Enhancing Your Students’
Conceptual Understanding of
Statistics
Michael Sullivan
Joliet Junior College
[email protected]
[email protected]
GAISE
(Guidelines for Assessment and Instruction in Statistics Education)
http://www.amstat.org/education/gaise/
• Six Recommendations
– Emphasize statistical literacy
• I am going to expand this to statistical literacy, statistical
reasoning, and statistical thinking
– Use real data
– Stress conceptual understanding rather than knowledge of
procedures
– Foster active learning
– Use technology for developing conceptual understanding
and analyzing data
– Use assessments to improve and evaluate student learning
A Recommendation on the First Day of
Class
You are not in a math class!!!!!!
What is Conceptual Understanding?
– The ability to interpret and explain results
– The ability to determine which techniques are
appropriate
– De-emphasis of procedural methods
– Teach fewer concepts, and dig deeper
We are trying to teach too much material in our statistics courses.
Conceptual Understanding
• De-emphasize the use of procedures and emphasize the “big picture” idea.
Two simple examples:
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n 1
2
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x
2
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n 1
n
2
Conceptual Understanding
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 x  x  y  y 
x y 

r
 
 sx   s y
n 1
i
i


i
r
i
i
2


x


i
2
  xi 

n




i
n
2


y

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  yi 
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Test question: Draw two
scatter diagrams for
which r is close to 0.
The Learning Pyramid
Lecture
5%
Reading
10%
Audio Visual
20%
Demonstration
30%
Discussion Group
50%
Practice by Doing
75%
Teach Others/Immediate Use
80%
Adapted from The Learning Triangle: National Training Laboratories, Bethel Maine
©mindServegroup 2005
The Golden Ratio
1. Measure the height of your partner in centimeters.
Call this y.
2. Measure the height to your partners naval. Call this x.
(We could also measure arm span)
3. Draw a scatter diagram of the data treating height to
naval (arm span) as the explanatory variable.
4. Find the least-squares regression line treating height to
naval as the explanatory variable.
5. What is the slope? Interpret the slope.
6. Does it make sense to interpret the intercept? Why?
Emphasize Statistical Literacy
Statistical literacy involves understanding and using the basic language and
tools of statistics: knowing what statistical terms mean, understanding the use
of statistical symbols, and recognizing and being able to interpret
representations of data.
Source: ARTIST website
Examples of Problems/Questions that Demonstrate Statistical Literacy
Suppose you have just constructed a 95% confidence interval. Name two options
for increasing the precision of the interval.
Draw a scatter diagram for which r = 1.
According to popcorn.org, the mean consumption of popcorn annually by
Americans is 54 quarts. The marketing division of popcorn.org unleashes an
aggressive campaign designed to get Americans to consume even more popcorn.
After two months, it was concluded that the marketing campaign was effective.
Suppose, in fact, that the actual mean consumption of popcorn after the marketing
campaign is 53.4 quarts. What type of error was committed? Why?
What is an observational study? What is a designed experiment? Which allows the
researcher to claim causation between an explanatory variable and a response
variable?
Explain what is meant by confounding. What is a lurking variable?
What is resistance? Is the mean resistant? The median? Is the standard deviation
resistant?
Emphasize Statistical Reasoning
Statistical reasoning is the way people reason with statistical
ideas and make sense of statistical information. Statistical
reasoning may involve connecting one concept to another
(e.g., center and spread) or may combine ideas about data
and chance. Reasoning means understanding and being able
to explain statistical processes, and being able to fully
interpret statistical results.
Source: ARTIST website
Examples of Problems/Questions that Demonstrate Statistical Reasoning
In clinical trials of Nasonex, 3774 adult and adolescent allergy patients
(patients 12 years old and older) were randomly divided into two
groups. The patients in group 1 (experimental group) received 200 µg of
Nasonex, while the patients in group 2 (control group) received a
placebo. Of the 2103 patients in the experimental group, 547 reported
headaches as a side effect. Of the 1671 patients in the control group,
368 reported headaches as a side effect. Is there significant evidence to
conclude that the proportion of Nasonex users that experience
headaches as a side effect is greater than the proportion in the control
group? Are the results practically significant?
The Food and Drug Administration sets Food Defect Action Levels
(FDALs) for some of the various foreign substances that inevitably end
up in the food we eat and liquids we drink. For example, the FDAL for
insect filth in peanut butter is 3 insect fragments (larvae, eggs, body
parts) per 10 grams. A random sample of 50 ten-gram portions of
peanut butter is obtained and results in a sample mean of 3.6 insect
fragments per ten-gram portion. Describe the sampling distribution of
the sample mean.
Examples of Problems/Questions that Demonstrate Statistical Reasoning
Stocks may be categorized by industry. The following data represent the 5-year rates of
return (in percent) for a sample of financial stocks and energy stocks ending December
3, 2007. Which sector is riskier? Does the sector with the higher risk, reward its
investors? Why?
Examples of Problems/Questions that Demonstrate Statistical Reasoning
The following data represent the weights of plain M&Ms candies. Describe the
distribution of weights. Which measure of central tendency is most appropriate to
report? Which measure of dispersion is most appropriate to report? Justify your
recommendations.
Examples of Problems/Questions that Demonstrate Statistical Reasoning
Suppose 100 different researchers wish to estimate the mean amount of time (in
hours) 18 – 24 year old males spend watching television each week. Each
researcher surveys a random sample of forty 18 – 24 year old males and constructs
a 95% confidence interval for the mean time (in hours) 18 – 24 year old males
watch television each week. How many of these intervals do we expect to capture
the population mean?
Emphasize Statistical Thinking
Statistical thinking involves an understanding of why and how statistical
