Box Plots: Interpreting Spread

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Students learn to use box plots to describe the spread of a quantitative column, and then deepen their perspective on shape by matching box plots to histograms.

Lesson Goals

Students will be able to…

Describe box plot shape including skew, symmetry and potential outliers.
Interpret a box plot to answer questions about a dataset and its spread.
Compare and contrast information displayed in a box plot and a histogram, identifying that box plots have variable intervals and histograms have fixed bins.
Recognize the shape of various distributions, and identify that shape in box plots and histograms.

Student-facing Lesson Goals

Let’s describe the spread of different box plots.
Let’s compare and contrast box plots and histograms.

Materials

Supplemental Materials

Preparation

For the Box Plots Card Sort activity in this lesson you will need to print and cut one set of cards for each pair of students in your class. We recommend storing the cards in envelopes.
There is an activity during this lesson that requires tape, markers, and flip chart posters created during an earlier lesson, Histograms: Visualizing "Shape". Before class, tape the partially completed flip chart posters to the walls of your classroom—one poster per team of three students.

🔗Describing Spread

Overview

Students develop and practice using the vocabulary needed to describe the spread of box plots.

Launch

The Box Plots Card Sort activity students are about to complete requires some teacher preparation. Make sure you’ve printed and cut out a set of cards for each pair of students before proceeding.

If that preparation is unrealistic for you, project the images for students to refer to as they work through this section and modify the directions accordingly. (Viewing all of the images at once, rather than as individual cards, requires a higher cognitive load for students, so we don’t recommend it.)

Your teacher will give you and your partner an envelope containing lettered box plot cards. Lay out the cards in front of you.

What do you Notice about the box plots?
What do you Wonder about the box plots?

Investigate

Let’s sort some box plots!

With your partner, sort the cards into 2-3 logical piles of your choice.
- Not sure how to group the cards?
- We recommend thinking about the different features represented on the cards, including: range, IQR, median, and whisker length.
Record your groupings in section 1 of Sorting Box Plots.
Respond to questions 4 and 5. Be prepared to share your groupings and reflections with the class.

As you circulate, make note of the different groupings that students use; this will help with facilitation of class discussion and debrief. Encourage students to use appropriate vocabulary (i.e., IQR, range, median, quartile, whisker, etc).

Students should be familiar with the concept of "symmetry", given that they sorted histograms into symmetrical and asymmetrical groupings during Histograms: Visualizing "Shape". They may even be able to correctly use the terms "skew left" and "skew right", depending on their depth of understanding of those concepts.

What are the groupings you and your partner developed for this first round? Let’s hear about two different options.
symmetrical / skew left / skew right
IQR is evenly split (with the median at the center) / IQR is asymmetrically split
there are two whiskers / there is a different amount of whiskers
the IQR occupies all of the range / most of the range / just a small portion of the range
the median is greater than 50 / the median is less than 50
Note: Other groupings are certainly possible! The list above is just a sampling.

Do not invite every group to share; there are two more rounds of card sorting, and we want students to develop some of their own ideas.

You probably noticed that some box plots have long whiskers to the left, and while others have long whiskers to the right.

Box plots, like histograms, can be characterized as "skew". Let’s compare skew-left, skew-right, and symmetric box plot shapes.

Symmetric	Skew-left	Skew-right

Values are balanced on either side of the center.	Values are clumped around what’s typical, but trail off to the left.	Values are clumped around what’s typical, but trail off to the right.

Symmetric

Skew-left

Skew-right

A box plot

Values are balanced on either side of the center.

Values are clumped around what’s typical, but trail off to the left.

Values are clumped around what’s typical, but trail off to the right.

As you debrief with students, encourage them to make sense of their groupings. In other words, rather than simply noticing that sometimes the whiskers are the same length, invite students to articulate what that can tell them about the distribution of the data.

Now let’s practice matching box plots to descriptions of the data’s distribution.
Complete Reading Box Plots.

Synthesize

Envision a skew-left box plot. If there are outliers, where would they be? Explain how you know.
Most of the data in a skew-left box plot is clustered at the high end of the range, with points trailing off to the left. Possible outliers would fall at the low end of the range.
The median is the middle value in a list of data points. Knowing this: How come the median is not always positioned in the middle of the IQR?
If the median is not in the middle, we know that quarters 2 and 3 do not span the same distance on the number line - but they do still contain the same amount of data! That data is simply more densely packed in one quarter compared to the other.