A CURRICULUM FRAMEWORK FOR
PREK12 STATISTICS EDUCATION
Writers
Christine Franklin
Gary Kader
Denise S. Mewborn
Jerry Moreno
Roxy Peck
Mike Perry
Richard Schaeffer
Advisors
Susan Friel
Landy Godbold
Brad Hartlaub
Peter Holmes
Cliff Konold
Presented to the American Statistical Association
Board of Directors for Endorsement
March 2005
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Level A . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 21
Level B . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Level C . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
References. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
Appendix for Level A . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Appendix for Level B . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Appendix for Level C . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A CURRICULUM FRAMEWORK FOR
PREK12 STATISTICS EDUCATION
Introduction
The Ultimate Goal: Statistical Literacy
Citizenship
Personal Choices
Statistical literacy is required for daily personal choices. Statistics provides information on the composition of foods and thus inform our choices at the grocery store. Statistics helps to establish the safety and effectiveness of drugs to help us choose a treatment. Statistics helps to establish the safety of toys to assure that our little ones are not at risk. Our investment choices are guided by a plethora of statistical information about stocks and bonds. The Nielsen ratings decide which shows will survive on television and thus affect what is available. Many products have a previous statistical history and our choices of products can be affected by awareness of this history. The design of an automobile is aided by anthropometrics, the statistics of the human body, to enhance passenger comfort. Statistical ratings of fuel efficiency, safety and reliability are available to help us select a vehicle.
The Workplace and
Professions
Efforts to improve quality and accountability are prominent among the many ways that statistical thinking and tools can be used to enhance productivity. The competitive marketplace demands quality. Quality control practices such as the statistical monitoring of design and manufacturing processes identify where improvement can be made and lead to better product quality. Systems of accountability can help produce more effective employees and organizations, but many accountability systems now in place are not based on sound statistical principles and may, in fact, have the opposite effect from the one desired. Good accountability systems require proper use of statistical tools to determine and apply appropriate criteria.
Science
The Federal Drug Administration requires extensive testing of drugs to determine effectiveness and side effects before they can be sold. A recent advertisement for a drug designed to reduce blood clots stated ÒPLAVIX, added to aspirin and your current medications, helps raise your protection against heart attack or strokeÓ. But the advertisement also warns that ÒThe risk of bleeding may increase with PLAVIX...Ó
This was determined by a clinical trial involving over 12,000 subjects. Among the 6259 taking PLAVIX + aspirin 3.7% showed major bleeding problems while only 2.7% of the 6303 taking the placebo had major bleeding. This is viewed as a Òstatistically significantÓ result.
Statistical literacy involves a healthy dose of skepticism about ÒscientificÓ findings. Is the information about side effects of PLAVIX treatment reliable? A statistically literate person should ask such questions and be able to answer them intelligently. A statistically literate high school graduate will be able to understand the conclusions from scientific investigations and to offer an informed opinion about the legitimacy of the reported results. To quote from Mathematics and Democracy: The Case for Quantitative Literacy (Steen, 2001), such knowledge Òempowers people by giving them tools to think for themselves, to ask intelligent questions of experts, and to confront authority confidently. These are skills required to survive in the modern worldÓ.
Summary
The Case for Statistics Education
Over the past quarter century, statistics (often labeled
data analysis and probability) has become a key component of the K12
mathematics curriculum. Advances in
technology and in modern methods of data analysis of the 1980Õs, coupled with
the data richness of society in the information age, led to the development of
curriculum materials geared toward introducing statistical concepts into the
school curriculum as early as the elementary grades. This grassroots effort was given
sanction by the National Council of Teachers of Mathematics (NCTM) when their
influential document Curriculum and
Evaluation Standards for School Mathematics (NCTM, 1989), included Data
Analysis and Probability as one of the five content strands. As this document and its 2000
replacement entitled Principles and
Standards for School Mathematics (NCTM, 2000) became the basis for reform
of mathematics curricula in many states, the acceptance of and interest in
statistics as part of mathematics education gained strength. In recent years many mathematics
educators and statisticians have devoted large segments of their careers to the
improvement in statistics education materials and pedagogical techniques.
NCTM is not the only group calling for improved statistics education beginning at the school level. The National Assessment of Educational Progress (NAEP, 2005) is developed around the same strands as in the NCTM Standards, with data analysis and probability questions playing an increasingly prominent role in the NAEP exam.
The emerging quantitative literacy movement calls for greater emphasis on practical quantitative skills that will help assure success for high school graduates in life and work; many of these skills are statistical in nature. To quote from Mathematics and Democracy: The Case for Quantitative Literacy (Steen, 2001):
á
Quantitative
literacy, also called numeracy, is the natural tool for comprehending
information in the computer age. The expectation that ordinary citizens be
quantitatively literate is primarily a phenomenon of the late twentieth
century.
á Unfortunately, despite years of study and life experience in an environment immersed in data, many educated adults remain functionally illiterate.
