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 9-12
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 K-12 Overview, are: probability terms, the
concept of the probability of an event, predicting and determining
probabilities, expected value, 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
The field of statistics is relatively new. Beyond the work of
scientists, Florence Nightingale was the great pioneer in gathering
and analyzing statistical data for public health questions. During
the great cholera epidemic of 1854 in London, England, statistics on
the prevalence of cholera cases in various London neighborhoods were
used to deduce that the cholera originated with a single well. In our
own century statistics touches all of us through such diverse means as
statistical quality control in industry, advertising claims,
pre-election polls, television show ratings, and weather forecasts.
To be successful members of present day society, high-school graduates
need an understanding of statistics and probability which formerly was
rare even among college graduates.
By the time students enter high school, they should have mastered
basic descriptive statistical methods. On the basis of their varied
experience, they should be able to set up a study, gather the data,
and appropriately analyze and report their findings. Throughout
grades 9 to 12, students should have numerous opportunities to
continue to practice these skills in a variety of ways, and also to
extend these skills, in connection with their growth in other
mathematical areas. As students learn new algebraic functions, they
might revisit a problem they had previously modeled linearly and apply
a different model. For example, they may have linearly modeled the
series of winning times of the men's Olympic marathon but now
understand that there would probably be a limiting time and so attempt
to fit a quadratic or logarithmic curve instead. Where appropriate,
the content should be developed through a problem-centered approach.
For example, if students are required to generate a report on two sets
of data which have the same measures of central tendency only
to find later they have very different variance, they should
recognize the need for some way to identify that difference.
John Allen Paulos, in his book, Innumeracy, cites numerous
problems associated with a lack of understanding of probability. If
people are to make appropriate decisions, then they must understand
the relationship of probability to real situations and be able to
weigh the consequences against the odds. As with statistics,
probability needs to be experienced, not memorized. Work done at this
level should provide insight into the use of probability and
probability distributions in a variety of real-world situations. The
normal curve presents interesting opportunities to examine uses and
abuses of mathematics.
Students should have access to appropriate technology for their
work in probability and statistics, not only to simplify calculation
and display charts and graphs, but also to generate appropriate data
for activities and projects. They should make use of data taken from
the Internet and CD-ROMs, and simulate experiments with Calculator
Based Laboratories. Whenever possible, real data gathered from
school, the community, or cooperating businesses should be used.
Probability and statistics offers a rich opportunity to integrate
with other mathematics content and other disciplines. This content
provides the opportunity to generate the numbers and situations which
should be used in other areas such as geometry, algebra, functions,
and discrete mathematics. The goal to have students become effective
members of a democratic society requires them to practice and
participate in decision-making experiences. The ability to make
intelligent decisions rests on an understanding of statistics and
probability, and students should regularly integrate this content with
their experiences in social studies, science, and other
The topics that should comprise the probability and statistics
focus of the mathematics program in grades 9 through 12 are:
- designing, conducting, and interpreting statistical work to solve problems
- analyzing data using range, measures of central tendency, and dispersion
- applying probability dispersions in real situations
- evaluating arguments based upon their knowledge of sampling and data analysis
- interpolating and/or extrapolating from data using curve fitting
- using simulations to estimate probabilities
- determining expected values
- using the law of large numbers
Standard 12 - Probability and Statistics - Grades 9-12
Indicators and Activities
The cumulative progress indicators for grade 12 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 9, 10,
11, and 12.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 9-12 will be such that all students:
17. Estimate probabilities and predict outcomes from
- In a standard class test, students are asked to compute
the probability that a given raffle ticket for a senior class raffle
to raise money for the senior trip will win a prize. The class will
be printing 500 tickets that they will sell for $1 each. First prize
is a stereo worth $150. Second prize is a $100 shopping spree in the
local Gap store. Third prize is a $50 gift certificate to The Golden
Goose restaurant. There are ten fourth prizes of a commemorative
T-shirt worth $8 each. Students also compute the expected value of
- Students determine the area of an irregular closed figure
drawn on a large sheet of paper using the Monte Carlo method: Each
person in the group drops a handful of pennies over their shoulder
(without looking) onto the paper containing the figure. They count
the number of coins on the paper (total shots) and the number within
the figure (hits). They thus produce the ratio of hits to total shots
and multiply this fraction by this area of the paper to estimate the
area of the figure.
