Box Plots and Interquartile Range
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.
Then, students use the five-number summary to construct a new type of data display: a box plot. Similar to their first encounter with the median, students are introduced to the structure of a box plot through a kinesthetic activity. Using the class data set that contains the numbers of letters in their names (from an earlier lesson), they first identify the numbers that make up the five-number summary. Then, they use their numbers to position themselves on a number line on the ground, and are guided through how a box plot would be constructed with them as the data points.
- Comprehend that “interquartile range (IQR)” is another measure of variability that describes the span of the middle half of the data.
- Create a box plot to represent a data set.
- Describe (orally) the parts of a box plot that correspond with each number in the five-number summary, the range, and the IQR of a data set.
For the Human Box Plot activity:
- Each student will need the index card that shows their name and the number of letters in their name (used for the Finding the Middle activity), as well as a class data set.
- Compile the numbers on the cards into a single list or table. Prepare one copy of the data set for each student.
- Have some extra index cards available for students who might have been absent in that earlier lesson.
- Prepare five index cards that are labeled with "minimum," "maximum," "Q1," "Q2," and "Q3."
- Make a number line on the ground using thin masking tape (0.5 inch). It should show whole number intervals and span at least from the lowest data value to the highest. The intervals should be at least a student's shoulder's width.
- Prepare a roll of wide masking tape (2- or 3-inch wide) to create a box and two whiskers on the ground.
- I can use IQR to describe the spread of data.
- I know what information a box plot shows and how it is constructed.
A box plot is a way to represent data on a number line. The data is divided into four sections. The sides of the box represent the first and third quartiles. A line inside the box represents the median. Lines outside the box connect to the minimum and maximum values.
For example, this box plot shows a data set with a minimum of 2 and a maximum of 15. The median is 6, the first quartile is 5, and the third quartile is 10.
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
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
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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