Use of computer lab exercises can extensively enrich learning
experiences in statistics courses. The use of computers allows
us to utilize larger, richer data sets relevant to students' experiences,
and to provide visual explanations of key concepts, such as sampling
distributions. In this paper, I present motivation for these
lab exercises, examples of student experiences, and suggestions
for implementation and assessment of these computer activities.

**1. Introduction**

*Statistics should be taught as a laboratory science, along
the lines of physics and chemistry rather than traditional mathematics.
Students must get their hands dirty with data. The laboratory
must be a requirement and must contain more than a few computers.
. . having students discuss and write about their understandings
and interpretations of the problems: - *R. Scheaffer, in Cobb
(1992)

Recent calls for change in statistics education (e.g. Singer &
Willet, 1990; Tanner & Wardrop, 1992; Cobb, 1992) have emphasized
that statistics students should be working with real data in a
hands-on environment, learning first hand how and why data are
collected. In this way, students are *doing* statistics
instead of just reading and hearing about statistics. I have
found that incorporating computer lab activities into my statistics
courses accomplishes just this. Computers allow us to provide
students with access to larger, more relevant data sets, and to
visual, interactive demonstrations of key concepts. Furthermore,
I require students to incorporate the computer results into written
explanations of statistical concepts, providing them with valuable
technical writing experience and practice using the language of
statistics. Students are required to explain their understanding
of the concepts and to discuss issues in data collection and model
assumptions, giving me more insightful assessment of their understanding.
I further assess these skills by including a take-home, open-ended
component on the final exam, requiring individual exploration
and development of solutions using the computer. Such assessment
techniques directly measure student ability to do statistics.
Through these exercises students not only achieve a high level
of computer literacy, but also appreciation, and a higher level
of understanding of statistics. Still, these exercises need to
be carefully implemented into the course to be successful.

**2. Use of Real Data and Data Collection Experiences
**

As Cobb (1992) states, students should view statistics as a process
of scientific inquiry, discovering information from real life
data. If the students are able to connect to the data, they will
better retain the information from the example. Also, by collecting
data themselves, students actively participate in the process
and are better able to understand the issues involved. Thus,
I wanted to create laboratory exercises that would utilize actual
data sets of immediate relevance to the students and exercises
requiring students to personally collect data.

To incorporate the first type of data sets, I have adapted several data sets for use by the students. For example, a set of baseball statistics compiled by Chris Albright (albright@big.bus.indiana.edu) is used to allow the students to explore the different behavior of means and medians in the presence of skewness or outliers. Students are asked to identify individual players that appear as outliers on such variables as home runs and batting average. Outliers are thus connected with actual people they have heard about. Another data set was compiled by Robin Lock from AAUW and US News and World Report data on US colleges and universities (available from Journal of Statistics Education data archive). Due to memory limitations with our computer program (student version of Minitab) I reduced this list to a set of 100 western colleges and universities. Still, students are interested in this data and are able to see how their university compares to others. Thus, students want to know the answers and see the computer as a means to obtain those answers. This increases student involvement and appreciation, as they are eager to explore the relationships among the variables. Students also have definite preconceptions they can bring to the data, and hypotheses they want to test.

Secondly, in several of the labs students gather the data themselves.
These activities range from measuring the diameter of tennis
ball to understand variability, to a taste test between Coke and
Pepsi, to recording prices on a set of items at two different
grocery stores, to observing the proportion of each M&M color.
Since they have collected the data themselves, they will feel
an ownership for the data, establishing an even stronger curiosity
about the results. Students also need to make certain decisions
along the way. For example, one lab asks students to time the
wait of cars at two corners, one with a stop light and one with
a stop sign. Students must precisely define when a car has begun
and ended its wait. They learn that many of these decisions are
not automatic and can have an effect on the final results. They
are also directly involved with issues about independence and
randomization. These shorter experiences are good models for
their longer term project. Thus, by using realistic and self-collected
data students gain a better understanding of the use of statistics
and its relevance to their own experiences.

