STANDARD 12  PROBABILITY AND STATISTICS
All students will develop an understanding of statistics and
probability and will use them to describe sets of data, model
situations, and support appropriate inferences and arguments.

Standard 12  Probability and Statistics  Grades 34
Overview
Students can develop a strong understanding of probability and
statistics from consistent experiences in classroom activities where a
variety of manipulatives and technology are used. The key components
of this understanding in probability for elementary school students,
as identified in the K12 Overview, are: probability terms, the
concept of the probability of an event, predicting and determining
probabilities, and the relationship between theoretical
and experimental probabilities. In statistics the key components
for elementary school students are data collection, organization,
and representation, central tendency, and analysis and
inference.
Based on their earlier experiences with data, third and
fourthgraders should strengthen their ability to collect,
organize, and represent data. They should build on their informal
discussions of data by developing their ability to analyze data,
formulate hypotheses, and make inferences from the data. As their
numerical skills increase, they should begin to understand and to use
the mean and median, as well as range and mode, as
measures of central tendency. Frequent probability experiments
should help students extend their ability to make predictions
and understand probability as it relates to events around them, and
should provide the intuition they will need in order to determine
probabilities in simple situations.
As in the previous grade levels, probability and statistics
understanding is best developed through frequent opportunities to
perform experiments and gather and analyze data. Such activities are
most valuable when students choose a topic to investigate based on a
real problem or based on an attempt to answer a question of interest
to them. Children should experience new activities, but they should
have the opportunity to revisit problems introduced in grades K2 when
doing so would allow them to practice or develop new understandings.
Probability and statistics are closely related. Students should
use known data to predict future outcomes and they should grapple with
the concept of uncertainty using probability terms such as
likely, not likely, more likely, and less
likely. Developing an understanding of randomness in probability
is crucial to acquiring a more thorough understanding of
statistics.
Third and fourth grade is a wonderful time for students to see
connections among subjects. Most science programs at this level
involve collection and analysis of data as well as a focus on the
likelihood of events. Social studies programs usually ask children to
begin to develop ideas of the world around them. Discussions might
focus on their school, neighborhood, and community. Such explorations
can be enhanced through analysis and discussion of data such as
population changes over the last century. Thirdand fourthgraders
are more attuned to their environment and are more sensitive to media
information than early elementary school children. Discussions about
such things as the claims in TV advertisements or commercials, or
newspaper articles on global warming, help students develop the
ability to use their understandings in real situations.
At all grade levels, probability and statistics provide students
with rich experiences for practicing their skills in content areas
such as number sense, numerical operations, geometry, estimation,
algebra, patterns and functions, and discrete mathematics.
The topics that should comprise the probability and statistics
focus of the mathematics program in grades three and four are:
 collecting, organizing, and representing data
 analyzing data using the concepts of range, mean, median, and mode
 making inferences and formulating hypotheses from their analysis
 determining the probability of a simple event assuming outcomes are equally likely
 making valid predictions based on their understandings of probability
Standard 12  Probability and Statistics  Grades 34
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 3 and
4.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 34 will be such that all students:
1. Formulate and solve problems that involve
collecting, organizing, and analyzing data.
 Students wish to study the differences in temperature
between their hometown and a school they have connected with in Sweden
through the Internet. They exchange highs and lows for each Monday
over the threemonth period from January through March. They note
whether the temperatures are given in degrees Celsius or Fahrenheit,
and use a thermometer with both markings to change from one to the
other if necessary. They organize and represent the data and develop
questions about possible differences in lifestyle that are prompted by
the temperature. They then exchange their questions with their sister
school to learn more about their culture.
 While studying about garbage and recycling, children
notice the amount of waste generated in the cafeteria each day. A
variety of questions begin to surface such as: What types
of waste are there? How much of each? Can we measure it? How? How
often should we measure it to get an idea of the average amount
of waste generated each day? How can we help make less
waste? The class considers how it can find answers to these
questions, designs a way to obtain the data, and finds answers to
their questions.
 Students perform experiments such as rolling a toy car
down a ramp and measuring the distance the car rolled beyond the
bottom of the ramp. This experiment is repeated, holding the top of
the ramp at various heights above the ground. Students discuss the
patterns and relationships they see in the data and use their
discoveries to predict the distances obtained for ramps of other
heights.
2. Generate and analyze data obtained using
chance devices such as spinners and dice.
 Each child in the class rolls a die 20 times and records
the outcomes in a frequency table. The class combines the results in a
class frequency table. They discuss which outcome occurred most often
and least often and then whether the class results differ from their
individual results and why that might be.
 Students make their own cubes from cardstock and label the
sides 1, 2, 2, 3, 4, 5. They roll their cubes 20 times each,
recording the results. After combining their results, the class
discusses the experiment and the reasons the results differ from the
results obtained when using a regular die.
 As a question on a class test, students
are told that Sarah rolled a die 20 times and she got twelve 1s,
two 2s, three 3s, and three 6s. They are
