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.
|
Time involved
30 to 40 minutes
Student
directions
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
- Indicate whether one of the
graphs has a larger standard deviation than the other or if the
two graphs have the same standard
deviation.
- 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:
- How many groups were correct
for all 15 pairs of histograms?
- What approaches did you use to
decide which graph had the larger standard
deviation?
- What features of a histogram
seem to have no bearing on the standard deviation?
- 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:
- Provide written descriptions of
the characteristics of a distribution that affect the standard
deviation.
- 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".
