Standard Deviation

5 min

Narrative

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.

Launch

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. Follow that with a whole-class discussion.

Student Task

What do you notice? What do you wonder?

mean: 10, MAD: 1.56, standard deviation: 2

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 1, 1, 2, 3, 4, 3, 2, 1, 1, 0.</p>

mean: 10, MAD: 2.22, standard deviation: 2.58

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0.</p>
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mean: 10, MAD: 2.68, standard deviation: 2.92

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 11, 10, 8, 5, 3, 5, 8, 10, 11, 0.</p>

mean: 10, MAD: 1.12, standard deviation: 1.61

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 1, 2, 3, 7, 15, 7, 3, 2, 1, 0.</p>

mean: 10, MAD: 2.06, standard deviation: 2.34

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 2, 4, 6, 4, 3, 4, 6, 4, 2, 0.</p>

mean: 10, MAD: 0, standard deviation: 0

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0.</p>

 

Sample Response

Things students may notice:

  • All of the sets have the same mean.
  • The MAD and standard deviation were both large for the same dot plots and small for the same dot plots.
  • The standard deviation is always greater than the MAD (except for the last plot where both are zero).

Things students may wonder:

  • Are MAD and standard deviation both measures of variability?
  • What is standard deviation?
Activity Synthesis (Teacher Notes)

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.

Math Community

After the Warm-up, display the revisions to the class Math Community Chart that were made from student suggestions in an earlier exercise. Tell students that over the next few exercises, this chart will help the class decide on community norms—how they as a class hope to work and interact together over the year. To get ready for making those decisions, students are invited at the end of today’s lesson to share which “Doing Math” action on the chart is most important to them personally.

Standards
Building On
  • HSS-ID.A.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
  • S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
  • S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
  • S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
Addressing
  • HSS-ID.A.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
  • S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
  • S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
  • S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

20 min

10 min