Quartiles

The figure is a simple number line drawn horizontally. A single continuous line represents the full range of the data, from the smallest value on the left to the largest value on the right. The line is marked with three vertical tick marks that divide it into four distinct segments. These three tick marks are labelled, from left to right, as $Q_1$, $Q_2$, and $Q_3$. The segment from the left end to $Q_1$ is the first quarter of the data, from $Q_1$ to $Q_2$ is the second quarter, from $Q_2$ to $Q_3$ is the third quarter, and from $Q_3$ to the right end is the fourth quarter. The middle tick mark, $Q_2$, is also labelled as the median.

The physical idea is straightforward: just as the median splits the ordered data into two equal halves, quartiles split it into four equal parts. Each of the four intervals contains exactly one-quarter (25%) of the observations. The three dividing points are the first quartile ($Q_1$), the second quartile ($Q_2$, which is the median), and the third quartile ($Q_3$). The figure makes it visually clear that the median is not a separate concept but is actually the middle quartile.

The textbook uses this figure to introduce the formulas for locating quartiles in a discrete data set. For ungrouped data arranged in ascending order, the position of the $k$-th quartile ($k = 1, 2, 3$) is given by:

where $N$ is the total number of observations. For example, the first quartile ($k=1$) is at position $\frac{N+1}{4}$, the median ($k=2$) is at position $\frac{2(N+1)}{4} = \frac{N+1}{2}$, and the third quartile ($k=3$) is at position $\frac{3(N+1)}{4}$.