Transcript What is the mean?

Guidelines for Assessment and Instruction
in Statistics Education
A Curriculum Framework for Pre-K-12 Statistics Education
The GAISE Report (2007)
The American Statistical Association
http://www.amstat.org/education/gaise/
Christine Franklin & Henry Kranendonk
NCSM Conference
March 19, 2007
Outline of Presentation
•
Overview of the GAISE Report
•
The Evolution of a
Statistical Concept
- The Mean as Fair Share/Variation from Fair Share
- The Mean as the Balance Point/Variation from the Mean
- The Sampling Distribution of the Mean/Variation in Sample Means
•
Summary
Benchmarks in Statistical Education in
the United States (1980-2007)
• The Quantitative Literacy Project (ASA/NCTM Joint Committee,
Early 1980’s)
• Curriculum and Evaluation Standards for School Mathematics
(NCTM, 1989)
• Principles and Standards for School Mathematics (NCTM, 2000)
• Mathematics and Statistics College Board Standards for College
Success (2006)
• The GAISE Report (2005, 2007)
GOALS of the GAISE Report
• Promote and develop statistical literacy
• Provide links with the NCTM Standards
• Discuss differences between Mathematics and Statistics*
• Clarify the role of probability in statistics*
• Illustrate concepts associated with the data analysis
process*
• Present the statistics curriculum for grades Pre-K-12 as a
cohesive and coherent curriculum strand*
• Provide developmental sequences of learning experiences*
Stakeholders
• Writers of state standards
• Writers of assessment items
• Curriculum directors
• Pre K-12 teachers
• Educators at teacher preparation programs
STATISTICAL THINKING
versus
MATHEMATICAL THINKING
•
The Focus of Statistics on
Variation in Data
•
The Importance of Context
in Statistics
PROBABILITY
Randomization
• Sampling -- "select at random from a
population"
• Experiments -- "assign at random to a
treatment"
THE FRAMEWORK
Underlying Principles
PROBLEM SOLVING PROCESS
Formulate Questions
•
•
clarify the problem at hand
formulate one (or more) questions that can be answered with data
Collect Data
•
•
design a plan to collect appropriate data
employ the plan to collect the data
Analyze Data
•
•
select appropriate graphical or numerical methods
use these methods to analyze the data
Interpret Results
•
•
interpret the analysis taking into account the scope of inference based on the
data collection design
relate the interpretation to the original question
Developmental Levels
•
The GAISE Report proposes three
developmental levels for evolving
statistical concepts.
Levels A, B, and C
The Framework Model
A Two-Dimensional Model
•
One dimension is the four components of the
statistical problem-solving process, along with
the nature of and the focus on variability
•
The second dimension is comprised of three
developmental levels (A, B, and C)
THE FRAMEWORK MODEL
Process
Component
Level A
Level B
Level C
Formulate
Question
Beginning awareness of
the statistics question
distinction
Increased awareness of
the statistics question
distinction
Students can make the
statistics question
distinction
Collect
Data
Do not yet design for
differences
Awareness of design for
differences
Students make designs for
differences
Analyze
Data
Use particular properties
of distributions in context
of specific example
Learn to use particular
properties of distributions
as tools of analysis
Understand and use
distributions in analysis as
a global concept
Interpret
Results
Do not look beyond the
data
Acknowledge that looking
beyond the data is feasible
Able to look beyond the
data in some contexts
THE FRAMEWORK MODEL
Nature of
Variability
Measurement
variability
Natural variability
Induced variability
Sampling variability
Chance variability
Focus on
Variability
Variability within a
group
Variability within a
group and variability
between groups
Variability in model
fitting
Co-variability
Activity Based Learning
•
The GAISE Report promotes active
learning of statistical content and
concepts
Two Types of Learning Activities
•
•
Problem Solving Activities
Concept Activities
The STN article illustrates a
Problem Solving Activity
across the three
developmental levels.
The evolution of a statistical
concept --
•
What is the mean?
•
Quantifying variation
in data from the mean
Level A Activity
The Family Size Problem
A Conceptual Activity for:
•
Developing an Understanding of the Mean as
the “Fair Share” value
•
Developing a Measure of Variation from “Fair
Share”
A Question
How large are families today?
• Nine children were asked how many
people are in your family.
• Each child represented her/his family size
with a collection snap cubes.
Snap Cube Representation for Nine Family Sizes
How might we examine the data on
the family sizes for these nine
children?
2
3
3
4
4
5
6
7
Ordered Snap Cube & Numerical Representations
of Nine Family Sizes
9
Notice that the family sizes vary.
What if we used all our family
members and tried to make all
families the same size, in which
case there is no variability.
How many people would be in each
family?
How can we go about
creating these new families?
We might start by separating
all the family members into
one large group.
All 43 Family Members
Step 1
Have each child select a
snap cube to represent
her/him-self.
These cubes are indicated in
red.
Create Nine “New” Families/Step1
Step 2
Next have each child select
one family member from the
remaining group.
These new family members
are shown in red.
Create Nine “New” Families/Step2
Continue this process until
there are not enough family
members for each child to
select from.