**3. Use of Visual Explorations
**

Since many students are visual learners, computers also play an
essential role in allowing students to efficiently investigate
concepts visually. To provide the students with visual explanations,
I use a series of programs developed by Robert delMas, University
of Minnesota. These programs currently run on Macintosh computers.
Several similar programs are also being developed, such as *ExplorStat*
(Wackerly, University of Florida), *HyperStat *(Lane, Rice
University)*,* and *Teaching Statistics Visually*(Tracy,
Doane, & Mathieson, Oakland University). These programs allow
students to explore concepts and discover properties on their
own. For example, the Standard Normal Distribution program by
delMas allows students to adjust the population mean and standard
deviation to see how the z-score and probabilities respond. They
can adjust the X values, Z value, and probabilities independently,
seeing for themselves how each affects the other. Another delMas
program allows them to simulate sampling distributions while changing
the population curve, sample size, and number of samples. Students
directly experience the Central Limit Theorem, providing them
with a visual image to refer back to. Another delMas program,
a simulation of the Monte Hall problem, allows students to experience
long-run probabilities in a fun setting where the answer is not
initially intuitive for them. Many statistical properties are
most apparent "in the long run" and computers allow
students to quickly execute these simulations on a larger scale
in order to properly discover the properties. I see these simulations
as a compliment to smaller scale simulations done in class, whereas
often smaller in-class examples on their own can be misleading.

These are just a few examples of how these programs can enhance
student understanding of concepts by giving them a visual representation
to accompany the verbal representation and to discover properties
of these ideas on their own. By constructing their own knowledge
in this way students achieve a deeper understanding and longer
retention of the ideas.

**4. Student Writeups
**

It is crucial for students to not only see these activities, but
to be asked to explain what they learned, further internalizing
the experience and making them responsible for the knowledge.
After each activity, students have one-week to complete a writeup
about the activity. These writeups are either answers to a series
of questions I pose, or a "Full Writeup" technical report.
In these questions, students are asked to explain the concepts
in their own words. This provides me with the most accurate picture
of what they understand and allows me to provide feedback on their
interpretations. They also learn that there can be multiple interpretations
for the same experience. In the Full Write-ups, students are
to provide an introduction, explanation of data collection procedures,
presentation of results, discussion of the results, and their
final conclusions. Students are also asked to verify any assumptions
required by statistical procedures. In this way, students must
develop a logical argument based solely on the data they observed.
Guidelines for these lab writeups (similar to those given by
Egge et. al.,1995 and Spurrier, Edwards, and Thombs, 1995) are
also given. I want students to learn how to effectively summarize
their results and develop their own interpretations. Students
are also asked to provide recommendations for future studies so
they may learn to critically evaluate the procedures that were
used. Students also see that gathering information from data
is a continuous process. These writeups allow us to clearly see
what the student learned from the activity, as well as providing
the students with valuable practice utilizing the language of
statistics which is often quite foreign to them initially.

**5. Implementation
**

To ensure successful implementation of lab activities, care has
to be taken in their construction and use, especially with a computer
phobic audience. For our labs, I have utilized the MINITAB statistical
software in conjunction with Microsoft Word for the reports.
Students find the Minitab software easy to learn and I feel it
provides them with sufficient background to transfer to other
computer packages. Students are also able to easily incorporate
Minitab output and graphics into a word processing program. The
student version of Minitab can have some memory limitations, but
I feel the statistical procedures are sufficient for my introductory
statistics courses and is able to surpass many student anxieties.

Furthermore, the instructions need to be well laid out for the students. I require students to purchase a laboratory manual I put together that includes detailed instructions for using Minitab and Word, as well as instructions for the labs for the semester. This gives students a permanent reference manual and has facilitated their learning of the software and given them a clear idea of what is expected over the course of the semester. The manual also includes pictures of the menus they will need to use as they execute commands. Instructions are quite detailed initially, but then become less directed, as I try to encourage the students to become more independent analysts. Students do often exhibit poor memory of earlier labs, and thus an index, glossary of commands, and a frequently asked questions section have been added. Thus, I want the instructions to be clear enough students can follow them independently, but I also want to insure that the students are learning to help themselves solve problems as well.

Our lab currently has 25 machines. To ensure each student has
individual access to a machine I break the class into two sections
of 20-25 students each to meet in the lab with the instructor
once per week. The immediate access to the instructor has also
eased computer anxiety as students work through the lab instructions.
The lab is also open in the afternoons and evenings to provide
students with the opportunity to finish their lab writeups. Ideally
the lab assistant employed at this time will have an understanding
of the programs involved. It is important that the students don't
become overwhelmed by the computer exercises. My goal is for
them to finish the software instructions during the hour we met
together so that this is not their stumbling block.

