## Comparing Averages

3-5
1

Students create a box and whisker plot and compare the mean, median, and mode of a set of data.

To assess prior knowledge, provide a small data set and ask them to find these statistics. You may wish to use the number of vowels in their last names.

Give each student an index card. Ask them to write their first and last names on their cards and then record the number of letters in each name and the total letters. Tell them that they are going to make a human box plot, and that this special graph shows the range and median for a set of data.

Ask students to hold their index cards in front of them. Help students order themselves, starting with the student whose name has the fewest letters. Students who have the same name length should stand side by side.

 Sample Ordered List of Student Names Diana Rigg 9 Susan Sarandon 12 Anthony Hopkins 14 Paul Newman 10 Nicholas Cage 12 Katharine Hepburn 16 Meryl Streep 11 Michael Caine 12 Christopher Reeve 16 Ben Kingsley 11 Jack Nicholson 13 Valerie Bertinelli 17 Maximum: 17 Range: 8 Mode: 12 Minimum: 9 Median: 12 Mean: 12.9

Give the student with the smallest number (for example, 9) a card on which you have written "Minimum." Give the student with the longest name (for example, 17) a card on which you have written "Maximum." Ask students to find the range of the data. To find the range, subtract the minimum from the maximum. Record the range on the board.

### Finding the Mode

The mode, which cannot be determined from a box plot, is the data point that occurs most often. The mean is the arithmetic average. The median is the halfway point or 50th percentile in the ordered data—one half the observations are above it and one half are below it.

To find the mode, have students determine which value occurs more times than all the others. Identify that value as the mode and record it on the board.

### Finding the Median

To find the median, or middle number, have students at each end of the line say "1" at the same time and sit on the floor. Ask students next to them to say "2." Have students count off in this fashion until only one or two students are standing.

If there are an odd number of students, there will be one student; if there is an even number, there will be two students. If there is one student, identify this number as the median. If there are two students, the arithmetic average of their numbers is the median.

Ask a student to write the median on the board under the mode and label both measures of center. Give student(s) who represent the median a card on which you have written "median" and invite them to stand again. Tell students that another name for median is "50th percentile," because 50% of the students have shorter names, and 50% have longer names.

### Creating Box and Whisker Plots

Tell students that they will next find the median, or middle number, on each of the two sides. Have the student with the shortest name and the person in the median position count off as before. Ask the group on the other side of the median to find the middle point of the upper half of the set. Provide a card that says "75th percentile" to the center student on the upper end and a card that says "25th percentile" to the center student on the lower end.

As this terminology may be new to students, you may wish to explain that the 25th percentile is that point greater than 25 percent of the score. To use a money analogy, it is like a quarter. Similarly, the 75th percentile is the point greater than 75 percent of the scores. In the money analogy, it is like 75 cents.

You may wish to line the index cards up on the blackboard tray so the data is visible to all the students. If you have done so, you could indicate with sticky notes the low and high values, the median, the mean, and the mode with labeled index cards used in the human box and whiskers plot.

Tell students that they will become part of a graph called a box plot. Have students holding the 25th percentile and 75th percentile cards to stretch their arms out to make the ends of the box. Put one end of a long piece of yarn in the right hand of the student holding the 75th percentile card. Then, holding the yarn, walk back to the student who holds the 25th percentile card and place yarn in his or her right hand. As you pass in front of that student, ask the student to grab the yarn with the left hand, so that a line of yarn is stretched across the front of his or her body.

Complete the fourth side of the box by carrying the yarn back to the student holding the 75th percentile card. Ask the student to grab the yarn in his or her left hand. To make the "whiskers," stretch a piece of yarn between the student holding the "minimum" card and the student at the 25th percentile. This creates the lower "whisker." Similarly, stretch a piece of yarn between the student holding the "maximum" card and the student at the 75th percentile. This creates the upper "whisker."

Invite students, a few at a time, to step out of the line to see that a yarn "box" with “whiskers” is formed. Explain that they have made a human box and whiskers plot. The maximum and minimum points are the endpoints of the "whiskers" and the 25th and 75th percentile parts are called the lower and upper hinges, respectively, of the box. Copy the plot on the board, then collect the yarn and the cards and ask the students to take their seats.

