# Quartiles

The *quartiles* of an ordered data set are the $$3$$ points that split the data set into $$4$$ equal groups. The $$3$$ quartiles are defined as follows:\ <br>

1. $$Q\_1$$: The first quartile is the middle number between the smallest number in a data set and its median.
2. $$Q\_2$$: The second quartile is the median ($$50^{th}$$ percentile) of the data set.
3. $$Q\_3$$: The third quartile is the middle number between a data set's median and its largest number.

#### Computing the First and Third Quartile

We will use the [first method described in the Wikipedia](https://en.wikipedia.org/wiki/Quartile#Method_1):

We will split the data into two halves, *lower half* and *upper half*:

* If there are an odd number of data points in the original ordered data set, do not include the median (the central value in the ordered list) in either half.
* If there are an even number of data points in the original ordered data set, split this data set exactly in half.

The value of the first quartile ($$Q\_1$$) is the median of the lower half and the value of the third quartile ($$Q\_3$$) is the median of the upper half.

#### Example 1

We will consider the following ordered dataset for this example:

> 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49

The median of the dataset is $$40$$. As there are an odd number of data points, we do not include the median (the central value in the ordered list) in either half:

> Lower half: 6, 7, 15, 36, 39
>
> Upper half: 41, 42, 43, 47, 49

The median of the lower half is $$15$$, so the value of the first quartile is $$15$$, and the median of the upper half is $$43$$, so the value of the third quartile is $$43$$.

#### Example 2

We will consider the following ordered dataset for this example:

> 7, 15, 36, 39, 40, 41

As there are an even number of data points in the original ordered data set, we will split this data set exactly in half:

> Lower half: 7, 15, 36
>
> Upper half: 39, 40, 41

The median of the lower half is $$15$$, so the value of the first quartile is $$15$$, and the median of the upper half is $$40$$, so the value of the third quartile is $$40$$.