# Understanding the standard deviation: What makes it larger or smaller?

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

## Summary

Using cooperative learning methods, this activity helps students develop a better intuitive understanding of what is meant by variability in statistics. Emphasis is placed on the standard deviation as a measure of variability. This lesson also helps students to discover that the standard deviation is a measure of the density of values about the mean of a distribution. As such, students become more aware of how clusters, gaps, and extreme values affect the standard deviation.

## Learning Goals

Students will come to understand the standard deviation as a measure of density about the mean. They should also become more aware of how clusters, gaps, and extreme values affect the standard deviation.

## Context for Use

This activity is usually done in the middle, or end of a unit on distribution and measures of spread. The lesson takes 30—40 minutes and may be used with any size classroom. The activity may be easily adapted for junior high, high school, and college-level instruction.

## Description and Teaching Materials

This activity uses a set of 15 pairs of histograms presented in the prototype activity. The items were constructed to control for characteristics that students might attend to when judging variability (e.g., location of center, number of bars, range, variability in bar heights, one graph is the mirror image of the other, gaps). A set of answers for the 15 histogram pairs (also presented in the prototype activity) are also needed. The original activity can be downloaded as a Microsoft Word document from the following website:

http://www.gen.umn.edu/faculty_staff/delmas/stat_tools/. Once you are at the Stat Tools website, click the MATERIALS button. Scroll down to the Variability Activity and select an operating system format (Macintosh or Windows) for the downloaded file. Detailed lesson plan of how to use cooperative learning for this activity. (Acrobat (PDF) PRIVATE FILE 88kB Jun30 06)

http://www.gen.umn.edu/faculty_staff/delmas/stat_tools/. Once you are at the Stat Tools website, click the MATERIALS button. Scroll down to the Variability Activity and select an operating system format (Macintosh or Windows) for the downloaded file. Detailed lesson plan of how to use cooperative learning for this activity. (Acrobat (PDF) PRIVATE FILE 88kB Jun30 06)

## Teaching Notes and Tips

Cooperative Learning:

 See "Understanding the SD Using CL" file for specific directions about teaching this lesson using Cooperative Learning strategies.

Time Involved:

- 5 minutes to introduce the activity

- 20-25 minutes for students to work in groups, sorting graphs

- 10 minutes for instructor-led discussion of graphs

- 5 minutes for follow up questions

Practical Tips:

As they learn about the standard deviation, many students focus on the variability of bar heights in a histogram when asked to compare the variability of two distributions. For these students, variability refers to the "variation" in bar heights. Other students may focus only on the range of values, or the number of bars in a histogram, and conclude that two distributions are identical in variability even when it is clearly not the case.

- Make sure you have enough copies of the 15 pairs of graphs for each group of students.

- Student pairs should compare answers with another pair of student after completing each page of graphs. Waiting any longer risks reinforcing possible misconceptions. Early comparison also forces greater conceptual processing as students must describe their answers (and therefore their thinking) with the new group of two.

- So that every individual participates in the group of four, teachers may recommend that students take turns explaining their answer and leading any subsequent discussion.

- Throughout the lesson, instructors should encourage students to be as specific as possible when identifying the characteristics of the graphs that make the standard deviation larger or smaller (e.g., What criteria are you using? What characteristics of each graph are you focusing on?).

- Students should be encouraged to speak precisely about these characteristics—e.g., describing a distribution as having a larger "range" rather than "it looks bigger".

- If students are confident of an incorrect answer, have them compare their answer with another group. Specifically, help them identify characteristics that differ between the graphs in a pair, but that have no bearing on differences in variability (e.g., different locations of center, variability in bar height).

- Students may focus on a characteristic of a distribution that does affect variability (e.g., distance of the scores from the mean), but neglect other characteristics (e.g., a different number of values in each distribution, gaps around the mean). These students may incorrectly decide that the characteristic they identified does not affect variability. They may require some guidance to attend to other characteristics.

- Encourage students to develop a "visual" understanding or representation of variability instead of trying to identify a single rule or set of rules that provides a correct response for every pair (see the accompanying video clip for examples).

- After all groups have completed the task, bring the class together for a debriefing. Ask the class questions such as:

o How many groups were correct for all 15 pairs of histograms?

o What approaches did you use to decide which graph had the larger standard deviation?

o What features of a histogram seem to have no bearing on the standard deviation?

o What features do appear to affect the standard deviation?

