Unfortunately, there is no strict rule for computing quartiles. Different methods give different results. One of the most commonly used methods is the following:

Let's calculate the quartiles for the following set:

We already know how to calculate the median (the middle quartile); it's 7. Below this point, we have half the data \{1, 2, 3, 4, 5, 6\}. The median of this subset (and therefore the lower quartile of the entire set) is 3.5. The median of the upper subset (and thus the upper quartile of the original set) is 10.5.