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 78
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.
Students should enter the seventh grade with a strong intuitive
understanding of probability and statistics as a result of their
activities in grades K6, and should have a basic understanding of the
more formal methods which were introduced in grades 56. They will
build on this foundation in grades 7 and 8.
Students in grades 7 and 8 present unique challenges. They are
turning to their peer group for leadership and support and, at the
same time, placing a strain on the relationships between themselves
and significant adults in their lives. Some students begin to
experiment with things they associate with being an adult: smoking,
alcohol, drugs, and sex. The quantity of statistics in all of these
areas provides an ideal opportunity to weave together statistical
activities which dovetail with information provided by the the health
and physical education department.
Students at these ages also become more aware of community issues.
Integrating statistics activities with topics in the social studies
curriculum can enhance their work in both areas as well as fit in with
their growing interests. Handson science activities require good
statistical methods and understanding in order to develop accurate and
appropriate conclusions. At the same time, students need to
understand how often statistics and probability statements are
incomplete, misunderstood, or purposely used to mislead. Having
students read books such as How to Lie with Statistics by
Darryl Huff or Innumeracy by John Allen Paulos provides
excellent opportunities to discuss how statistics and probability are
misused.
In statistics, students continue to collect, organize, and
represent data and to use various measures of central
tendency to describe their data. But they should now become more
focused on sampling techniques that justify making
inferences about entire populations. Examples of this appear
frequently in the news media. They also begin to explore variance
and correlation as additional tools in describing sets of
data.
Many of the probability experiments should continue to be related
to games and other fun activities. Students in these grades should
continue to develop their understanding of compound events and
their related probabilities, and should continue to consider and
compare experimental and theoretical probabilities.
Furthermore, the connection between probability and statistics should
help them understand issues such as sampling and
reliability. Students need to develop a sense of the
application of probability to the world around them as well. Everyday
life is rich with "coincidences" which are actually likely
to occur. For example, they should examine the probability that two
people in their class, or any group of 25 or more people, have the
same birthday. The results always stir up considerable interest and
disbelief.
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 7 and 8 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
 analyzing probability situations theoretically
 predicting events based on realworld data
Standard 12  Probability and Statistics  Grades 78
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 7 and
8.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 78 will be such that all students:
9. Generate, collect, organize, and analyze data and
represent this data in tables, charts, and graphs.
 Most students in grades 7 and 8 have major physical growth
activity. Students can continue to maintain the statistics related to
their body that they began to collect in the fifth and sixth grades.
They should continually update what the average person in the grade
would look like in terms of this data.
 In the spring, the social studies teacher and the
mathematics teacher plan a unit on the school board elections.
Students are broken into groups to study questions such as What
percent of the registered voters can be expected to vote?
Will the budget pass? and Who will be elected to the
board of education? Students plan their survey, how they will
choose the sample, how best to gather the data, and how best to report
the information to the class.
10. Select and use appropriate graphical
representations and measures of central tendency (mean, mode,
and median) for sets of data.
 Students study the sneakers worn by students in the
school. They form into a human histogram based upon their brand of
sneakers. The data is recorded and a discussion is encouraged about
the distribution of sneakers throughout the school. Students discuss
in their journals which of the measures of central tendency a sporting
goods store would use in determining which brands to stock and in what
proportion. The students gather prices for a variety of brands and
styles and enter the data into a spreadsheet. They respond in their
journal as to whether and why the mean, median, or mode would be most
useful to discuss sneaker prices.
 Presented with a list of OPEC countries and their
estimated crude oil production in a recent year, students determine
how best to report the data. Some present their graphs as box plots,
others use histograms, and others use circle graphs. They use the
three measures of central tendency and discuss what each result would
mean in this situation and which would be best to use in other
situations.
 Students work on problems like this one from the
New Jersey Department of Education's Mathematics Instructional
Guide (p. 799):
 A set of test scores in Mrs. Ditkof's class of 20
students is shown below.
62 77 82 88 73 64 82 85 90 75
74 81 85 89 96 69 74 98 91 85
Determine the mean, median, mode, and range for the data.
Suppose each student completes
an extracredit assignment worth 5 points, which is then added to
his/her score. What is the mean
of the set of scores now if each student received the extra five
points? Explain how you
calculated your answer.
11. Make inferences and formulate and evaluate
arguments based on data analysis and data displays.
 Students are presented with data from The World
Almanac showing the number of cigarettes smoked per year per adult
and the rate of coronary heart disease in 21 countries. They produce
a scatterplot and recognize a relatively high correlation between the
two factors. They write an essay on the possible causes of this
relationship and their interpretation of it.