Create Nine “New” Families/Step4
Discuss results
•
The fair share value
Note that this is developing the
division algorithm and
eventually, the algorithm for
finding the mean.
A New Problem
What if the fair share value for
nine children is 6?
What are some different snap
cube representations that
might produce a fair share
value of 6?
Snap Cube Representation of Nine Families,
Each of Size 6
Have Groups of Children
Create New Snap Cube
Representations
For example, following are two
different collections of data
with a fair share value
of 6.
Two Examples with
Fair Share Value of 6.
Which group is “closer”
to being “fair?”
How might we measure “how
close” a group of numeric data
is to being fair?
Which group is “closer”
to being “fair?”
The upper group in
blue is closer to fair
since it requires only
one “step” to make it
fair. The lower group
requires two “steps.”
How do we define a “step?”
•
One step occurs when a snap cube is
removed from a stack higher than the
fair share value and placed on a stack
lower than the fair share value .
•
A measure of the degree of fairness in a
snap cube distribution is the “number of
steps” required to make it fair.
Note -- Fewer steps indicates closer to fair
Number of Steps to
Make Fair: 8
Number of Steps to
Make Fair: 9
Students completing Level A understand:
•
the notion of “fair share” for a set of
numeric data
•
the fair share value is also called the
mean value
•
the algorithm for finding the mean
•
the notion of “number of steps” to make fair as a
measure of variability about the mean
•
the fair share/mean value provides a basis for
comparison between two groups of numerical data
with different sizes (thus can’t use total)
Level B Activity
The Family Size Problem
•
How large are families today?
A Conceptual Activity for:
•
Developing an Understanding of the Mean as
the “Balance Point” of a Distribution
•
Developing Measures of Variation about the
Mean
Level B Activity
How many people are in your family?
Nine children were asked this question.
The following dot plot is one possible
result for the nine children:
-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
Have groups of students create
different dot plot representations
of nine families with a mean of 6.
-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
In which group do the data
(family sizes) vary (differ) more
from the mean value of 6?
1
4
2
1
2
0
1
2
3
-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
0
0
4
3
2
0
2
3
4
-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
In Distribution 1, the Total Distance
from the Mean is 16.
In Distribution 2, the Total Distance
from the Mean is 18.
Consequently, the data in Distribution
2 differ more from the mean than the
data in Distribution 1.
1
4
2
1
2
0
1
2
3
-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
Note that the total distance for the values below
the mean of 6 is 8, the same as the total distance
for the values above the mean. For this reason,
the distribution will “balance” at 6 (the mean)
The SAD is defined to be:
The Sum of the
Absolute Deviations
Note the relationship between SAD and
Number of Steps to Fair from Level A:
SAD = 2xNumber of Steps
Number of Steps to
Make Fair: 8
Number of Steps to
Make Fair: 9
An Illustration where the SAD
doesn’t work!
4
4
-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
1
1
1
1
1
1
1
1
-+--+--+--+--+--+--+--+--+2 3 4 5 6 7 8 9 10
The SAD is 8 for each distribution, but
in the first distribution the data vary
more from the mean.
Why doesn’t the SAD work?
Adjusting the SAD for group sizes
yields the:
MAD = Mean Absolute Deviation
Measuring Variation about the Mean
•
SAD = Sum of Absolute Deviations
•
MAD = Mean of Absolute Deviations
•
Variance = Mean of Squared
Deviations
•
Standard Deviation = Square Root of
Variance
Summary of Level B and Transitions
to Level C
•
Mean as the balance point of a
distribution
•
Mean as a “central” point
•
Various measures of variation about
the mean.
The Mean at Level C
•
At Level C, the notion of the
“Sampling Distribution of the Sample
Mean” is Developed.
•
This development connects probability
and statistics and provides the link
between the descriptive statistics
students have learned at Levels A and
B and concepts of inferential statistics
they will learn at Level C.
Eighty Circles/What is the Mean Diameter?
Activity
• Students select samples of 10 circles they
considered to be representative of the 80
circles. The mean for each sample is
determined.
• Students select simple random samples of
10 circles. The mean for each sample is
determined.
How do the results from
self-selection compare with
random sampling?
•
Following are results from two
introductory-level statistics classes
(50 students).
Dotplot of Random Selection versus Self Selection
Random Selection
Self Selection
1.0
1.2
1.4
1.6
1.8
Sample Means
Population Mean = 1.25
2.0
2.2
Sampling Distributions provide
the link to two important concepts
in statistical inference.
•
Margin of Error
•
Statistical Significance
The STN article in your packet provides
an illustration of how a sampling
distribution is used to develop these
statistical concepts.
SUMMARY: GOALS of GAISE Report
• Promote and develop statistical literacy
• Provide links with the NCTM Standards
• Discuss differences between Mathematics and Statistics
• Clarify the role of probability in statistics
• Illustrate concepts associated with the data analysis process
• Present the statistics curriculum for grades Pre-K-12 as a
cohesive and coherent curriculum strand
• Provide developmental sequences of learning experiences