Call on a volunteer to label the 25th, 50th, and 75th points. Call on another volunteer to draw a line from the left-hand side of the box and label the end of the line with the lowest value in the data set.

Have another student draw a line from the right-hand side of the box and label the end of the line with the highest value in the data set. Ask students to copy the figure from the board, naming the highest and lowest values, the 25th and 75th percentiles, and the median.

Invite students to find the mean by computing with paper and pencil or using their calculators to add all the values and dividing by the number of students in the class. If any value occurs more than once, it should be entered into the sum as many times as it appears.

Tell students that they will next construct a box plot on the computer. Go to the Illuminations Box Plotter Tool, where students should follow the directions for entering their own data and drawing the box plotter.

This site allows you to print out the box plot, so you may wish to complete several box plots while you are here.

• Crayons
• Paper
• Index cards
• Yarn
• Calculators
• Computers with internet access

Assessment Options

1. At this stage of the unit, students should be able tog:
• Construct and read a box and whisker plot
• Identify the mean, median, mode, and range in a set of data
2. Students may raise other questions that will enrich the discussion. Follow their lead, which may result in increased understanding of the box plot and the statistics used to generate it.
3. After the lesson, you may wish to add more comments to the Class Notes. When revisited later in the year, this information may suggest ways to apply this learning.

Extensions

1. To develop their understanding further, ask students how the box and whiskers plot would change if the teacher's name were included in the data set.
2. Suppose a new student came into the class. How would that change the plot we made? (Repeat with other names.)
3. Suppose (student name) moved away. How would that change the plot? (Repeat with other names.)
4. Challenge - Pose the following questions:
• Suppose the median is like a half dollar. What amount is the 25th percentile like? [25 cents; a quarter.]
• What does 25th percentile mean? [25 percent of the class is accounted for when we get to this piece of data.]
• How about the 75th percentile? How were these points shown on the plot? [They form the ends of the box.]
• Using this analogy, how about the lower whisker? [0 %, 0 cents] The upper whisker? [100%, 100 cents or 1 dollar.]
5. Move on to the last lesson, Describing Data.

Questions for Students

1. What graph did we make today?

[Box and whisker plot (or simply box plot).]

2. What length was the most common name length in our class? What measure of central tendency is that?

[Answers may depend upon the student data; The Mode.]

3. What was the difference between these numbers? What do we call that difference?

[Answers will depend upon the student data; The Range.]

4. What were the mean and median of the data set? What does each term tell about the data? How did we find the mean? The median?

[Answers will depend upon the student data; Students should share acceptable methods for finding the mean and medain.]

5. How many students in the class had names longer than the name at the 75th percentile? How many students had names shorter than the length of the name at the 25th percentile?

[Answers will depend upon student data.]

6. How can you locate the range on the box plot?

[Locate the largest and smallest data points, and subtract them.]

Teacher Reflection

• Which students were able to understand the features of the human box and whiskers plot?
• Which students easily found the range and mode? The median? The mean?
• Which students could compare the measures of center with understanding?
• Which students were not yet able to draw a box and whiskers plot? What did they have trouble with? What were they able to do without prompting?
• What data could we collect to extend this instructional experience?
• What will I do differently the next time I teach this lesson?

### First Names First

3-5
Students create bar graphs and find the range and mode of a set of data. They explore the relationship of vowels to consonants in their names.

### Last Names Next

3-5
Students create bar graphs and circle graphs and find the range and mode of two data sets.

3-5

### Describing Data

3-5
Students create glyphs based on the characteristics of their names. They also enter data into a spreadsheet and sort it in several ways.

### Learning Objectives

Students will:

• Create a box and whiskers (or box) plot.
• Find and compare the measures of center (mean, median, and mode) for a set of data.

### NCTM Standards and Expectations

• Represent data using tables and graphs such as line plots, bar graphs, and line graphs.
• Describe the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed.
• Use measures of center, focusing on the median, and understand what each does and does not indicate about the data set.
• Compare different representations of the same data and evaluate how well each representation shows important aspects of the data.