- Students work through the On the Boardwalk
lesson that is described in the Introduction to the Framework.
In this lesson they explore the probability that a quarter thrown onto
a rectangular grid will land entirely within one of the squares on the
grid, and then discuss how changing the size of the squares will
affect the probability.
- A point P inside a square is selected at random
and is used to form a triangle with vertices A and B of the square.
Students determine the probability that the triangle is acute using a
simulation and a theoretical calculation.
18. Understand sampling and recognize its role in
- While studying United States history, students read about
the prediction in the 1936 presidential race that Alfred Landon would
defeat incumbent President Franklin Delanor Roosevelt. They raise
questions as to why that prediction was so far off and research how TV
stations can forecast winners of some elections with a very small
percentage of the voting results reported. Students contact local
radio and TV stations and newspapers to discover how they determine
their population sample.
19. Evaluate bias, accuracy, and
reasonableness of data in real-world contexts.
- After reading the chapter on sampling in the book How
to Lie With Statistics by Darryl Huff, students bring in ads,
graphs, charts, and articles from newspapers which all makestatements
or claims allegedly based on data. Students examine the articles for
information about the sample and identify those claims which may have
little or no substantiation. They also discuss how the sample
populations chosen could have influenced the outcomes.
- Students take statements such as "50% of the
students failed the test," and "4 out of 5 dentists
recommend" and discuss what data they would need to know in order
to judge if the conclusions were reasonable. How many students took
the test? How many dentists were queried? How were the students or
dentists selected? What factors can be identified which would bias
20. Understand and apply measures of dispersion and
- Students are presented with data gathered by an
archaeologist at several sites. The data identifies the number of
flintstones found at each site and the number of charred bones. The
archaeologist claimed that the data showed that the flintstones were
used to light the fires that charred the bones. Students produce a
scatterplot, find the correlation between the two sets of figures, and
use their work to support or criticize the claim.
- As an assessment activity using their journals, students
respond to the claim that children with bigger feet spell better.
They discuss whether they believe the claim is true, how statistics
might have led to this claim, and whether it has any importance to a
philosophy of language teaching.
21. Design a statistical experiment to study a
problem, conduct the experiment, and interpret and communicate
- Based on a discussion among some members of the class, a
question arises as to which are the most popular cars in the
community. The students work in cooperative groups to design an
experiment to gather the data, analyze the data, and design an
appropriate report format for their results.
- Intrigued by the question How long would it take
dominoes set up one inch apart all the way across the room to
fall?, the class designs an experiment to gather data on smaller
sets of dominoes and then extrapolates to estimate the answer.
- Students have just finished a unit in which they discussed
the capturerecapture method for estimating the population of wildlife.
Part of their assessment for the unit is a project where they work in
groups to design and conduct a simulation of the capturerecapture
method. One group uses the method to determine the number of
lollipops in a large bag.
22. Make predictions using curve fitting and numerical
procedures to interpolate and extrapolate from known
23. Use relative frequency and probability, as
appropriate, to represent and solve problems involving
- After a unit where dependent and independent events were
detailed, students are challenged by a problem containing this
excerpt from The Miami Herald of May 5, 1983.
An airline jet carrying 172 people between Miami and Nassau lost
its engine oil, power, and 12,000 feet of altitude over the
Atlantic Ocean before a safe recovery was made.
When all three engines' low oil pressure warning
lights all lit up at nearly the same time, the
crew's initial reaction was that something was wrong with
the indicator system, not the oil pressure.
