Lesson 12

The purpose of this warm-up is to elicit the idea that calculating the standard deviation is very similar to calculating the MAD, which will be useful when students explore standard deviation in a later activity. While students may notice and wonder many things about these dot plots, the similarities and differences between standard deviation and the MAD as measures of variability are the important discussion points.

Display the dot plots and statistics for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the images. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the concept of variability does not come up during the conversation, ask students to discuss this idea.

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 standard deviation by the same value.

Monitor for students who

1. Check values without much thought until the mean or standard deviation is correct.
2. Use a starting set of values, then modify a few of them up or down to match the statistics given.
3. Use symmetry to get the mean correct.
4. 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.

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.”

Demonstrate how to find standard deviation and mean using the technology available. In Algebra 1, students are dealing with the entire population, not sampling, so the population standard deviation is used.

Open the Spreadsheet & Statistics tool from Math Tools, or navigate to https://www.geogebra.org/classic/spreadsheet. Enter the values in column A. Select all of column A, and then choose the button that looks like a histogram and “One Variable Analysis.” Click the button that says \(\Sigma x\). The population standard deviation is labeled \(\sigma\). Additionally, the spreadsheet command =SD(A1:A10) will compute the population standard deviation for data in cells A1 through A10.

Arrange students in groups of 2. Give students time to work through the first two questions, followed by a whole-class discussion.

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.

Select previously identified students in the order listed in the narrative to share their distributions and methods for finding values that worked.

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 so that it has a mean of 1. (Shift the data up or down so that it has a mean of 1.)

Here are some questions for discussion.

• “What do you think that 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: 1) 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 2) 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 the values in the second data set. That makes the standard deviation for 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?” (Answers vary. Sample response: My mean was much different from my partners. Mine was 5 and theirs 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.

Open the Spreadsheet & Statistics tool from Math Tools, or navigate to https://www.geogebra.org/classic/spreadsheet. Demonstrate how to add, change, and remove data points from a data set using technology. Provide access to statistical technology. Give students time to work through the questions then pause for a brief whole-class discussion.

Students who compute a different standard deviation may be using the sample standard deviation statistic. Tell these students to use the value for \(\sigma\) rather than \(s\) 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 impacted 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 display less variability and the standard deviation increases when outliers are added because the data in the distribution display more variability.

Add standard deviation to the display of measures of center and measures of variability created in an earlier lesson.

• Standard deviation: A measure of variability often used with mean that describes the spread of the data. It is a more mathematically useful measure of variability than MAD.
• 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.)