investigations are conducted. This includes recognizing and understanding
the entire investigative process (from question posing to data collection to
choosing analyses to testing assumptions, etc.), understanding how
models are used to simulate random phenomena, understanding how data
are produced to estimate probabilities, recognizing how, when, and why
existing inferential tools can be used, and being able to understand and
utilize the context of a problem to plan and evaluate investigations and to
draw conclusions.
Source: ARTIST website
Examples of Problems/Questions that Demonstrate Statistical Thinking
(a) What makes this a designed
experiment? What type of
experimental design is this?
(b) What is the response
variable? Is it qualitative or
quantitative?
(c) What factors are controlled in
the experiment?
(d) In many experiments, the
researcher will recruit
volunteers and randomly
assign the individuals to a
treatment group. In what
regard was this done for this
experiment?
(e) Did the students perform
better on the final exam in
the fall semester?
(f) Can you think of any factors
that may confound the
results?
Examples of Problems/Questions that Demonstrate Statistical Thinking
What does GAISE say about being statistical literate?
• See the GAISE report pages 5 – 7
• Consider their carpentry analogy:
• In week 1 of the carpentry (statistics) course we learned to use
various kinds of planes (summary statistics). In week 2 we learned
to use different kinds of saws (graphs). Then we learned about using
hammers (confidence intervals). Later we learned about the
characteristics of different types of wood (tests). By the end of the
course we had covered many aspects of carpentry (statistics). But I
wanted to learn how to build a table (collect and analyze data to
answer a question) and I never learned how to do that.
“Many introductory courses contain too much material and students end up with a
collection of ideas that are understood only at a surface level, are not well integrated
and are quickly forgotten.”
- Page 10, the GAISE report
Using Technology to Enhance Students’ Conceptual Understanding - Applets
• Correlation by eye applet
• Draw a scatter diagram where the correlation of the data is about 0.8.
• Draw two scatter diagrams where the correlation of the data is close to 0.
• Draw a scatter diagram with about 6 points where the correlation is about
0.25. Now add another point and move it around the Cartesian plane. How
does this single point impact the value of the correlation coefficient?
• Regression by eye applet
• Draw a scatter diagram where the explanatory and response variable are
negatively associated. Compare SSE for the regression line to an “eye-balled”
line.
• Draw a scatter diagram where the explanatory and response variable are
positively associated. Add another point that may be influential. How does this
point impact the slope and/or intercept?
• Confidence interval applet
• Using either the proportion or mean confidence interval applet, illustrate
the meaning of “level of confidence”
• Using either the proportion or mean confidence interval applet, illustrate
the impact of not meeting the model requirements on the proportion of
intervals that capture the parameter.
Using Technology to Enhance Students’ Conceptual Understanding - Simulations
Illustrate the sampling distribution of the sample mean using MINITAB.
Advantages of Personal Response Systems
• Increased attention
• Increased attendance
• Increased retention
• Draper and Brown
– Students are twice as likely to attempt to
construct an answer to a question using a PRS
compared to a question that required them to
raise their hand.
PRS Transmitter
Types of Questions
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Multiple Choice
True/False
Numeric
Series
Short Answer
Survey
Multiple Choice
From what kinds of variables would side-by-side boxplots be generated?
(a)
(b)
(c)
(d)
(e)
Qualitative only
Quantitative only
One qualitative and one quantitative
Two quantitative
Not sure
Free Response
The reading speed of sixth-grade students is approximately normal, with a mean
speed of 125 words per minute and a standard deviation of 24 words per minute.
What is the probability that a randomly selected sixth-grade student reads less
than 100 words per minute?
Series
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The following represent the steps in the statistical
process. Put them in the correct order.
A)
B)
C)
D)
Draw conclusions from the information
Identify the research objective
Organize and summarize the information
Collect the information needed to answer the
research questions
Camtasia Videos
• Ask students to watch the lecture at
home…then class can be dedicated to
developing the students conceptual
understanding
Sources
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ARTIST website (https://app.gen.umn.edu/artist/index.html)
Chance, Beth L. (2002) Components of Statistical Thinking and Implications for Instruction and
Assessment. Journal of Statistics Education 10, No. 3
delMas, Robert C. (2002) Statistical Literacy, Reasoning, and Learning: A Commentary. Journal of
Statistics Education 10, No. 3
Draper, S. and Brown, M. (2004) Increasing interactivity in lectures using an electronic voting
system. Journal of Computer Assisted Learning 20, 81 – 84
Draper, S. , Cargil, J. and Cutts, Q. (2002) Electronically enhanced classroom interaction. Australian
Journal of Educational Technology 18, 13 – 23
Ebert-May, D., Brewer, C. and Allred, S. (1997) Innovation in large lectures—teaching for active
learning Bioscience 47 601-608
GAISE College Report (www.amstat.org/education/gaise/)
Garfield, Joan (2002) The Challenge of Developing Statistical Reasoning. Journal of Statistics
Education 10, No. 3
Hake, R. (1997) Interactive-engagement vs traditional methods: A six-thousand student survey of
mechanics test data for introductory physics courses. American Journal of Physics
Kennedy, G.E. and Cutts, Q.I. (2005) The association between students’ use of an electronic voting
system and their learning outcomes. Journal of Computer Assisted Learning 21 260 – 268
Rumsey, Deborah (2002) Statistical Literacy as a Goal for Introductory Statistics Courses. Journal of
Statistics Education 10, No. 3
West, J. (Dec. 9, 2005) Learning outcomes related to the use of personal response systems in large
science courses. Academic Commons.
Wit, E. (2003) Who wants to be… The use of a personal response system in statistics teaching.
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