Skill: Construct box-and-whisker plots

A box-and-whisker plot shows the five-number summary of a data set:

• minimum
• lower quartile, $Q_1$
• median
• upper quartile, $Q_3$
• maximum

The box represents the middle $50\%$ of the data. The whiskers extend to the minimum and maximum.

Example

Construct a box-and-whisker plot for this data set:

$$3,\ 5,\ 6,\ 7,\ 8,\ 10,\ 12,\ 13,\ 15,\ 16,\ 20$$

Step 1: Make sure the data are in order

The data are already arranged from least to greatest.

Step 2: Find the median

There are $11$ values, so the median is the middle value:

$$3,\ 5,\ 6,\ 7,\ 8,\ \boxed{10},\ 12,\ 13,\ 15,\ 16,\ 20$$

Thus, the median is $10$.

Step 3: Find the lower quartile

Use the values below the median:

$$3,\ 5,\ 6,\ 7,\ 8$$

The middle value is $6$, so:

$$Q_1=6$$

Step 4: Find the upper quartile

Use the values above the median:

$$12,\ 13,\ 15,\ 16,\ 20$$

The middle value is $15$, so:

$$Q_3=15$$

Do not include the median in either half.

Step 5: Identify the five-number summary

$$\begin{aligned} \text{Minimum} &=3\\ Q_1&=6\\ \text{Median}&=10\\ Q_3&=15\\ \text{Maximum}&=20 \end{aligned}$$

Step 6: Draw the plot

Mark the five values on a number line. Draw:

• a box from $Q_1=6$ to $Q_3=15$,
• a line inside the box at the median, $10$,
• whiskers from the box to the minimum, $3$, and maximum, $20$.

3          6          10          15          20
|----------[-----------|-----------]-----------|
minimum   Q₁        median        Q₃        maximum

The box, from $6$ to $15$, shows the middle half of the data. The box edges are quartiles, not averages.