How to make a boxplot (five-number summary)

What a five-number summary is

A five-number summary describes a data set with five values listed in order: the minimum, the first quartile, the median, the third quartile, and the maximum. Together they mark where the data begins and ends and how the middle is divided into quarters.

The median is the middle value once the data is sorted. It splits the data into a lower half and an upper half. The first quartile, written $Q_1$, is the median of the lower half, and the third quartile, written $Q_3$, is the median of the upper half. About 25% of the data is at or below $Q_1$, 50% at or below the median, and 75% at or below $Q_3$.

The distance between the quartiles is the interquartile range, or IQR, defined as $\text{IQR} = Q_3 - Q_1$. The IQR measures the spread of the middle 50% of the data, and you will use it to flag outliers. A boxplot is built from the median and quartiles rather than the mean, so it is resistant to a few extreme values.

How to make a boxplot step by step

To make a boxplot, turn the five-number summary into a box with two whiskers. Here is the procedure for a basic boxplot.

1. Sort the data from smallest to largest. Every quartile depends on the data being in order.
2. Find the five-number summary. Locate the minimum, $Q_1$, the median, $Q_3$, and the maximum. For an odd number of values, the median is one data point, so leave it out when you find $Q_1$ and $Q_3$ from the two halves.
3. Draw and scale an axis that covers the minimum through the maximum, with even spacing.
4. Draw the box from $Q_1$ to $Q_3$, and draw a line inside it at the median. The box shows the middle 50% of the data.
5. Draw the whiskers as lines from the box out to the minimum and the maximum.

The box always spans the IQR, and the line inside it is the median, not the mean. Each whisker represents about 25% of the data, and the box holds the middle 50%.

Modified boxplots plot outliers as separate points

A modified boxplot is the version to draw whenever the data might contain outliers, and it is the version the AP course expects. Instead of stretching the whiskers all the way to the minimum and maximum, a modified boxplot plots outliers as separate points and pulls the whiskers back to the most extreme values that are not outliers.

To decide which points are outliers, use the 1.5 times IQR rule. A value is an outlier if it is more than $1.5 \times \text{IQR}$ above $Q_3$ or more than $1.5 \times \text{IQR}$ below $Q_1$. The two fences are:

Any value beyond a fence is plotted as its own point, often marked with an asterisk or a dot. The whisker on that side then stops at the largest or smallest value that is still inside the fence. The 1.5 IQR rule for outliers works through the fences in more detail.

A modified boxplot carries more information than a basic one, because a single far-off value is shown for what it is instead of being hidden at the end of a long whisker.

How to read skew from a boxplot

A boxplot shows the shape of a distribution through the position of the median and the length of the two sides. Compare the two halves of the box and the two whiskers to read the skew.

• Roughly symmetric: the median sits near the center of the box, and the two whiskers are about the same length.
• Skewed right: the median sits closer to $Q_1$ on the left, and the right whisker or a high outlier stretches farther. The longer tail points toward the larger values.
• Skewed left: the median sits closer to $Q_3$ on the right, and the left whisker stretches farther. The longer tail points toward the smaller values.

The mean gives a second clue when you know it. The AP course notes that a right-skewed distribution usually has a mean larger than its median, because the high tail pulls the mean up, while a left-skewed distribution usually has a mean smaller than its median. For a roughly symmetric distribution the mean and median are close together.

What a boxplot cannot show you

A boxplot summarizes a distribution with just five numbers, so it hides some features. Two data sets with the same five-number summary can look very different in a histogram or dotplot.

A boxplot cannot show:

• Gaps and clusters inside the box or whiskers, because the space between quartiles is drawn as a solid box.
• Multiple peaks, since there is no way to mark where values pile up into a bimodal shape.
• The sample size, since the picture looks the same for 20 values or 2,000.

When those details matter, pair the boxplot with a histogram or dotplot. Boxplots earn their keep when you compare several groups side by side, where the shared five-number structure makes centers and spreads easy to line up.

Check your work with a calculator

On the AP exam a graphing calculator with statistical capabilities is expected, and its one-variable statistics output reports the five-number summary directly. Read the labels for minX, Q1, Med, Q3, and maxX.

To get the five numbers for any small data set, enter it in the five-number summary calculator. To watch a boxplot change shape as you drag points and add outliers, open the descriptive statistics sandbox. Boxplots are Unit 1 topic 1.8 in the AP Statistics course.