asked what they would conclude about Sarah's experiment and what
might have accounted for her
results.
3. Make inferences and formulate
hypotheses based on data.
 Students read A Three Hat
Day by Laura Geringer. They use concrete objects (different
colored beans, hats, or pattern blocks) to show different orders for
wearing three different hats. They investigate how many different
ways there are to wear four different hats.
 After collecting, organizing, and analyzing data on the
favorite sport of the fourth graders in their school, third graders
are asked to interpret the findings. Why do you suppose
soccer was chosen as the favorite sport? How close were other
sports? What if we collected data on the same question from
fourth graders in another county or another state? Do you
think first graders would answer similarly? Why?
 Students read Mr. Archimedes' Bath and
Who Sank the Boat by Pamela Allen and discuss what happens to
the water level in a container as things are added and why.
 The fourth grade class is planning a walking tour of a
local historic district in February. They want to take hot chocolate
but don't know which type of cup to take so that it stays warm as
long as possible after being poured. In the science unit on the
cooling of liquids, the students discussed notions of variables and
constants. They set up an experiment using cups of the same size but
of different materials and measure the temperatures in each at equal
intervals over a 30minute period. They plot the data and use their
graphs to discuss which cup would be best.
4. Understand and informally use the
concepts of range, mean, mode, and median.
 Before counting the number of raisins contained in each
of 24 individual boxes of raisins, students are asked to estimate the
number of raisins in each box. They count the raisins and compare the
actual numbers to their estimates. Students discover that the boxes
contain different numbers of raisins. They construct a frequency
chart on the blackboard and use the concepts of range, mean, median,
and mode to discuss the situation.
 In a fourth grade assessment, students are asked to
prepare an argument to convince their parents that they need a raise
in their allowance. Students discuss what type of data would be
needed to support their argument, gather the data, and use descriptive
measures as a basis for their argument. In a cooperative effort,
sixth grade students play the part of parents and listen to the
arguments. The sixth graders provide feedback as to whether the
students had enough information to convince them to raise the
allowance and, if not, what more they might use.
5. Construct, read, and interpret displays of data
such as pictographs, bar graphs, circle graphs, tables, and
lists.
 Presented with a display of data from USA TODAY,
students generate questions which can be answered from the display.
Each child writes one question on a 3x5 card and gives it to the
teacher. The cards are shuffled and redistributed to the students.
Each student then answers the question he or she has been given and
checks the answer with the originating student. Disagreements are
presented to the class as a whole for discussion.
 Following a survey of favorite TV shows of students in the
entire third grade, groups of students develop their own pictographs
using symbols of their choosing to represent multiple children.
6. Determine the probability of a simple event assuming
equally likely outcomes.
 Children toss a coin fifty times and record the results as
a sequence of Hs and Ts. They tally the number of heads and tails.
Are there the same number of heads and tails? The children
discuss situations that often lead to misconceptions such as If
three tosses in a row come up heads, what is the chance that
the next toss is a head? Is there a better chance than
there would have been before the other tosses took place? After
what is a lively discussion, the children review their sequence of Hs
and Ts to see what happened on the next toss each time that three
consecutive heads appeared. This analysis should demonstrate that
each result does not depend upon the previous ones.
 Students discuss the probability that a particular number
will come up when a die is thrown, and predict how many times that
number will appear if the die is rolled 50 times. They then toss a die
50 times and compare the results with their predictions.
7. Make predictions that are based on
intuitive, experimental, and theoretical probabilities.
 Fourthgraders are presented with a bag in which there are
marbles of three different colors, the same number of two of the
colors, and twice as many of the third. They are asked what they
would expect to happen if a marble were drawn twelve times and placed
back in the bag after each time. The experiment is performed and the
children discuss whether their estimates of the outcome made sense in
light of the actual outcome.
 During an ecology unit, students discuss the
capturerecapture method of counting wildlife in a local refuge. A
number of animals, say 30 deer, are captured, tagged, and released;
later another group of deer is captured. If five of the twentyfive
recaptured deer are tagged, then you might conclude that about one in
five deer have been tagged, and therefore that the total number of
deer in the refuge is about 5 x 30 or 150. The students perform a
capturerecapture experiment using a large bag of lollipops to
determine the number of lollipops in the bag.
8. Use concepts of certainty, fairness, and
chance to discuss the probability of actual events.
 Students discuss the probability of getting a zero or a
seven on the roll of one die or picking a blue bead from a bag full of
blue beads, and use this as an introduction to a discussion about the
probability of certain events and impossible events.
 Students discuss the relationship between events such as
flipping a coin, a newborn baby being a girl, guessing on a truefalse
question, and other events which have an approximately equal chance of
occurring.
References

Allen, Pamela. Mr. Archimedes' Bath. New
York: Lothrop, Lee, and Shepard Books, 1980.
Allen, Pamela. Who Sank the Boat. New York: Putnam,
1990.
Lindquist, M., et al. Making Sense of Data.
Curriculum and Evaluation Standards for School
Mathematics Addenda Series, Grades K6. Reston,
VA: National Council of Teachers of Mathematics, 1992.
General References

Burton, G., et al. Third Grade Book. Curriculum and
Evaluation Standards for School
Mathematics Addenda Series, Grades K6. Reston, VA:
National Council of Teachers of Mathematics, 1992.
Geringer, Laura. A Three Hat Day. New York: Harper and
Row, 1985
OnLine Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as gradespecific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
