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:

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:

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 $40$.