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 56
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 middle school students, as
identified in the K12 Overview, are: probability terms, the
concept of the probability of an event, predicting and determining
probabilities, the relationship between theoretical and
experimental probabilities, and compound events. In
statistics, the key components are: data collection, organization,
and representation, sampling, central tendency, variance and
correlation, and analysis and inference.
In grades K4, students explored basic ideas of statistics by
gathering data, organizing data, and representing data in charts and
graphs, and then using this information to arrive at answers to
questions and raise further questions. Students in grades 5 and 6 are
keenly interested in movies, fashion, music, and sports. These areas
provide a rich source of real problems for students at this age. The
students should make the decision on how to sample and then
collect and organize data. They should determine how best to
represent the data and begin to develop a more formal
understanding of summary statistics of central tendency such as
the mean, the median, and the mode. They should recognize that for
certain types of data, such as height, the mean is an appropriate
measure, but it is inappropriate for other types of data, such as hair
color. These activities should provide opportunities for students to
analyze data and to make inferences regarding the data
and to communicate their inferences in a convincing manner. They
should further develop their understanding of statistics through the
evaluation of arguments by others, whether they come from classmates,
advertising, political rhetoric, or news sources.
While statistical investigations can be similar to those in earlier
grades, fifth and sixthgraders should have access to statistical
software on computers or calculators which have statistical
capability. This will allow them to carry out statistical work using
real data without becoming mired in tedious calculations. The
technology will be used to do the manipulation of the data and the
students will focus on developing their skills in interpreting the
data.
Students enter these grades having participated in a wide variety
of activities designed to help them understand the nature of
probability and chance. The emphasis in grades K4 was primarily on
simple events such as the roll of a die or the flip of one coin. Even
when compound events such as the roll of two dice were considered, the
outcomes were looked upon as a simple event. In grades 5 and 6,
students begin to experiment with compound events such as flips
of several coins and rolls of dice and to predict and evaluate their
theoretical and experimental probabilities. As they develop
their understanding of fractions, ratios, and percents, they should
use them to represent probabilities in place of phrases such as
"three out of four." They begin to model probability
situations and to use these models to predict events which are
meaningful to them.
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. Because
most of the activities are handson and students are constantly
dealing with numbers in a variety of ways, it assists the development
of number sense as well.
The topics that should comprise the probability and statistics
focus of the mathematics program in grades 5 and 6 are:
 collecting, organizing, and representing data
 analyzing data using range and measures of central tendency
 making inferences and hypotheses from their analysis of data
 evaluating arguments based upon data analysis
 interpolating and/or extrapolating from data using a line of best fit
 representing probability situations in a variety of ways
 modeling probability situations
 predicting events based on realworld data
Standard 12  Probability and Statistics  Grades 56
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 5 and
6.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 56 will be such that all students:
9. Generate, collect, organize, and
analyze data and represent this data in tables, charts, and
graphs.
 Students recognize that this is a time of growth for many
of them. The class measures various lengths associated with a person,
such as height, length of forearm, length of thigh, handspan, length
of foot, and armspan. They enter the data into a spreadsheet and
produce various graphs as well as statistical analyses of their
measures. They update their data every month and discuss the change,
both individually and as a class.
 Students survey another class to determine data of
interest, such as the last movies seen, and then organize the data and
produce reports discussing the interests of the grade level.
 Students work on problems like this one from the
New Jersey State Department of Education's Mathematics
Instructional Guide (p. 795):
A fair spinner with 4 congruent regions labeled A, B, C, and D
is spun 20 times by each member of a class of 23 students.
Assume that your class conducted the experiment and obtained
the expected results. Make a bar graph illustrating the
combined class results. Explain why you drew your bar
graph the way you did.
Explain why an individual
student's results might be different from the class
results.
10. Select and use appropriate graphical
representations and measures of central tendency (mean, mode,
and median) for sets of data.
 Students demonstrate understanding of measures of central
tendency by writing a letter to a fictional classmate explaining how
the mean, the median, and the mode each help to describe data. They
then extend their discussion by presenting a picture of an
"average student" in their grade. The picture discusses
height, color of hair, preference in movies, etc. In creating the
picture, the students must choose the appropriate measures of central
tendency based upon the type of data and justify their choice. (For
example, the mean is not appropriate in discussing hair color.) They
will likely want to present pictures of both an "average
boy" and an "average girl" in the class.
 During a social studies unit, students determine a method
to ascertain the value of the homes in their community. They
determine the mean, the median, and the mode for the data and decide
which provides the most accurate picture of the community. They
include in their study homes from different sections of the town.
 Students perform an experiment where one group is
given 10 words in a jumbled order while another group is given them in
a sequence which facilitates remembering them.After giving each group
one minute to study the words, the students are asked to turn their
papers over and write as many of the words as they remember. The
papers are graded by fellow students and the scores reported. After
considering various graphing methods, the students determine that a
boxandwhiskers plot would be the best way to illustrate the
results and compare the two groups.
11. Make inferences and formulate and evaluate
arguments based on data analysis and data displays.
 Students are asked to develop a generalization about their
classmates. They are allowed to make any hypothesis which is
appropriate. For example, some boys might suggest that boys are
stronger than girls or others might say that girls are taller than
boys. They should determine how they would determine the validity of
their hypotheses by designing a data collection activity related to
it.
