# Interquartile Range

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## Objective

SWBAT measure variability within a data distribution by calculating the range and interquartile range of a data set and describe what the range or IQR represent given the context of the data.

#### Big Idea

We use the interquartile range to measure variability when the median is the best measure to describe the data set.

## Think About It

7 minutes

Students work in pairs on the Think About It problem.  First, students decide which measure of central tendency would be the best to use for each of the students, and justify the choices.  They are able to quickly decide that the median makes sense for Mario's data, while the mean makes sense for Yoshi's data, because of the work we did in the previous lesson.

The second question asks students to decide who is the more consistent test-taker.  Students may not be familiar with the word 'consistent,' so it is a good idea to quickly talk through and define the word before starting the lesson.  Making decisions about consistency/inconsistency is an important part of this lesson, so it is imperative that kids can access the vocabulary.

It's okay here if students decide Mario is inconsistent because of the one outlier in the data set; in the next section of the lesson, students will have the chance to see a visual representation of the data that will show Mario is actually consistent with his test scores.

## Intro to New Material

15 minutes

To begin the Intro to New Material section, students work independently to create a box plot with Mario's data, for Example 1.  This is a skill they've mastered in a previous lesson, so they are able to create the visual representation on their own.  After 4 minutes, I display a student's work on the document camera and have students verify that they've used a median of 91, lower quartile of 87, upper quartile of 93, minimum of 65 and maximum of 98.

I ask students what they notice about the box plot.  I take hands until someone names that the whisker from 65 to the box is really long.  We talk about what that means - that the 65 is an outlier, because it is so far away from the other data in the set.  I then ask students where most of the data in the set is displayed.  Students should name that 50% of the data is found in the box part of the plot.  I use white board markers to draw a new, much longer box over the plot and ask what we might be able to tell about Mario's scores if the box were like this.  I want students to discuss that a box with a greater range would mean that Mario's scores were much more varied.

I tell students that we're going to use a numerical way to determine the range of the middle 50% of the data, so that we don't always have to draw box plots.

To begin Example 2, students find the median and quartiles on their own.  I then ask students to turn and talk to their partners about how the can determine the size of the box, without actually drawing it.  If students struggle to articulate that we can subtract the quartiles (because they are the edges of the box, and we can find the distance between them), I'll quickly sketch the pot to help guide them.  I'll let them know that this distance, 6, is the interquartile range.

We talk about the context of this problem and decide that 6 is a pretty big range for miles ridden on a bike.  We conclude that Lance is inconsistent during the month of April.

The key points for this section:

• The interquartile range is a measure of variability of data represented by the distance between the first and third quartiles of a data set.
• The larger the IQR, the more variability or less consistency a data set has.  The smaller the IQR, the less variability or more consistency a data set has.
• IQR is best used when the median is the best measure to describe a data set because the data set contains an outlier.

## Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set.   As they are working, I circulate around the room and check in with each group.  I am looking for:

• Are students ordering their data before finding the quartiles?
• Are students finding the quartiles accurately?
• Are students showing work, to find the IQR?
• Are students writing complete conclusions about the consistency, based on the context of the problem?

I'm asking:

• What is the IQR?
• How did you find it?
• What does the IQR mean, in the context of the problem?
• What's the outlier in this data?
• Why is IQR the best measure to determine variability in this data?

After 10 minutes of of partner work time, students complete the Check for Understanding problem independently.  This CFU sample gives a sense of what students' written responses might look like.

## Independent Practice

15 minutes

Students work on the Independent Practice problem set.

One thing I am careful to look for is students who might struggle with finding a quartile that is not a number on the list of data points (ie - the quartile is between two data points).  I keep a list of the quartiles on my packet as I circulate, so that I can quickly compare student responses with my answer key.

I also take the time to read a written response from every student, and give in-the-moment feedback on their words.  I want my students to be clear communicators, and expect them to write thoughtful answers.  If a student gives a response along the lines of 'The IQR is 4 so he's consistent,' I insist that the student re-does the work.

## Closing and Exit Ticket

8 minutes

After Independent Practice time, I have students share their responses to Problem 4 with their partners.  Each student needs to name one way to make the response stronger.  I then ask for people to let us know if his/her partner had an exemplary response.  We hear 1-2 outloud, and offer both positive and critical feedback.

Students then complete the Exit Ticket to close the lesson This exit ticket sample shows what student responses might look like.