set:

${4,4,5,6,7,8,8,9,10,11,15}$

Firstly, you find the median of the data which is the middle value. In this case, 8. Then you consider the right half of data values and find the median value of that half. This determines the third quartile which is 10 in this case. Then considering the left half the median value of this range of values is 5. Hence, we have $Q_3 = 10$ and $Q_1 = 5$. Therefore, our IQR is just 5:

$IQR = Q_3 - Q_1 = 10 - 5 = 5$