Box Plots — Quartiles, IQR & How to Draw

A box plot, also called a box-and-whisker plot, shows where a data set is centered and how spread out it is. It highlights the median, the middle $50\%$ of the data, and the values near the ends, so you can read the overall shape quickly.

The main landmarks are the first quartile $Q_1$, the median, the third quartile $Q_3$, and the interquartile range $IQR = Q_3 - Q_1$. One condition matters right away: quartiles are not defined by one universal rule. If your class, textbook, or software uses a specific quartile method, keep that method consistent from start to finish.

What A Box Plot Shows At A Glance

The box runs from $Q_1$ to $Q_3$, so it contains the middle $50\%$ of the data. The line inside the box is the median.

The whiskers show how far the data extends beyond the box. In some box plots, they go to the minimum and maximum. In others, they stop at the most extreme values that are not treated as outliers. You need that rule before you decide what the whiskers mean.

How Quartiles And IQR Work

The interquartile range measures the spread of the middle half of the data:

$$IQR = Q_3 - Q_1$$

A larger $IQR$ means the middle half is more spread out. A smaller $IQR$ means it is more tightly grouped.

How To Draw A Box Plot Step By Step

Use the same order each time:

1. Sort the data from least to greatest.
2. Find the median.
3. Find $Q_1$ and $Q_3$ using the quartile convention you are expected to use.
4. Draw a number line and mark $Q_1$, the median, and $Q_3$.
5. Draw the box from $Q_1$ to $Q_3$ and the median line inside it.
6. Add whiskers using the rule your class or software expects.

Worked Example: Finding Quartiles For A Box Plot

Start with the ordered data set

$$2,\ 4,\ 5,\ 7,\ 8,\ 9,\ 12,\ 15,\ 18$$

There are $9$ values, so the median is the fifth value:

$$\text{median} = 8$$

For this example, use the common classroom rule that excludes the overall median when finding the lower and upper halves.

The lower half is

$$2,\ 4,\ 5,\ 7$$

so

$$Q_1 = \frac{4 + 5}{2} = 4.5$$

The upper half is

$$9,\ 12,\ 15,\ 18$$

so

$$Q_3 = \frac{12 + 15}{2} = 13.5$$

Now find the interquartile range:

$$IQR = 13.5 - 4.5 = 9$$

That gives the key markers for the box:

$$Q_1 = 4.5,\quad \text{median} = 8,\quad Q_3 = 13.5$$

If the whiskers go to the minimum and maximum, they extend to $2$ and $18$. So the box stretches from $4.5$ to $13.5$, the median line sits at $8$, and the full plot runs from $2$ to $18$.

How To Read A Box Plot Quickly

Start with the median to locate the center of the data.

Then check the width of the box. A narrow box means the middle half is tightly grouped. A wide box means it is more spread out.

Finally, compare the whiskers and the median's position inside the box. If one side is noticeably longer, the distribution may be more stretched on that side.

Common Mistakes With Box Plots

Do not skip the sorting step. If the data is not in order, the median and quartiles will be wrong.

Do not assume every box plot uses the same quartile rule or the same whisker rule. Two correct plots can look different if they were built with different conventions.

Do not read the edges of the box as the minimum and maximum. They usually mark $Q_1$ and $Q_3$ instead.

Do not assume a wider box means "more data" in that region. It means the values there cover a wider interval on the number line.

When Box Plots Are Useful

Box plots are useful when you want a quick view of center and spread without listing every value. They are especially helpful for comparing two or more groups side by side.

They are common in statistics classes, lab reports, and any setting where the median and the middle half of the data matter more than a detailed list of every value.

Try Your Own Version

Take a short sorted data set, find the five-number summary, and sketch the box plot by hand. Then compare it with a graphing tool to check whether your quartile rule and whisker rule match the result.