Intent This activity, and the discussion that leads into it, introduces students to the normal distribution.

Mathematics The normal distribution (also called the Gaussian distribution) is the technical name for what many call the bell curve. Of the many ways that data may be distributed, the normal distribution is of particular interest and is useful in many statistical situations. For example, many types of data related to people—such as the heights or shoe sizes of adult men or women—are approximately normally distributed. The normal distribution is a specific type of bell-shaped frequency pattern, with a precise, technical mathematical definition. Additionally, measurement variation is approximately normally distributed. It is for this last reason that the normal distribution is introduced in this unit.

Progression After a teacher-led introduction to the normal distribution, students work individually to create graphs of surmised data from several situations, including labeled axes and their own choices for intervals, units of measurement, and frequency of data within each interval. They then share their results in groups.

Approximate Time 30 minutes for introduction 20 minutes for activity (at home or in class) 10 minutes for discussion

Classroom Organization Whole-class introduction, then individuals, then groups, followed by whole-class discussion

Materials Frequency bar graphs from Time Is Relative, What’s Your Stride?, and Pulse Analysis Transparencies of the graphs [link to pdf of What’s Normal, p. 1–2]

Doing the Activity Before assigning the activity, lead a discussion to introduce the normal distribution. To begin, draw students’ attention to the frequency bar graphs made earlier of the following data sets. Timing of five seconds (from Time Is Relative) Stride length (from What’s Your Stride?) Pulse rates (from Pulse Analysis) Ask students, What features do these graphs have in common? They will probably focus on two key features. The graphs are highest “in the middle.” (Students may or may not use the term mean.) The graphs gradually go down toward the ends. Using a diagram like the one below, explain that curves with this general appearance are called bell shaped and that there is a very special bell-shaped curve called the normal distribution. [link to Blackline Masters.doc.]

Bring out the connection between the area under such a curve and the probability of various results. For example, if the shaded area on the next diagram is, say, 20 percent of the total area under the curve, then 20 percent of all measurements are between points a and b. Students may recognize that a similar idea applies to frequency bar graphs. Point out the similarity between this shaded area under the curve and the area of a bar in a frequency bar graph. It’s as if the tops of all the bars in a frequency bar graph were connected to draw a smooth curve.

You can also show the next diagram, which depicts three different normal curves on the same set of axes, and ask students what they think the differences indicate. The goal is for them to recognize that the amount of variation from one measurement to another is different in each graph. The exact shape of a normal curve depends on the scales being used and the specific situation.

Tell students that you are giving them a simplified description of the normal distribution. They will not be able to determine for sure whether a data set is normally distributed. The precise definition involves a complex formula for the graph—one that most people encounter only if they study statistics in college. [Link to math maps] You may want to clarify that in the term normal distribution, the word “normal” is being used in a special, technical sense. It does not mean “ordinary,” although the normal distribution is one that occurs in many situations. Ask, What features do these normal curves have in common? Students should see that, as with the frequency bar graphs under consideration, the normal curves are highest in the middle and decrease gradually toward both ends. Make sure they note one more specific phenomenon: The normal curve is symmetric. Introduce the term line of symmetry for the vertical line that divides a normal curve into two equal parts. Then ask, What does the location of the line of symmetry represent? Students should realize that values to the right of the line of symmetry “balance out” values to the left. Help them as needed to use this observation to reach an important conclusion: The line of symmetry represents the mean of the data. If students mention the median (in addition to or instead of the mean), explain that for symmetric data, the mean and median are the same, and perhaps ask why. A more subtle observation on the shape of the graph concerns concavity. You can bring this out by asking, What changes in the way the normal curve “curves”? You might use the following diagram to illustrate the ideas. Introduce the terms concave up and concave down to describe the different portions of the curve.

Note that the change of concavity provides an important visual image of standard deviation. For example, the point on the horizontal axis that corresponds to the first point of concavity to the right of the mean is one standard deviation above the mean. The significance of concavity in relation to standard deviation will be discussed in the activity The Best Spread. Ask, Do you think the frequency bar graphs of our experimental data resemble the normal distribution? Students’ response may depend on how much data they collected for each experiment and on how they grouped the results. Whatever their response, tell them that if they were to record more and more data, their graphs would probably begin to look more and more like the normal distribution. The normal curve is generally considered a reasonable expectation for results of measurement variation. Then tell them that, based on this general experimental phenomenon, this unit makes the following assumption: Normality assumption: If you make many measurements of the period of any given pendulum, the data will closely fit a normal distribution. Post this assumption in the room, as it will be referred to later in the unit. Now ask, How does the idea of normal distribution relate to the unit problem? Students should recognize that, according to the normality assumption, the normal distribution describes the kind of measurement variation they should expect in pendulum experiments. Therefore, familiarity with the normal distribution is moving them along in the process of determining which variables are important.

Discussing and Debriefing the Activity In their groups, have students compare the frequency bar graphs they sketched. Then discuss, as a class, which situations they think are normally distributed. For Question 1, some students may have arranged the categories so that the tallest frequency bars are in the middle and then concluded that the distribution is approximately normal. If so, point out that a normal distribution requires that the data items be numeric in nature. Tell the class that nonnumeric data, like shoe type, is sometimes called categoricaldata. Of Questions 2 through 4, only the situation in Question 2 might be approximately normally distributed (and even that might not be), although students may not have the facts on which to make this judgment. For Question 3, bring out the fact that far more people have incomes below the mean than above it (due to the effect on the mean of a small number of people with very high incomes). In particular, this means that the distribution of incomes is not symmetric around the mean. As needed, review that symmetry is one of the key characteristics of the normal distribution. However, income distribution does resemble the normal distribution in at least one respect: It trails off toward the extremes (at least at the upper end). You can review here that in the normal distribution, values farther from the mean are less likely (that is, occur less often) than values closer to the mean. For Question 4, help students to see that, assuming either constant or increasing birthrates, the population of different age groups decreases gradually toward the higher age groups. For example, there are generally more people between the ages of 0 and 10 than between 10 and 20, more between 10 and 20 than between 20 and 30, and so on. (States like Hawaii and Florida, which attract many retirees, might be exceptions to this pattern.) You might mention that many properties of people and objects are distributed normally or close to normally, but many are not. It isn’t necessarily easy to decide, in theory, which are which. You may want to summarize several key aspects of the normal distribution that were brought out in this activity. Normally distributed data must be numeric. Normally distributed data are symmetric about the mean. For normally distributed data, results farther from the mean are less likely than results closer to the mean. Finally, review the assumption that is being made in this unit: that measurements of a given pendulum’s period are normally distributed.

Key Questions What features do these graphs have in common? What does the location of the line of symmetry represent? What changes in the way the normal curve “curves”? Do you think the frequency bar graphs of our experimental data resemble the normal distribution? How does the idea of normal distribution relate to the unit problem? Which situations do you think are normally distributed? Why?