 align=middle width=80% Student's version HTML Format Word Format Video Demo*: QuickTime Format *Need QuickTime player

What Makes the Standard Deviation Larger or Smaller?

Robert C. delMas
University of Minnesota
354 Appleby Hall
128 Pleasant Street SE
Minneapolis, MN 55455

Statistics Teaching and Resource Library, June 30, 2001

© 2001 by Robert C. delMas, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.

The activity is designed to help 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. 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. This activity can help students discover that the standard deviation is a measure of the density of values about the mean of a distribution and to become more aware of how clusters, gaps, and extreme values affect the standard deviation.

Key words: Variability, standard deviation

## Objective

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.

## Materials needed

• 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).
• The activity can also 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.

## Students are presented 15 pairs of graphs numbered one through fifteen. The mean for each graph (m) is given just above each histogram. For each pair of graphs presented students are asked to do the following

1. Indicate whether one of the graphs has a larger standard deviation than the other or if the two graphs have the same standard deviation.
2. Try to identify the characteristics of the graphs that make the standard deviation larger or smaller.

## Students can check your answers against the instructor’s answer key as you complete each page.

Two of the fifteen graph pairs are presented below. A has a larger standard deviation than B B has a larger standard deviation than A Both graphs have the same standard deviation A has a larger standard deviation than B B has a larger standard deviation than A Both graphs have the same standard deviation

Instructor notes

Break the class into groups of three or four students. Circulate from group to group and encourage the groups to debate which graph in each pair has the larger standard deviation. Ask students for the basis of their decisions (e.g., What criteria are you using? What characteristics of each graph are you focusing on?).

If students are confident of an incorrect answer, have them check the answer. 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:

1. How many groups were correct for all 15 pairs of histograms?
2. What approaches did you use to decide which graph had the larger standard deviation?
3. What features of a histogram seem to have no bearing on the standard deviation?
4. 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). An example of a summary is provided in the video clip.

Suggestions for assessment

A shorter version of the activity can be accomplished by using the following 10 pairs of graphs:

1, 4, 5, 6, 8, 9, 12, 13, 14, 15

The first seven pairs are typically more straightforward for students whereas the last three typically give students more difficulty because they require students to balance several characteristics simultaneously. Students will tend to form opinions about the factors that affect the standard deviation when working on the first seven pairs that can become challenged when they work on the last three.

The remaining pairs (2, 3, 7, 10, and 11) can then be used as assessment items in a follow-up to the activity.

Students can also 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.

Video clip

An excerpt from a class where Joan Garfield and the author lead students through the activity.

Editor's note: Before 11-6-01, the "student's version" of an activity was called the "prototype".

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