An Average Activity

Est. Class Sessions: 2

Developing the Lesson

Part 1: Exploring Medians Using Data About Us

Describe Averages. Begin by exploring medians using data about the students in your class as described below. Then use the discussion about Room 204 on the An Average Activity pages in the Student Guide to review terms and formalize procedures. Begin the lesson by discussing students' current understanding of the term average.

Ask:

When have you heard the term average?
What do you think it means?

Generate a list of responses, which may include batting average, average rainfall, grade point average, average student, and average height. These averages indicate what is representative or typical in a given situation. Advise students that they will learn how to use one number to describe what is typical in a set of numbers.

Knowing what is typical helps us to predict what will happen with new data. For example, when a student says that she averages about two goals per game in soccer, she does not mean that she scores two goals every game. She means that the typical number of goals she scores is two, but that sometimes she scores fewer and sometimes more. Someone may reasonably predict that in the next game the student might score about two goals.

Remind students of the term mode that was defined in Lesson 2. The mode is one kind of average and is sometimes used as the typical number. Review the definition of the mode as the number that appears most frequently in a data set.

Tell students that in this activity, they will learn about another kind of average, the median. They will find the median of a data set for several variables. Define the median as the value exactly in the middle of the data.

Average. In everyday language we use the term “average” to describe what is normal or typical. In mathematics, the average is a single value that is used to represent a set of numbers. For example, the average grade for a student is one number that is used to represent all of his or her grades. In fourth grade, students will revisit mode and find two additional types of averages: the mean and the median.

The arithmetic mean is the most commonly used average. In fact, the term average is frequently used interchangeably with mean. However, in Math Trailblazers we use average to describe measures of central tendency. The mean is one measure of central tendency. For example, if a student spells 18, 19, 10, 14, and 19 words correctly on a series of five spelling tests, the student's mean number correct is (18 + 19 + 10 + 14 + 19) ÷ 5 which equals 16. Students will learn to find means in Unit 5.

In this unit, students find the median values of sets of data. The median is the number exactly in the middle of the scores. To find the median of an odd number of scores, arrange the scores from smallest to largest and choose the middle number. The median score for the above data (10, 14, 18, 19, 19) is 18 words, since there are two values smaller than 18 and two values larger than 18. If a student takes an even number of tests, the median is not as obvious since there is not one middle piece of data. If a student earns scores of 10, 14, 18, and 19 on four tests, his or her median score is 16. In this case, we look at the two middle pieces of data (14 and 18). The median is the number halfway between these two numbers. If a student earns scores of 10, 13, 18, and 19 on the four tests, the median score is 15.5 since it is midway between the 13 and 18.

The third measure of central tendency, the mode, is briefly discussed in this and previous lessons. However, median is the only average that is assessed in this unit.

Find Medians of Data Sets. Review the data your class collected and graphed in Lesson 2 along with the list of numerical variables your class generated at the beginning of that lesson. Choose two or three numerical variables and tell students that they are going to find the median and use it as a typical value for each variable. Choose numerical variables with values that students can easily report and that have a relatively wide range of values. Height is a good variable for this activity, since it is easy to recognize the median height of a group of students when they are standing in a line in order from shortest to tallest. In this discussion we will use the following variables as examples: height, number of pets, and number of pencils in your desk.

If height is a sensitive issue for any of the students in your class, you can choose another variable. For example, students can compare the length of their hands or the length of their arm spans instead of their heights.

Ask students to write as large as possible the number of pets they have on one side of a sheet of paper and the number of pencils in their desks on the other side. (Make sure students label the numbers so they know which is which.) Define the variables precisely. For example, based on the decision of your class, the number of pencils in a desk may not include pens or markers.

Ultimately, you will find medians for the whole class, but first demonstrate the procedure using five students lined up in front of the room. Choose five students with varying heights and ask them to arrange themselves in order from shortest to tallest. The third student in line—the student in the middle—is the student with the median height. Note that this is one example where we have to measure only the middle data point.

Next, find the median value for the other variable, number of pencils. Have the five students show the number of pencils in their desk by holding up the data they wrote on sheets of paper. The students rearrange themselves in order with the student with the smallest number of pencils at one end of the line and the student with the largest number of pencils at the other end of the line. The student in the middle of the line is holding the median number of pencils. Note that more than one student may have the median number of pencils. If the students in the line have 1, 1, 1, 2, and 2 pencils, the median is 1 pencil. (The mode is also 1 pencil.) Repeat the procedure to find the median number of pets at home.

Median of an Even Number of Values. Demonstrate how to find the median of an even number of values. Ask six students to show their data for one variable, such as the number of pencils in their desks. Here are three possible data sets and the corresponding medians.

Data Set A: 0,1,3,5,6,10

The median is 4 pencils. Since there is not one middle data point, look at the numbers the two students in the middle of the line are holding (3 and 5). The median (4 pencils) is the number halfway between these two numbers. Note that in this example, there is no mode because no number appears more than once.

Data Set B: 0,1,2,2,5,6

Both the median and the mode are 2 pencils, since the two middle data points are both 2 pencils and no other number appears more than once.

Data Set C: 0, 1, 3, 4, 6, 10

The median is 3 1/2 pencils, since 3 1/2 is midway between the two middle data points (3 and 4). Again, there is no mode.

Medians of Larger Sets of Data. Once the students understand the process, ask the entire class to stand with their data. First, students find the median height of the entire class by comparing heights with one another, arranging themselves in order from smallest to largest, and identifying the student or students with the median height. Then, students find the medians for the other two variables in a similar fashion using the numbers they have written down.

Demonstrate or have a student demonstrate the process of counting off equal numbers from the two ends until the middle is reached.

Student Guide: 13 14 15 16

Answers