- The instructor can offer a summary of the key features of a distribution that determine the standard deviation of a distribution (e.g., that a distribution is centered, and that variability measures the extent to which values cluster about the center).

 See "Understanding the SD Using CL" file for specific directions about teaching this lesson using Cooperative Learning strategies.

Time Involved:

- 5 minutes to introduce the activity

- 20-25 minutes for students to work in groups, sorting graphs

- 10 minutes for instructor-led discussion of graphs

- 5 minutes for follow up questions

Practical Tips:

As they learn about the standard deviation, many students focus on the variability of bar heights in a histogram when asked to compare the variability of two distributions. For these students, variability refers to the "variation" in bar heights. Other students may focus only on the range of values, or the number of bars in a histogram, and conclude that two distributions are identical in variability even when it is clearly not the case.

- Make sure you have enough copies of the 15 pairs of graphs for each group of students.

- Student pairs should compare answers with another pair of student after completing each page of graphs. Waiting any longer risks reinforcing possible misconceptions. Early comparison also forces greater conceptual processing as students must describe their answers (and therefore their thinking) with the new group of two.

- So that every individual participates in the group of four, teachers may recommend that students take turns explaining their answer and leading any subsequent discussion.

- Throughout the lesson, instructors should encourage students to be as specific as possible when identifying the characteristics of the graphs that make the standard deviation larger or smaller (e.g., What criteria are you using? What characteristics of each graph are you focusing on?).

- Students should be encouraged to speak precisely about these characteristics—e.g., describing a distribution as having a larger "range" rather than "it looks bigger".

- If students are confident of an incorrect answer, have them compare their answer with another group. Specifically, help them identify characteristics that differ between the graphs in a pair, but that have no bearing on differences in variability (e.g., different locations of center, variability in bar height).

- Students may focus on a characteristic of a distribution that does affect variability (e.g., distance of the scores from the mean), but neglect other characteristics (e.g., a different number of values in each distribution, gaps around the mean). These students may incorrectly decide that the characteristic they identified does not affect variability. They may require some guidance to attend to other characteristics.

- Encourage students to develop a "visual" understanding or representation of variability instead of trying to identify a single rule or set of rules that provides a correct response for every pair (see the accompanying video clip for examples).

- After all groups have completed the task, bring the class together for a debriefing. Ask the class questions such as:

o How many groups were correct for all 15 pairs of histograms?

o What approaches did you use to decide which graph had the larger standard deviation?

o What features of a histogram seem to have no bearing on the standard deviation?

o What features do appear to affect the standard deviation?

- The instructor can offer a summary of the key features of a distribution that determine the standard deviation of a distribution (e.g., that a distribution is centered, and that variability measures the extent to which values cluster about the center).

## Assessment

The Cooperative Learning lesson plan emphasizes several ways that instructors may incorporate positive interdependence (i.e., collective assessment) and individual accountability. See the Cooperative Learning module for more details (http://serc.carleton.edu/introgeo/cooperative/index.html).

Students may be asked to:

1. Provide written descriptions of the characteristics of a distribution that affect the standard deviation.

2. Write a descriptive definition of the standard deviation that does not involve mathematical symbols.

Another sample assessment is included with the original activity at (http://www.causeweb.org/repository/StarLibrary/activities/delmas2001). This assessment utilizes a shorter version of the same activity.

Students may be asked to:

1. Provide written descriptions of the characteristics of a distribution that affect the standard deviation.

2. Write a descriptive definition of the standard deviation that does not involve mathematical symbols.

Another sample assessment is included with the original activity at (http://www.causeweb.org/repository/StarLibrary/activities/delmas2001). This assessment utilizes a shorter version of the same activity.

## References and Resources

More information about Cooperative Learning is available at:

http://serc.carleton.edu/introgeo/cooperative/index.html

CAUSEweb includes the original version of the of this activity: (http://www.causeweb.org/repository/StarLibrary/activities/delmas2001).

An excerpt from a class where Joan Garfield and Bob delMas lead students through the activity is available on the STAR library on CAUSEWeb (http://www.causeweb.org/repository/StarLibrary/activities/delmas2001).

http://serc.carleton.edu/introgeo/cooperative/index.html

CAUSEweb includes the original version of the of this activity: (http://www.causeweb.org/repository/StarLibrary/activities/delmas2001).

An excerpt from a class where Joan Garfield and Bob delMas lead students through the activity is available on the STAR library on CAUSEWeb (http://www.causeweb.org/repository/StarLibrary/activities/delmas2001).