The quartiles divide the ordered set of data into 4 equal groups. The median is the second quartile, and the first quartile is the median of the first half of the data while the third quartile is the median of the second half of the data.

A quartile will be part of the distribution if one or more of the data points is equal to the quartile. For example:

Suppose the distribution is 1 2 3 4 5 6 7 8 9 10 11 12

The quartiles are 3.5,6.5,9.5 none of which are part of the distribution.

Suppose the distribution is 1 1 2 2 3 3 5 5 6 6 7 7

The first quartile is 2 which is in the distribution.

The second quartile is 4 which is not in the distribution.

The third quartile is 6 which is in the distribution.

So you tell if a quartile is in the distribution by computing the quartile and seeing if it is in the distribution.