They considered the possibility of a malfunction in the
indication system because it's such an unusual
thing to see all three with low pressure indications. The odds are so
great that you won't get three indications like
this. The odds are way out of sight, so the
first thing you would suspect is a problem with the indication
Aviation records show that the probability of an engine failure
in any particular hour is about 0.00004. If the failures of three
engines were independent, what would the probability be of them
failing within one hour? Discuss why the speaker in the article
would refer to such a probability as "way out of sight."
Discuss situations which might make the failures of three engines
not independent events.
- Students keep a record of their trips through the town and
whether or not they have to stop at each of the four traffic lights.
After one month, the data is grouped and studied. Theyuse their data
to determine whether the timing of the lights is independent or
- While discussing the issue of mandatory drug testing in
social studies, students examine the probability of misdiagnosing
people as having AIDS with a test that would identify 99% of those who
are true positives and misdiagnose 3% of those who don't have
AIDS. They examine situations where the prevalence of the disease is
50%, 10%, and 1% using 100,000 people as a base. They discuss the
fact that, at the 1% level, 75% of the people identified as having
AIDS would be false positives, the implications that fact has on
mandatory testing, and potential ways to improve the predictive value
24. Use simulations to estimate
- Students derive the theoretical probability of winning the
New Jersey Pick 6 lottery and then write a computer program to
simulate the lottery. The students enter the winning numbers and the
computer generates sets of 6 numbers until it hits the winning
combination. The computer prints out the number of sets generated
including the winning one. Students run the program several times,
attempting to verify experimentally the theoretical probability they
25. Create and interpret discrete and continuous
probability distributions, and understand their application to
- Students work on a project where they pick one form of
insurance (life, car, home), and determine the variables which affect
the premiums they would need to pay for this type of insurance and
what it would cost for them to obtain it. Using their research, they
write an essay summarizing how insurance companies use statistics and
probabilities to determine their rates.
- An article in Consumer Reports indicates that 25%
of 5lb bags of sugar from a particular company are underweight. The
class works with the local supermarket to develop and perform a
consumer research project. Each group is given a commodity to study
(e.g., potato chips, sugar). They design a method for randomly
selecting and testing whether the product matches the claimed
specifications or not. They use their data to determine the
probability that a randomly selected bag would be underweight.
- Students repeatedly extracted five marbles from a
bag containing 10 red and 10 blue marbles, and each time record the
number of marbles of each color obtained. They combine the data for
the entire class, tabulating the number of times there were 0, 1, 2,
3, 4, and 5 red marbles, and the percentages for each number. They
compared their percentages to the theoretical percentages for this
binomial distribution, and make the connection to the fifth row of
26. Describe the normal curve in general terms, and
use its properties to answer questions about sets of data that
are assumed to be normally distributed.
27. Understand and use the law of large numbers (that
experimental results tend to approach theoretical probabilities
after a large number of trials).
- Students are given two dice, each a different color and
roll them repeatedly. For each roll, they record the result for each
individual die as well as the total. After a large number of rolls
they compare their relative frequencies to the expected outcomes.
Then they combine the totals for the entire class and compare the
experimental results with the theoretical predictions.
- Students are presented with a paper containing the
following gambler's formula: When playing roulette,
bet red. If red does not win, double the bet on red. Continue in
this manner. They evaluate whether the formula makes
sense, identify potential problems, and limitations, and discuss the
fallacy that the odds improve for red to appear on the next roll every
time red doesn't win.
Burrill, Gail, et al. Data Analysis and Statistics Across the
Curriculum. A component of the Curriculum and Evaluation
Standards for School Mathematics Addenda Series, Grades 9-12.
Reston, VA: National Council of Teachers of Mathematics, 1992.
Huff, D. How to Lie with Statistics. New York: Norton,
Paulos, J. A. Innumeracy: Mathematical Illiteracy and its
Consequences. New York: Hill and Wang, 1988.
The North Carolina School of Science and Mathematics.
Contemporary Precalculus Through
Applications. Providence, RI: Janson Publications,
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 grade-specific activities submitted
by New Jersey teachers, and to provide a forum to discuss the