 The teacher in one fifthgrade class is especially alert
for generalizations that students make about any topic. She writes
them on slips of paper, and keeps them in a box. As an assessment of
the students' ability to develop statistical activities to
validate hypotheses, groups of students pull slips from the box,
develop data collection activities, collect the data, analyze it, and
make reports to the class about the validity of the generalizations
originally made.
 Students are shown a newspaper article which states that
25% of fifth graders have smoked a cigarette. They discuss their
reaction by indicating whether they believe the figure to be correct,
too high, or too low. They then design a survey which they use to
poll their fellow fifth graders in an effort to check the validity of
the claim for the population of their school. They also send a letter
to the newspaper requesting the sources of data for the article and
compare the data in the article with their data.
12. Use lines of best fit to interpolate and predict
from data.
 Given a jar with straight sides and half filled with
water, students drop marbles in five at a time. After each group of
five, they measure the height of the water and record in a table the
number of marbles in the jar and the height of the water. The
students then represent their data in a scatterplot on an xy plane
and find that the points lie almost exactly in a straight line. They
draw a line through the data and use it to determine answers to
questions like: How high will the water be after 25 marbles have
been added? and How many marbles will it take to have
the water reach the top? Activities like this one form the
foundation for understanding graphs in algebra.
13. Determine the probability of a compound
event.
 Students create a table to show all possible results of
rolling two dice. At the left of the rows are the possible rolls of
the first die and at the top of the columns are the possible rolls of
the second die. They complete the table by putting in each cell the
appropriate sum of the number in the top row and the left column.
Counting the number of times each sum appears in the table, they
determine the probability of getting each possible sum. They then roll
two dice 100 times and compare the sums they get with the sums
predicted from the table.
 Students make a list of all possible outcomes when four
coins are tossed and determine thetheoretical probability of having
exactly two heads and two tails.
14. Model situations involving probability, such as
genetics, using both simulations and theoretical
models.
 Students examine the probability of a family with four
children having two boys and two girls by simulating the situation
using four coins. They first choose which side of the coin will
represent males and which will represent females. They toss the set
of coins 50 times and record their results as the number of boys and
the number of girls in each "family." They compare the
results of their experiment with the prediction based on
probability. They also survey a large sample of students in the school
and record the family composition of all families with four children.
All of these are used to discuss the likelihood of an evenlymatched
family.
 A 25cent "prize" machine in the grocery store
contains an equal number of each of six plastic containers with Power
Ranger tattoos. Students are asked to determine how many containers
they need to buy to have a good chance of getting all six. They
simulate this situation with a bag containing an equal number of six
different colored marbles. They draw out, record, and replace one
marble at a time until they have drawn marbles of all six colors,
recording the number of times that took. They repeat the simulation
three times. The class results are gathered and discussed. One issue
discussed is whether the model is a good one for the situation or
whether it should be modified in some way to better represent
reality.
 Students read Caps for Sale by Esphyr
Slobodkina. The peddler in the story sells caps and wears his entire
inventory on his head: a checked cap and four each of identical blue,
gray, and brown hats. Students use concrete objects to model some of
the different orders in which the hats can be worn. They come to
realize that there are many ways and try to discover the total number
of different ways. They search for an efficient way to determine the
number of permutations.
 Students work through the TwoToned Towers
and Pizza Possibilities lessons that are described in the First
Four Standards of the Framework. They make a systematic list
of all the towers built out of four red and blue cubes (or of all the
pizza combinations) and calculate the probability that a tower has
three or four blue cubes.
15. Use models of probability to predict
events based on actual data.
 Students examine weather data for their community from
previous years, and then use their analysis of the data to predict the
weather for the upcoming month. They compare the actual results with
their predictions after the month has passed and then use the
comparison to determine ways to improve their predictions.
 Using data from previous years, students determine the
number of times their favorite professional football team scored a
number of points in each of six ranges of scores (05, 610, 1115,
1620, 2125, and 26 or more). They determine the fraction or
percentage of games the score was in each of those ranges and make a
spinner whose areas are divided the same way. Each Friday during
football season, they spin their spinners to predict how many points
the team will score and who will win the game. Toward the end of the
season, they discuss the success or failure of their efforts and the
probable causes.
16. Interpret probabilities as ratios and
percents.
 The students are introduced to the game Pass The Pigs
(Milton Bradley) where two small hardrubber pigs are rolled. Each
pig can land on a side where there is a dot showing, a side where the
dot does not show, on its hooves, on its back, leaning forward
balancing itself on its snout, and balancing itself on its left
foreleg, snout, and left ear. The students determine the fairness of
the distribution of points on the sides of the pig by rolling the pig
numerous times, recording the results, and using the ratios of
successes for each, divided by the total number of rolls, to represent
the probability of obtaining each situation.
 Students examine uses of probability expressed as
percentages in such situations as weather forecasting, risks in
medical operations, and reporting the confidence interval of
surveys.
References

New Jersey State Department of Education,
Mathematics Instructional Guide: Linking Classroom
Experiences to Current Statewide Assessments.
D. Varygiannes, Coord. Trenton, N.J., 1996.
Slobodkina, Esphyr. Caps for Sale. New York: W.R. Scott,
1947.
General References

Stenmark, J. K., et al. Family Math. Berkeley, CA:
Regents, University of California, 1986.
Zawojewski, Judith, et. al. Dealing with Data and Chance.
A component of the Curriculum and
Evaluation Standards for School Mathematics Addenda Series,
Grades 58. Reston, VA: National Council of Teachers of
Mathematics, 1991.
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.
