The Meaning of the Mean
By Michael Naylor

Statistics involves representing data graphically and numerically and drawing conclusions from studying data. Three measures are commonly used to describe data: the mean, the median and the mode. The median is the score that is physically in the middle of the data when the numbers are put in order. The mode is the score that occurs most often.

The mean, commonly called the "average," is the trickiest to understand of the three. The algorithm for finding the mean is simple: take the sum of the data and divide by the total number of figures. Consequently, avoid providing that formula until students have had plenty of experiences exploring data with manipulatives, graphs and numbers.

Hair compare (Grades K-2)
Gather yarn in "hair colors" (black, brown, yellow, red). Have students choose which color matches their hair then measure and cut the yarn to match their hair length. Place all of the pieces of yarn next to each other (be sure to include your own hair in the display). Ask students for their observations. Who has the longest hair? The shortest hair? Are any the same length? Also compare by color: the shortest blond, the longest brown, etc.

Laura Bethel-Sehn

Seasons (Grades K-2)
Give each child a sticky-note and ask the students to draw a picture to show the season in which they were born (i.e. winter snowflake, fall leaf, spring raindrop, summer flower, etc.). Stick the notes on the wall in vertical columns by season, as shown at right. Which season contains the most birthdays? Which season contains the fewest? If the class next door did the same thing, would their columns look the same as yours?

Hair to share (Grades 3-5)
Divide students into groups of four and have each child measure and cut yarn that is the same length as his or her hair. Then have each group cut one long piece that is equal to the total length of their smaller pieces. Ask the students how they could find the average hair length in their group. They'll need to fold the long piece of yarn into four equal sections. Compare average hair lengths between the groups.

Carefully tie the groups' long pieces together (with small knots so as not to diminish their lengths significantly) to make a piece of string as long as the total hair length of the entire class. Use this piece to find the average hair length in the class. You may find it difficult to evenly fold this string into the appropriate number of pieces. Try measuring the string itself, then dividing the measurement by the number of students in the class.

Leveling off (Grades 4-8)
Give students the graph below and ask them to reconstruct it with blocks. Challenge them to determine how many hours per day would be spent practicing if Keisha rearranged her schedule so that she practiced for the same amount of time every day, but kept the same total hours.

Students will find that each row can contain three blocks. This is a "leveling" model for finding the mean. Repeat this activity using other examples, being sure to have the results come out evenly, i.e. with no fractional blocks. Next, present data in a numerical format, such as:

Can students find the mean arithmetically?

Collect data from the class, such as the number of children in each student's family, and have students find the mean. The mean will probably have a fractional part, such as 1.75 children per household. Discuss the result. Does anyone really have 1.75 children?

Stepping out (Grades 4-8)
Have each of your students count the number of normal walking steps it takes them to walk from the door of your classroom to the nearest exit. Divide students into groups to record the data and find their group's average. The average number of steps in each group should be roughly the same.

Then tell them that a giant has joined their group. The giant takes steps that are five times bigger than the students' steps. Have your students calculate their group's mean, including the giant's steps.

The inclusion of the giant's steps is an example of an outlier, a data point that is far removed from the rest of the data. Discuss how the outlier affects the data – the average is not very representative of the group.

Give students the following problem: The owner of Clean-Cut Lawn Mowing pays her eight employees the following dollar amounts per day: 20, 20, 20, 20, 25, 25, 30 and 200. She says her employees' average pay is quite high.

What's wrong with her argument? What might be a more representative way to describe the salaries? (Either eliminate the outlier, or state what most employees make, which is $20 per day.)

Good luck showing your students that the mean is not so tough!

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

April 2004, Vol.34, No.7