Boxplots are helpful for visualizing the distribution of a quantitative variable. A boxplot clearly shows the median of the data and provides a summary at a glance of the bulk of the data and the presence of outliers. In the next activity, you will need to be able to identity and interpret the features of a boxplot, identify outliers in a dataset, and relate a boxplot of a quantitative variable to its distribution. In this section, you’ll learn to identify the key pieces of information needed to accomplish these tasks.

Features of a Boxplot

In order to interpret boxplots, you will need to identify the minimum, maximum, and median of a quantitative variable. You’ve done this in previous activities. If you need a refresher, take a look at the video below. A boxplot captures only the median of the dataset, not the mean, as a measure of center.

Five-Number Summary

You will also need to know the following definitions:

• the first quartile of a quantitative variable (sometimes denoted Q1) is the value below which one quarter of the data lies, and the first quartile is also equal to the 25th percentile;
• the third quartile of a quantitative variable (sometimes denoted Q3) is the value below which three quarters of the data lay, and the third quartile is also equal to the 75th percentile; and
• the interquartile range (sometimes denoted IQR) of a quantitative variable is the quantity Q3–Q1.

The collection of the minimum, first quartile, median, third quartile, and maximum form the five-number summary of the variable.

Identifying the Features of a Boxplot

As we explore the features of boxplots, we will work with part of a dataset that reports information about whether drivers involved in a fatal crash were impaired by alcohol.^[1] The dataset contains 51 entries corresponding to all 50 states, as well as Washington, DC.

The following table gives the five-number summary for the percentage of drivers involved in fatal collisions who were alcohol-impaired in all 50 states and Washington, DC.

One of the ways to visualize the data using the five-number summary is by creating a boxplot. For questions 1 – 4, refer to the following boxplot, which depicts data about the percentage of drivers involved in fatal collisions who were alcohol-impaired in all 50 states and Washington, DC. The boxplot is superimposed with the letters A – G labeling different features of the plot.

For questions 2 -4, complete each sentence using information from the boxplot above.

Interquartile Range and Outliers

Now, let’s define the idea of an outlier more precisely. Previously, we’ve seen that an outlier is a value that is unusual, given the other values in a dataset. But what does “unusual” mean? To be more precise, for data with only one variable, let’s define the define the following:

• upper outlier as an observation that is greater than Q3 + 1.5 × (IQR); and
• lower outlier as an observation that is less than Q1 – 1.5 × (IQR).

Use these definitions with the boxplot above question 1 to complete the sentences in questions 5 and 6.

It’s important to note that there are several good methods to use for determining an observation to be an outlier in the distribution. The IQR method commonly uses a distance 1.5 times IQR from Q1 or Q3, but certain applications use larger distances. The IQR method does work for skewed distributions, though. In the next section, you’ll learn about another method that doesn’t involve the IQR, and which works well for symmetrical distributions. Depending upon the application, it may be desirable to set the distance 2 or even 3 times IQR, but 1.5 times is commonly used and works well for our application, so we use it here.

Interpreting the Features of a Boxplot

The following table lists each state in the dataset, along with the corresponding percentages of drivers involved in fatal crashes who were impaired by alcohol, in order from lowest percentage to highest percentage. Use this table and the definition of outlier to answer questions 8 -9.

Now, let’s use technology to explore the dataset.

Identifying the Shape of a Distribution from a Boxplot

question 12

Just as histograms and dotplots can tell us about the distribution of a quantitative variable, so can a boxplot. For each boxplot below, choose the description that matches the shape of the data’s distribution. (Note that boxplots can be oriented vertically, as we saw previously, or horizontally, as we see below.)

 

Boxplot	Distribution
	a) left skewed b) symmetric c) right skewed include dropdown options similar to Question 10 in WTK 4C
	a) left skewed b) symmetric c) right skewed include dropdown options similar to Question 10 in WTK 4C
	a) left skewed b) symmetric c) right skewed include dropdown options similar to Question 10 in WTK 4C

Hint

What effect do outliers have on the skew of data?

Summary

In this section, you’ve learned about boxplots: how to calculate the five-number summary, how to read these numbers from a boxplot, and how to identify outliers in a dataset using the interquartile range. Let’s summarize where these skills showed up in the material.

• In Questions 1 and 7, you identified the features of a boxplot.
• In Questions 2 – 4, you interpreted the features of a boxplot.
• In Questions 5, 6, 8, and 9, you identified outliers in a dataset.
• In Questions 10 – 12, you related the boxplot of a quantitative variable to its distribution.

Being able to calculate and identify features of a boxplot and relate the boxplot and distribution of a quantitative variable are necessary statistical skills and will be used in the next activity. If you feel comfortable with these skills, please move on!