á Quantitative literacy empowers people by giving them tools to think for themselves, to ask intelligent questions of experts, and to confront authority confidently. These are the skills required to thrive in the modern world.
A recent study entitled Ready or Not: Creating a High School Diploma That Counts from the American Diploma Project recommends "must have" competencies needed for high school graduates "to succeed in postsecondary education or in highperformance, high growth jobs" include, in addition to algebra and geometry, aspects of data analysis, statistics, and other applications that are vitally important for other subjects as well as for employment in today's datarich economy.
Statistics education as proposed in this Framework can enable the "must have" competencies for graduates to Òthrive in the modern worldÓ.
NCTM Standards and the
Framework
The main objective of this document is to provide a
conceptual Framework for K12
statistics education. The foundation for this Framework rests on the NCTM Principles
and Standards for School Mathematics (2000).
The Framework is intended to support the objectives of the NCTM Principles and Standards. It is intended to complement the NCTM recommendations, not to supplant them.
The NCTM Principles and Standards describes the statistics content strand as follows.
Instructional programs from prekindergarten through grade 12 should enable all students toÑ
á formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them;
á select and use appropriate statistical methods to analyze data;
á develop and evaluate inferences and predictions that are based on data;
á understand and apply basic concepts of probability.
The Data Analysis and Probability Standard recommends that students formulate questions that can be answered using data and addresses what is involved in gathering and using the data wisely. Students should learn how to collect data, organize their own or others' data, and display the data in graphs and charts that will be useful in answering their questions. This Standard also includes learning some methods for analyzing data and some ways of making inferences and drawing conclusions from data. The basic concepts and applications of probability are also addressed, with an emphasis on the way that probability and statistics are related.
The NCTM Standards elaborates on these themes somewhat and provides examples of the types of lessons and activities that might be used in a classroom. More complete examples can be found in the NCTM Navigation Series on Data Analysis and Probability (20022004). Statistics, however, is a relatively new subject for many teachers who have not had an opportunity to develop sound knowledge of the principles and concepts underlying the practices of data analysis that they are now called upon to teach. These teachers do not clearly understand the difference between statistics and mathematics. They do not see the statistics curriculum for grades K12 as a cohesive and coherent curriculum strand. These teachers may not see how the overall statistics curriculum provides a developmental sequence of learning experiences.
This Framework provides a conceptual structure for statistics education which gives a coherent picture of the overall curriculum. This structure adds to but does not replace the NCTM recommendations.
The Difference between
Statistics and Mathematics
"Statistics is a methodological discipline. It exists not for itself but rather to offer to other fields of study a coherent set of ideas and tools for dealing with data. The need for such a discipline arises from the omnipresence of variability". (Cobb and Moore, 1997)
A major objective of statistics education is to help students develop statistical thinking. Statistical thinking, in large part, must deal with this omnipresence of variability; statistical problem solving and decision making depend on understanding, explaining and quantifying the variability in the data.
It is this focus on variability in data that sets statistics apart from mathematics.
The Nature of Variability
There are many different sources of variability in data. Some of the important sources are described below.
Measurement Variability
Natural Variability
Variability is inherent in nature. Individuals are different. When we measure the same quantity across several individuals we are bound to get some differences in the measurements. Although some of this may be due to our measuring instrument, most of it is simply due to the fact that individuals differ. People naturally have different heights, different aptitudes and abilities, or different opinions and emotional responses. When we measure any one of these traits we are bound to get variability in the measurements. Different seeds for the same variety of bean will grow to different sizes when subjected to the same environment because no two seeds are exactly alike; there is bound to be variability from seed to seed in the measurements of growth.
Induced Variability
If we plant one pack of bean seeds in one field, and another pack of seeds in another location with a different climate, then an observed difference in growth among the seeds in one location with those in the other might be due to inherent differences in the seeds (natural variability) or the observed difference might be due to the fact that the locations are not the same. If one type of fertilizer is used on one field and another type on the other, then observed differences might be due to the difference in fertilizers. For that matter, the observed difference might be due to a factor that we haven't even thought about. A more carefully designed experiment can help us to determine the effects of different factors.
This one basic idea, comparing natural variability to the variability induced by other factors, forms the heart of modern statistics. It has allowed medical science to conclude that some drugs are effective and safe, whereas others are ineffective or have harmful side effects. It has been employed by agricultural scientists to demonstrate that a variety of corn grows better in one climate than another, that one fertilizer is more effective than another, or one type of feed is better for beef cattle than another.
Sampling Variability
In a voter poll, it seems reasonable to use the proportion of voters surveyed (a sample statistic) as an estimate of the unknown proportion of all voters who support a particular candidate. But if a second sample of the same size is used, it is almost certain that there would not be exactly the same proportion of voters in the sample who support the candidate. The value of the sample proportion will vary from sample to sample. This is called sampling variability. So what is to keep one sample from estimating that the true proportion is .60 and another from saying it is .40 ? This is possible but unlikely if proper sampling techniques are used. Poll results are useful because these techniques and an adequate sample size can assure that unacceptable differences among samples are quite unlikely.
