Remind students of the different ways they have to express data in a single variable: bar graphs, circle graphs, line graphs, etc. (You may ask them to refer to the Representations handout from Lesson 2.) Explain that a bar graph can be used to represent data with non-numerical categories. For instance, to show how many people have dogs, cats, fish, and so on, you could create a bar graph where each bar represents a different kind of pet.
Activity 1: Creating a Dot Plot
Draw a number line on the board. Tell students that to illustrate a dot plot, they’ll help you construct a dot plot from data about them. For a data set to be represented by a dot plot, the data has to be numerical. For instance, the dot plot might represent how many wins a league of baseball teams has, or the heights of plants, or temperatures. Tell students that the first dot plot they will create is going to deal with how many brothers and sisters each of them has.
Below the number line, write “Brothers and Sisters.”
Say, “Now we are going to create a dot plot. We know that the data is the number of brothers and sisters we have. What would be a reasonable number to put at the left-most edge of the number line? In other words, what is the smallest answer someone could give to the question ‘How many brothers and sisters do you have?’”
Guide student responses to using zero as the smallest number and label the left-most vertical mark on the number line with a zero.
Say, “Now think about the largest reasonable answer someone could give when asked how many brothers and sisters s/he has.”
Work with the class to determine what a reasonable answer might be for the class. For example, if someone suggests ten, ask if anyone in the class has ten or more siblings. Once you determine the largest reasonable answer, put a vertical mark at the right end of the number line and label it, and then fill in and label the vertical marks between zero and the largest reasonable answer.
Ask students if the numbers on the number line would be the same for a different class. Discuss how the specific numbers on the number line depend not only on the question but on the population being asked.
Instruct students to come up to the board and place an X (representing the number of brothers and sisters the student has) over the appropriate spot on the dot plot.
When the plot is finished, point out to students the similarities between dot plots and bar graphs (the “height” above each value along the x-axis represents how many times that value appears in the data set) and differences (that a dot plot wouldn’t work to show how many people have different pets, whereas a bar graph would). Remind students that their number line is really just an x-axis and should therefore have consistent spacing between values.
Have students brainstorm a list of potential questions for which the resulting data could be expressed using a dot plot. Students should write down the suggestions.
Activity 2: Describing the Distribution
Say. “Imagine you are describing the data in the ‘Brothers and Sisters’ dot plot to people who haven’t seen it before. What would you say to them? How would you describe it?”
Guide students toward discussing how the data is “clumped” around particular values (probably 2 and 3) and how spread out (or not spread out) it is. Point out to students that telling someone how spread out data is can give a quick idea of the general “shape” of the data set. In mathematics, we like to quantify things – rather than just say something is “kind of spread out” or “clumped together,” we like to give these kinds of general statements specific numbers. In this case, we’re going to talk about the mean absolute deviation. (Write the term “mean absolute deviation.”)
Underline the word “mean.”
Ask students, “What does this word mean in mathematics? Can you explain it?” (The mathematical mean is the average value of the set of data.)
Accompany this with a quick review of how to calculate the mean.
Underline the word “absolute.”
Ask students, “What does this word mean in mathematics? How would you explain it?” (Absolute is used in reference to absolute value; absolute value represents the distance a number is from 0 on a number line.)
Underline the word “deviation.”
Ask students, “What does this word mean in mathematics? How would you explain it?” (Students may not have a mathematical context for what deviation means, so guide the discussion towards the fact that deviation deals with differences.)
Draw a circle around the words “absolute deviation” and tell students to think of those words as a group—the positive difference between two values. “When we talk about a difference, we need to talk about the difference between two values. In each case, one of those values is going to be the mean of the data set.”
Ask students to work in pairs to determine the mean of the data set represented by the “Brothers and Sisters” dot plot. Write the calculated mean on the board.
Now, return to the phrase “absolute deviation” and circle the first X on the dot plot, probably representing 0. Ask what the “absolute deviation” is between the value represented by the circled X and the mean. Guide students toward the idea that the difference is just the absolute value of the difference between the mean and the value. Write down this difference, and then move to the next X on the dot plot (which may be in the same column or in the next column). Write down the absolute deviation for this value. Point out that the absolute deviation will be the same for each value in the same column.
Find several absolute deviations so that students have the general idea. Working in pairs, have students write down all the absolute deviations for all the data.
Return to the phrase “mean absolute deviation” and put a check mark above “absolute deviation,” since all the absolute deviations have been found.
Say, “Now that we have found all of the absolute deviations, we are going to find the mean absolute deviation of the data. What do you think the mean absolute deviation will tell us? How do you think we find the mean absolute deviation?” (Students should guess they need to find the mean of those values.)
Have students continue working in pairs to find the mean absolute deviation of the data set. Write this value on the board and tell students it is a measure of how “spread out” the data is.
Pose the following questions:
1) “What would happen to the mean absolute deviation if a new student entered the class and that student had ten brothers and sisters?” (It would increase.)
2) “What would happen to the mean absolute deviation if every student in the class had the same number of brothers and sisters?” (It would be zero.)
The second question should lead to a discussion of the mean being identical to the number of brothers and sisters and the mean absolute deviation being zero, since the absolute deviations are all 0.
3) “What would happen to the mean absolute deviation if every student in the class had two siblings except one had three siblings?” (It would be very close to zero, just a tad higher.)
4) “What if every student in the class had two siblings except one had one sibling? How does that compare with the case in which one had three siblings?” (The mean absolute deviation would be very close to zero, just a tad lower.)
5) “What if every student in the class had one additional sibling?” (The mean absolute deviation would stay the same.)
Ask students to come up with questions that would lead to data sets with a mean absolute deviation greater than that of the Brothers and Sisters data set. Repeat with brainstorming questions that would lead to data sets with a mean absolute deviation less than that of the Brothers and Sisters data set.
Activity 3: Making Your Own Data Set
Have students work in small groups (4 to 5 groups in total) to come up with a question to ask of their classmates that would yield results appropriate to display in a dot plot. Have students survey their classmates and construct their own dot plots (one per group).
After each group has constructed their own dot plot, write the four or five survey questions on the board and ask each group to rank the results of the questions based on what they think the mean absolute deviation will be for the results.
Explore the predictions and the reasoning behind them before having each group calculate the mean absolute deviation for their data.
After all the mean absolute deviations have been calculated, have each group report their deviation and determine the order of the deviations. Did any groups get the order exactly correct? Did all the groups agree on the question that would yield data with the greatest (or least) deviation?
Explore some reasons why the actual results did or did not match the class’s predictions.
Extension:
- Routine: Ask students to devise and conduct surveys with mean absolute deviations greater (or less) than the mean absolute deviation of the Brothers and Sisters data set, or ask students to create a dot plot for ten hypothetical data values with a mean absolute deviation less than a given value.
- Small Group: Have small groups each write five survey questions that could yield data to be displayed in a dot plot. Have groups exchange lists of questions and rank the hypothetical results based on predictions of the mean absolute deviation. Have multiple groups rank the same set of five questions and compare results.
- Technology: This lesson involves use of scientific calculators for calculating the mean absolute deviation.
