The upper quartile, also referred to as the third quartile (denoted as $Q_3$), is determined by calculating the median of the upper half of a data set.

To compute the upper quartile, begin by organizing your data set in ascending order. Once the data is sorted, identify the median, which is the central value of the data set. If the total number of data points, denoted as $n$, is odd, the median is simply the middle number. Conversely, if $n$ is even, the median is the average of the two central numbers.

Following the identification of the median, divide the data set into two halves. The upper quartile is then derived as the median of the upper half of the data. If the number of data points in this upper half is odd, the upper quartile is the middle number of that half. If it is even, the upper quartile is calculated as the average of the two middle numbers within the upper half.

To illustrate this process, consider the following data set: $3, 7, 8, 12, 13, 14, 18, 21, 23, 27$. First, we find the median. In this case, the median is computed as the average of the 5th and 6th values, which is:

Next, we identify the lower half of the data set, which consists of $3, 7, 8, 12, 13$, and the upper half, which includes $14, 18, 21, 23, 27$. Now, we need to find the median of the upper half:

The upper half is $14, 18, 21, 23, 27$. The median of this upper half is $21$, thus the upper quartile ($Q_3$) is:

This systematic approach ensures that you accurately determine the upper quartile, providing valuable insights into the spread and distribution of your data.