Using Box Plots to Visualize Variation

Lesson Plan - Get It!

Audio:

• What does this picture look like to you?

What Is a Box Plot?

The picture above may look like pistons, but it’s actually a visual tool called a box plot, or more formally, a box-and-whisker plot.

A box plot helps you make sense of a data set by showing its center, spread, and outliers all at once—no long list of numbers required!

Take a closer look at how box plots work.

The Parts of a Box Plot

Imagine you’re looking at the populations of all 50 U.S. states and the District of Columbia. That’s a lot of data—but a box plot can help you quickly see the big picture.

Here's how to read each part of a box plot.

Interquartile Range (IQR): The shaded box shows the middle 50% of the data—from the first quartile (Q1) to the third quartile (Q3).

Median (Q2): A vertical line inside the box marks the middle of the data—half the values are lower, half are higher.

Whiskers: These lines extend from the edges of the box to the smallest and largest values that are not outliers.

Outliers: Any unusually high or low values are shown as dots outside the whiskers.

Tip: An outlier is usually a value more than 1.5 times the IQR above Q3 or below Q1.

How to Read a Box Plot

Now that you know the parts, learn what you can do with a box plot.

1. Compare Distributions: Which group has a higher median or more variation?

2. Spot the Center and Spread: The median shows the center. The IQR and whiskers show the spread.

3. Estimate Key Stats: Use the graph to estimate range, median, Q1, and Q3.

Box plots help you quickly see how spread out a data set is—when the box and whiskers are long, the data has more variation; when they're short, the data is more consistent.

Understanding the Five-Number Summary

Before you can draw a box plot, you need to find the five-number summary of your data set. This summary gives you five key values.

1. Minimum: the smallest number in the data.

2. Q1 (First Quartile): the value that marks the lower 25% of the data.

3. Median (Q2): the middle value of the data.

4. Q3 (Third Quartile): the value that marks the upper 75%.

5. Maximum: the largest number in the data.

These values divide your data into four equal parts. Once you have them, you're ready to sketch a box plot.

Tip: Put your data in order first—it makes finding quartiles much easier!

Look at a Real Example

Here’s a double box plot showing the number of wins for teams in the National League and American League during the 2005 MLB season. Let’s break it down.

Compare Distributions

By comparing the lengths of the boxes and whiskers, you can see the following.

The National League has a more balanced distribution—its quartiles and whiskers are fairly even in length, suggesting that team wins are consistently spread across the league.

The American League has a narrower left quartile (Q1 to Median) and a wider right quartile (Median to Q3), meaning more teams were clustered in the higher win range. This suggests the distribution is slightly skewed left, with more variation in the upper half of the data.

You should also notice this.

While both leagues cover the same overall range of wins (60 to 100), the National League’s wins are more evenly spread, while the American League shows a tighter clustering in the upper-middle quartiles.

This tells you that more American League teams had similar (and higher) win totals, while National League teams were more spread out across the range.

Spot the Center and Spread

Look at the line inside each box—that’s the median.

The National League’s median appears around 81 wins, right in the center of its box.

The American League’s median is higher—around 88 wins—and closer to the right edge of the box, suggesting more teams won more games.

The interquartile range (IQR)—the length of the box—shows where the middle 50% of data lies.

Both leagues have the same IQR, meaning the middle half of teams in each league had a similar range of win totals.

However, the American League's box is shifted slightly toward higher values, showing that more of its teams performed in the upper half of the win range.

Estimate Key Stats

From the graph, you can estimate the following.

Minimum wins: About 60 in both leagues.

Maximum wins: About 100 in both leagues.

Range (Max - Min):

National League: 100 - 60 = 40

American League: 100 - 60 = 40

Medians

National League: approximately 80

American League: approximately 85

You can also identify outliers by looking for any dots beyond the whiskers. In this case, there don’t appear to be any, which means most teams stayed within the expected range.

Give It a Try!

Try matching number summaries and box plots yourself.

First, look at each box plot shown below. Then, drag and drop the correct number summary into the matching data box.

Make sure you are on the correct box plot image for each one!

When ready, move on to the Got It? section for more practice.