Box Plots — Quartiles, IQR & How to Draw

A box plot, also called a box-and-whisker plot, turns a list of numbers into a compact picture of center and spread. It marks the median, the middle $50\%$ of the data, and the values near the ends, so the overall shape reads at a glance.

Quartiles, IQR, And The Whisker Rule

The main landmarks are the first quartile $Q_1$, the median, the third quartile $Q_3$, and the interquartile range:

A larger $IQR$ means the middle half is more spread out; a smaller one means it is tightly grouped. The box runs from $Q_1$ to $Q_3$, so it contains the middle $50\%$ of the data, with the median drawn inside.

One condition matters right away: quartiles are not defined by a single universal rule. If your class, textbook, or software uses a specific quartile method, keep it consistent from start to finish. The whiskers also depend on a rule. In some plots they reach the minimum and maximum; in others they stop at the most extreme values not treated as outliers.

Why Different Methods Give Different Quartiles

Because there is no universal definition of a quartile, different conventions disagree about whether to include the overall median when splitting the data into halves. That is not an error in either method; it is a choice. The practical consequence is that two correct box plots of the same data can look slightly different, so you must know which rule produced the one in front of you before you interpret it.

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

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

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

The lower half is

so

The upper half is

so

Now find the interquartile range:

That gives the key markers for the box:

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 $18$.

Practice And Quick Reading

Try the set $1, 3, 3, 6, 7, 9, 10, 12$ on your own. With eight values the median is $(6+7)/2 = 6.5$; the lower half $1, 3, 3, 6$ gives $Q_1 = 3$, and the upper half $7, 9, 10, 12$ gives $Q_3 = 9.5$. Check that your box runs $3$ to $9.5$ with the median at $6.5$.

To read a finished plot, start with the median to locate the center, then check the box width: narrow means the middle half is tightly grouped, wide means more spread. Finally compare the whiskers and the median's position. If one side is noticeably longer, the distribution is likely stretched on that side.

Calculation Pitfalls To Watch

• Do not skip the sorting step. Out-of-order data gives a wrong median and wrong quartiles.
• Do not assume every plot uses the same quartile rule or whisker rule. Two correct plots can differ when built with different conventions.
• Do not read the edges of the box as the minimum and maximum. They usually mark $Q_1$ and $Q_3$.
• Do not assume a wider box means more data there. It means the values cover a wider interval on the number line.

Box plots are most useful when you want a quick view of center and spread without listing every value, especially when comparing two or more groups side by side. They are common in statistics classes, lab reports, and any setting where the median and middle half matter more than a detailed list.