The purpose of this activity is to let students investigate what happens to the standard deviation using different data sets. The goal is for students to make conjectures about what standard deviation measures and how relative size of the standard deviation can be estimated from the shape of the distribution. In particular, students should recognize that adding or subtracting the same value from each value in the data set will change the mean by the same amount, but the standard deviation remains unchanged. Multiplying or dividing each value in the data set by the same value scales both the mean and the standard deviation by the same value.

Monitor for students who:

• Check values without much thought until the mean or standard deviation is correct.
• Use a starting set of values, then modify a few of them up or down to match the statistics given.
• Use symmetry to get the mean correct.
• Adjust the data by multiplying all of the values by a number to change the spread of the distribution to get close to the standard deviation.

Plan to have students present in this order from least to more abstract ways of thinking about the data.

The routine of Anticipate, Monitor, Select, Sequence, Connect (5 Practices) requires a balance of planning and flexibility. The anticipated approaches might not surface in every class, and there may be reason to change the order in which strategies are presented. While monitoring, keep in mind the learning goal and adjust the order to ensure that all students have access to the first idea presented (whether that be a common misconception or a different approach).

Identify students who create data sets with different types of distributions for “10 different numbers that have a mean of 1” and “10 different numbers with a standard deviation as close to 2.5 as you can get in one minute.”

The purpose of this discussion is to understand that the standard deviation is a measure of variability related to the mean of the data set. The discussion also provides an opportunity for students to discuss what they notice and wonder about the mean and standard deviation.

Invite previously selected students to share their distributions and methods for finding values that worked. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see.

For at least one student’s data set that had a standard deviation close to 2.5, ask them how they might adapt the data set so that it has a mean of 1. (Shift the data values up or down so that the set has a mean of 1.)

Connect the different responses to the learning goals by asking questions such as:

• “What do you think that the standard deviation measures? Why do you think that?” (I think that it measures variability because it behaved like MAD. When all the values were the same it was zero.)
• “Why is the standard deviation the same for {1,2,3,4,5} and {-2,-1,0,1,2}?” (For each data set: The values to the left of the mean are a distance of 2 and 1 from the mean and the values to the right of the mean are a distance 2 and 1 from the mean, and also, the middle value is 0 away from the mean. This results in the standard deviation being the same.)
• “Why is the standard deviation different for {-4, -2, 0, 2, 4} and {-4, -3, -2, -1, 0}?” (The values for the first data set are twice the distance from the mean as are the values in the second data set. That makes the standard deviation of the first set greater than the standard deviation of the second data set.)
• “When is the standard deviation equal to zero?” (It is zero when all the values are the same as each other or when there is no variability.)
• “Was your mean the same as your partner’s mean in the fifth match?” (My mean was much different from my partners. Mine was 5, and my partner’s was 10.)
• “How did using technology help or hinder your mathematical thinking about standard deviation and mean?” (It really helped me because I did not get bogged down with the calculations, and I could look for patterns in the data, the data displays, and the statistics. In particular, having to come up with my own data and then see what happened to the statistics was really helpful.)

The purpose of this mathematical activity is to let students investigate how the standard deviation and other measures of variability change when you add, change, or remove values in a data set. Monitor for students mentioning the concepts of shape, variability, and center. This activity works best when each student has access to statistical technology because it would take too long to do otherwise. If students don't have individual access, projecting the statistical technology would be helpful during the Launch.

Students who compute a different standard deviation may be using the sample standard deviation statistic. Tell these students to use the value for rather than for computations in this unit.

The purpose of this discussion is to talk about standard deviation as a measure of variability. The goal of this discussion is to make sure that students understand that the standard deviation behaves similarly to the MAD and that it is a measure of variability that uses the mean as a measure of center. Discuss how the standard deviation is affected by the addition and removal of outliers in the data set. The standard deviation decreases when outliers are removed because the data in the distribution then display less variability, and the standard deviation increases when outliers are added because the data in the distribution then display more variability.

Add standard deviation to the display of measures of center and measures of variability created in an earlier lesson. The blackline master provides an example of what this display may look like after all items are added.

The MAD already included in the example display is approximately 1.09, and the standard deviation is 1.194.

Here are some discussion questions.

• “What does the standard deviation measure? How do you know?” (It measures variability. Like the MAD, the standard deviation increases when outliers are included in the data set.)
• “The standard deviation is calculated using the mean. Do you think it is more appropriate to use with symmetric or skewed data sets?” (Symmetric, because it is calculated using the mean.)