Intent

Students will continue to build their understanding of mean and standard deviation.

Mathematics

When a nonzero constant is added to each value in a data set, the mean, a measure of the center of the distribution, will change by that amount. However, the standard deviation, a measure of the spread of the data, will not change. When each value in a data set is transformed by multiplication by a particular constant, both the mean and the standard deviation will change. In this activity, students explore these transformations and draw conclusions about their effects on these two descriptive statistics. They will also continue to create data sets that have given values for these statistics.

Progression

Students work on the activity in groups and discuss their results as a class.

Approximate Time

35 minutes

Classroom Organization

Groups, followed by whole-class discussion

Doing the Activity

This activity requires little or no introduction.

Question 3 is intended primarily as further work for groups that finish early. You might warn students that they will be able to create data sets with those exact means but will only be able to approximate the standard deviations.

Discussing and Debriefing the Activity

Focus the discussion on parts c and d of Questions 1 and 2.

Students’ explanations of the patterns they observe in Question 1d may take several forms. For example, they may picture the data points on the number line, so that adding the same thing to each data point just moves the points along and hence also moves the mean. Or they may see the change in the mean algebraically (although it’s unlikely they will have a full algebraic explanation involving the distributive law).

Students may attribute the lack of change in the standard deviation to the fact that the spread doesn’t change when the set of data points is moved along. Or they may recognize that when all the data are changed the same way, the mean also changes, so the spread from the mean remains the same.

The explanations for Question 2d will be similar.

Fathom can be used to provide a visual demonstration of the effects on a simple data set of adding or multiplying by a constant. [Link to the Fathom file MeanMedSD.ftm]

Key Question

What conclusions did you reach about how standard deviation is affected by changes in the data?

Supplemental Activity

Making Better Friends (extension) provides additional opportunities for students to explore standard deviation, continuing the challenges posed in this activity.