Intent Students connect their work in Statistics and the Pendulum and their experiments in The Standard Pendulum in anticipation of the activity Pendulum Variations.

Mathematics The theoretical normal distribution is a bell-shaped curve with a vertical line of symmetry at the mean, a section in the middle curved concave down, and tails curved concave up. The transition points in concavity—the inflection points of the graph—are the locations of one standard deviation above and below the mean. Experimental data derived from repeated measurements of some quantity will reflect measurement variation, and as a result the measured values will be approximately normally distributed. The mean and standard deviation of these data can be computed. The class has sketched a normal distribution with the mean and standard deviation computed for their standard pendulum data. Now students will construct a frequency bar graph of the actual data, overlay a bell-shaped curve that approximates the distribution, and visually note the approximate locations of the mean and a standard deviation above and below the mean. The final question in the activity—How different must an observation be before you can be confident it isn’t just measurement variation?—is the key to determining whether changing a variable has a real effect on the period.

Progression Students work on the activity individually and then share their work, including frequency bar graphs of their standard pendulum data, with the class.

Approximate Time 25 minutes for activity (at home or in class)10 minutes for discussion

Classroom Organization Individuals, followed by whole-class discussion

Materials Class data from the standard pendulum experiments

Doing the Activity This activity requires little or no introduction.

Discussing and Debriefing the Activity Question 1 provides an opportunity to review ideas about measurement variation. Ask for comments about the frequency bar graphs students created. If possible, get confirmation of the idea that the data items are approximately normally distributed and review the “normality assumption” stated in Standard Deviation Basics. Have volunteers share their frequency bar graphs and accompanying normal curves, and discuss how they used these items to estimate the mean and standard deviation. This discussion is also an opportunity to review the ideas in the section “Geometric Interpretation of Standard Deviation” in Standard Deviation Basics. The issue in Question 5 is best addressed by using the normal distribution graph with the mean and two standard deviations in either direction marked on the horizontal axis.