# Lesson 15

Quartiles and Interquartile Range

### Lesson Narrative

Previously, students learned about decomposing a data set into two halves and using the halfway point, the median, as a measure of center of the distribution. In this lesson, they learn that they could further decompose a data set—into quarters—and use the quartiles to describe a distribution. They learn that the three quartiles—marking the 25th, 50th, and 75th percentiles—plus the maximum and minimum values of the data set make up a five-number summary.

Students also explore the range and interquartile range (IQR) of a distribution as two ways to measure its spread. Students reason abstractly and quantitatively (MP2) as they find and interpret the IQR as describing the distribution of the middle half of the data. This lesson prepares students to construct box plots in a future lesson.

### Learning Goals

Teacher Facing

• Calculate the range and interquartile range (IQR) of a data set and interpret (orally and in writing) what they tell us about the situation.
• Comprehend that “interquartile range (IQR)” is another measure of variability that describes the span of the middle half of the data.
• Identify and interpret (in writing) the numbers in the five-number summary for a data set, i.e., the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

### Student Facing

Let's look at other measures for describing distributions.

### Student Facing

• I can use IQR to describe the spread of data.
• I know what quartiles and interquartile range (IQR) measure and what they tell us about the data.
• When given a list of data values or a dot plot, I can find the quartiles and interquartile range (IQR) for data.

Building Towards

### Glossary Entries

• interquartile range (IQR)

The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile.

For example, the IQR of this data set is 20 because $$50-30=20$$.

 22 29 30 31 32 43 44 45 50 50 59 Q1 Q2 Q3
• median

The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order.

For the data set 7, 9, 12, 13, 14, the median is 12.

For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. $$6+8=14$$ and $$14 \div 2 = 7$$.

• quartile

Quartiles are the numbers that divide a data set into four sections that each have the same number of values.

For example, in this data set the first quartile is 30. The second quartile is the same thing as the median, which is 43. The third quartile is 50.

 22 29 30 31 32 43 44 45 50 50 59 Q1 Q2 Q3
• range

The range is the distance between the smallest and largest values in a data set. For example, for the data set 3, 5, 6, 8, 11, 12, the range is 9, because $$12-3=9$$.

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