After each lab has been graded, an example lab, by one of the
students, is posted for students to refer to. The goal is for
students to review these papers to better understand what is expected
of them, feel pride if their paper is selected, see the work of
their peers (and thus what they are also capable of), and see
that there can be multiple interpretations/conclusions for the
same activity. This type of guidance and feedback is important.
I chose to do it after the lab is turned in so that students
generate their own reports and develop their own style of technical
exposition instead of following a template.

Student participation in development of these activities is also
important. I encourage student feedback and constantly update
the list of instructions. I have also employed a student assistant
to help in the development of the activities and to ensure clarity
of the instructions. It is important to understand that development,
implementation, and assessment of these lab activities add a significant
time commitment on the part of the instructor and that these activities
need to be well-integrated with the rest of the course to be successful.

**6. Assessment
**

When grading these writeups, less than 50% of the grade stems
from computer output, allowing me to demonstrate to the students
that the computer output is a small part of the analysis. Instead
more of the grade is composed of their explanations, their ability
to interpret the computer output, and their effective communication
of their knowledge. Students are encouraged to work in pairs
so they will discuss and debate the concepts with each other.
I feel that fostering this discussion greatly enhances the students'
learning experience. Still, since part of the final involves
computer work, students know that each of them is individually
responsible for knowing how to use the computer programs.

This component to the final is a take-home exam. Students are
given a series of questions, each requiring different statistical
analyses. They must identify and carry out the appropriate analyses,
graphs, and diagnostic checks, and interpret the results. They
are expected to identify the appropriate procedures based on the
question asked and types of variables involved. This part of
the exam is given out one week before a traditional in-class final
exam giving sufficient time to access the computer lab and think
about several approaches to the problem. They are encouraged
to ask the instructor questions about the computer commands, especially
if they need a technique that was not covered in the lab manual.
However, if they ask for statistical knowledge they are "charged"
by losing points if I tell them the answer. I indicate that this
is similar to having to pay an external consultant when they need
further statistical expertise. Students are graded on their ability
to identify and justify an appropriate analysis technique, perform
this technique and necessary diagnostic checks, and interpret
their results to formulate a conclusion about the problem, providing
me with an authentic performance assessment. I find this type
of complete, detailed analysis by the student is not feasible
on an in-class exam. To ensure individual assessment, I am developing
a list of suitable data sets (rich with multiple procedures) and
hope that a bank of such test questions will continue to be developed.
Current sources of test questions include the Journal of Statistics
Education and Statlib data archives and the casebook by Chatterjee,
Handcock, and Simonoff (1995).

**7. Conclusion
**

I feel these lab exercises are invaluable experiences for statistics
students at all levels, enhancing their understanding and increasing
their retention of statistics. Students learn the material mathematically
and concretely simultaneously, linking the two representations
together. These activities also enhance their writing and critical
evaluation skills. Students use the computer to answers questions
in which they have a vested interest and they also experience
the messiness of real life data. Thus they feel an ownership
for the data and see the relevance of statistics to their world,
while learning them to appreciate the power of computers as a
useful tool.

**8. References
**

Chatterjee, S., Handcock, M., and Simonoff, J. (1996)*, A Casebook
for a First Course in Statistics and Data Analysis*, New York:
John Wiley and Sons.

Cobb, G. (1992), ``Teaching Statistics,'' in *Heeding the Call
for Change: Suggestions for Curricular Action*, ed. L. Steen.
MAA Notes, No. 22.

Egge, E., Foley, S., Haskins, L., Johnson, R. (1995), ``Statistics
Lab Manual', Carleton University, Mathematics and Computer Science
Department, 3rd edition.

Mackisack, M. (1994), ``What is the Use of Experiments Conducted
by Statistics Students?,'' *Journal of Statistics Education*,
2(1).

Singer, J. and Willet, J. (1990), ``Improving the Teaching of
Statistics: Putting the Data Back into Data Analysis,'' *The
American Statistician*, 44(3), 223-230.

Spurrier, J.D., Edwards, D., and Thombs, L.A. (1995), *Elementary
Statistics Lab Manual*, Belmont: Wadsworth Publishing Co.

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