 Students are asked to predict how many drops of water will
fit on a penny. They write their prediction on a postit note along
with an explanation of their reasoning. The predictions are collected
and displayed on bar graphs or stemandleaf plots. Students perform
the experiment and record their results on another postit note. They
compare their hypotheses with the conclusions. A science lesson on
surface tension can easily be integrated with this lesson.
 Students are studying their community's recycling
efforts in an integrated unit. In getting ready for discussion in
this area, the mathematics teachers ask the students to predict how
many pounds of junk mail comes in to their community in a month. The
students collect all junk mail sent to their house over the course of
a month. They weigh the junk mail weekly and record the results. At
the end of the month, all the students bring in their data. The class
determines the mean, median, and mode for the collected data, decides
which of these measures would be the best to use, and agrees on a
method to use to estimate the amount of junk mail for the entire
community.
12. Use lines of best fit to interpolate and predict
from data.
 Presented with the problem of determining how long it
would take the wave to go around Giants Stadium, students
design an experiment to gather data from various numbers of students.
They produce a scatterplot and use it to determine a line of best fit.
They pick two points on the line and determine the equation for that
line. Last, they estimate the number of people around the stadium and
answer the question.
 Given some of the winning times for the Men's and
Women's Olympic 100 meter freestyle events during the past
century, students plot the data and produce a line of best fit for
each event. They use their equations to estimate the winning times in
those years for which the information was not recorded, and they
predict when the women's winning times will equal the men's
current winning times.
 Students are presented with an article that states that
police have discovered a human radius bone which is 25 centimeters
long. Students perform measurements of the lengths of radius bones of
varioussized people and their heights, produce a scatterplot, fit a
line to the data, and determine their prediction of the height of the
person whose bone was found. They write a letter to the chief of
police, predicting the height of the person, with justifications for
their conclusion.
13. Determine the probability of a compound
event.
 Students watch the longrange weekend weather forecast and
learn that the probability of rain is 40% on Saturday and 50% on
Sunday. They determine that the probability that it will rain on both
days is 20% by multiplying the two percentages together (.40 x .50 =
.20 or 20%), and similarly then find that the probability that it will
not rain on either day is 30%. Following the weekend, they discuss
the success or failure of their prediction methods.
 Two teams are in a playoff for the division title. If the
probability of the Eagles defeating the Falcons in an individual game
is 40%, what is the probability that they will win a three game
playoff? What about a fivegame playoff?
14. Model situations involving probability, such as
genetics, using both simulations and theoretical
methods.
 During an integrated unit with their science and health
classes, students discuss the various gender possibilities for
children within a family. For each large family, that is, number of
children, up to 6, they calculate the probability of each possible
gender mix. Three groups of students conduct simulations  one
with coins, one with dice (1, 2, or 3 on a die represent a female) and
one with spinners. They also collect this data for all of the
students in their school. They report their findings and compare the
theoretical possibilities, the simulated probabilities, and the actual
outcomes, and discuss the differences and similarities.
 Students study the chances of winning the New Jersey Pick
3 lottery. They model the problem by using spinners with 10 numbers
and calculate the theoretical probability. They may also use a
computer program to randomly generate a million 3digit numbers and
see how close to 1 out of 1000 times their favorite number comes
up.
15. Use models of probability to predict events based
on actual data.
 Students are presented with data collected by an ecologist
tallying the number of deer of one species that died at ages from 1 to
8 years. Students use the data to discuss the probability of living
to various given ages and what they would expect the life expectancy
of this species to be.
16. Interpret probabilities as ratios and
percents.
 Students examine uses of probability expressed as
percentages in such things as weather forecasting, risks in medical
operations, and reporting the confidence interval of surveys.
 Students work on problems like this one from the
New Jersey State Department of Education's Mathematics
Instructional Guide (p. 7103):
A dart board is composed of three concentric circles with radii
2 cm, 10 cm, and 20 cm [as indicated in an accompanying
diagram]. A grand prize is earned if a dart is thrown in the 2
cm circle (bullseye). Given that a person is blindfolded and throws
a dart somewhere on the board, find the probability that the
grand prize will be won when the first dart is thrown. Explain
the process you used to solve the problem.
References

Huff, D. How to Lie with Statistics. New York: Norton,
1954.
Paulos, J. A. Innumeracy: Mathematical Illiteracy and its
Consequences. New York: Hill and Wang, 1988.
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.
