Finding Common Ground, Indianapolis, March 2006
Statistics, Data Analysis, and Probability Group Report
Working Group:
Christine Franklin, University of Georgia
Bert Fristedt, University of Minnesota
Bonnie Hagelberger, Monroe Elementary School
Katherine Halvorsen, Smith College
Richard Scheaffer, University of Florida (emeritus)
Zsuzanna Szaniszlo, Valparaiso University
Dan Teague, NC School of Science and Mathematics
Every student at some time in their life will make decisions in which
statistical reasoning skills are necessary. This is true for all
students regardless of their plans for post-secondary education.
STATISTICS
Mathematical aspects of statistics, including presentation of data and
the development of statistical reasoning, should be part of the Pre
K-12 mathematics curriculum. The context of a statistical problem
treated in a mathematics class needs to involve appropriate mathematics
for the grade level in which it is used. Statistics is part of
mathematics but the context (story) should not crowd out the
mathematics. Statistical reasoning as taught in mathematics classes
provides an essential foundation for the increasing use of statistics
in other disciplines (sciences, social studies, English, etc.).
Statistical reasoning process
Statistics (Data Analysis in context)
1. Formulate questions (hypotheses)
that can be answered with data
2. Collect data (design investigations)
a. Categorical
b. Numerical (Measurement)
3. Represent data with tables, charts,
and graphs
4. Analyze data (turn data into useful
information)
a. Summarization of data:
quantification of variability
i. Categorical: counts, proportions,
bar graphs
ii. Measurements: plots on the real
number line, describing distributions (shape, center, and spread)
iii. Bivariate association: Two-way
tables and scatterplots; numerical summaries of association
b. Making conjectures, drawing
conclusions, making generalizations, inference
5. Interpret results: Relate the
inference to the original question
6. Probability’s role in data analysis
a. Random selection and margin of error
b. Random assignment and experimental
error
c. Sampling distributions
Items 1 through 5 describe a process that should be repeated, with
increasing sophistication through the grade levels. For example,
at the elementary school level students might collect data from the
second grade class and, based on the data, make conjectures about the
third grade class. At the middle school level students should use
random samples as the basis for evaluating conjectures. At the
high school level students may begin more formal inference using
sampling distributions and margin of error. At the high school
level, statistics should be integrated into the curriculum, not limited
to mathematics.
The study of statistics supports the learning of the arithmetic of
whole numbers, fractions, and decimals in the elementary grades, and it
supports the learning of proportional reasoning and algebra in the
middle grades and beyond.
A further discussion related to these topics may be found in the Pre K-12 Guidelines for Assessment and
Instruction in Statistics Education (GAISE) Report at www.amstat.org/education/gaise/.
(See pages 12 – 15.)
PROBABILITY
The introduction of probability should come after students are exposed
to fractions, although some preparatory notions of probability (more
likely, less likely) should be introduced beforehand. Probability at
the earlier stages meshes well with fractions; at later stages it
meshes well with certain aspects of algebra. Throughout, probability
supports important aspects of statistics.
Probability
0. Pre-probability
a. A sense of probability: more likely,
less likely
b. Informal experiences with counting
techniques: systematic listing of possibilities
1. Experimental (Empirical): Relative
frequency
2. Theoretical (Mathematical): Deriving
a probability from a model; e.g, equally likely outcomes.
a. Sample space
b. The arithmetic of probability
(Addition and Multiplication Rules, Independence)
3. Representation: Tree diagrams, Venn
diagrams, area models, interpreting odds as probabilities
4. Counting techniques based on the
multiplication and the addition principles
5. Probability distributions
a. Binomial
b. Normal
c. Empirical (experimental)
distributions
6. The role of probability in data
analysis
a. Random selection and margin of error
b. Random assignment and experimental
error
c. Sampling distributions
Allocation of Probability Topics to
the K-12 Curriculum
Items 1, 2, 3, and aspects of 4 and 6 are appropriate for introduction
at the middle school level. Item 5 and other aspects of items 4
and 6 belong in high school.
Issues to be continued
- How much context should be included in the mathematics curriculum
and how much belongs to other courses?
- Teachers are receptive to learning statistics and probability,
but because these may not have been a part of their teacher
preparation, they need in-service training and a lot of support in both
content and pedagogy.
Tangentially related issues
that deserve attention from the mathematics community
- Graph Theory – Discrete Math needs further discussion