An excellent discussion on the nature of variability is given in (Utts,1999).
The Role of Context
"The focus on variability naturally gives statistics a
particular content that sets it apart from mathematics itself and from other
mathematical sciences, but there is more than just content that distinguishes
statistical thinking from mathematics. Statistics requires a different kind of thinking, because data are not just numbers, they are numbers
with a context". (Cobb and Moore,1997)
Many mathematics problems arise from applied contexts, but the context is removed to reveal mathematical patterns.
Statisticians, like mathematicians, look for patterns, but the meaning of the patterns depends on the context.
"In mathematics, context obscures structure. In data analysis, context provides meaningÓ.
(Cobb and Moore, 1997)
A graph, which appears occasionally in the business section of newspapers, shows a plot of the Dow Jones Industrial Average (DJIA) over a tenyear period. The variability of stock prices draws the attention of an investor. This stock index may go up or down over some intervals of time, may fall or rise sharply over a short period. In context the graph raises questions. A serious investor is not only interested in when or how rapidly the index goes up or down, but also why. What was going on in the world when the market went up, what was going on when it went down? But strip away the context. Remove time (years) from the horizontal axis and call it "X", remove stock value (DJIA) from the vertical axis and call it "Y", and there remains a graph of very little interest or mathematical content!
Probability
Probability is a tool for statistics
Probability is an important part of any mathematical education. It is a part of mathematics that enriches the subject as a whole by its interactions with other uses of mathematics. Probability is an essential tool in applied mathematics and mathematical modeling. It is also an essential tool in statistics.
The use of probability as a mathematical model and the use of probability as a tool in statistics employ not only different approaches, but also different kinds of reasoning.
Two problems and the nature of the solutions will illustrate the difference.
Problem 1
Assume a coin is "fair" .
Question: If we toss the coin 5 times, how many heads will we get?
Problem 2
You pick up a coin.
Question: Is this a fair coin?
Problem 1 is mathematical probability problem.
Problem 2 is a statistics problem that can use the mathematical probability model determined in problem 1 as a tool to seek a solution.
The answer to neither question is deterministic. Coin tossing produces random outcomes, which suggests that the answer is probabilistic. The solution to problem 1 starts with the assumption that the coin is fair and proceeds to logically deduce the numerical probabilities for each possible number of heads 0,1, ....,5.
The solution to problem 2 starts with an unfamiliar coin; we don't know if it is fair or biased. The search for an answer is experimental  toss the coin and see what happens. Examine the resulting data to see if it looks like it came from a fair coin or a biased coin. There are several possible approaches, including: Toss the coin 5 times and record the number of heads. Then do it again: Toss the coin 5 times and record the number of heads. Repeat 100 times. Compile the frequencies of outcomes for each possible number of heads. Compare these results to the frequencies predicted by the mathematical model for a fair coin in problem 1. If the empirical frequencies from the experiment are quite dissimilar from those predicted by the mathematical model for a fair coin and are not likely to be caused by random variation in coin tosses, then we conclude the coin is not fair. In this case we induce an answer by making a general conclusion from observations of experimental results.
Probability and Chance Variability
Two important uses of "randomization" in statistical work occur in sampling and experimental design. When sampling we "select at random" and in experiments we randomly assign individuals to different treatments". Randomization does much more than remove bias in selections and assignments. Randomization leads to chance variability in outcomes that can be described with probability models.
The probability of something says about what percentage of the time it is expected to happen when the basic process is repeated over and over again.
Probability theory does not say very much about one toss of the coin; it makes predictions about the longrun behavior of the coin tosses.
Probability tells us little about the consequences of random selection for one sample but describes the variation we expect to see in samples when the sampling process is repeated a large number of times.
Probability tells us little about the consequences of random assignment for one experiment but describes the variation we expect to see in the results when the experiment is replicated a large number of times.
When randomness is present, the statistician wants to know if the observed result is due to chance, or something else. This is the idea of statistical significance.
The Role of Mathematics
in Statistics Education
The evidence that statistics is different from mathematics is not presented to argue that mathematics is not important to statistics education or that statistics education should not be a part of mathematics education. To the contrary, statistics education becomes increasingly mathematical as the level of understanding goes up.
But data collection design, exploration of data, and the interpretation of results should be emphasized in statistics education for statistical literacy. These are heavily dependent on context, but at the introductory level involve limited formal mathematics.
Probability plays an important role in statistical analysis, but formal mathematical probability should have its own place in the curriculum. Precollege statistics education should emphasize the ways that probability is used in statistical thinking; an intuitive grasp of probability will suffice at these levels.
The Framework
Underlying Principles
Statistical Problem
Solving
Statistical problem solving is an investigative process that involves four components:
Formulate Questions
á clarify the problem at hand
á formulate one (or more) questions that can be answered with data
Collect Data
á design a plan to collect appropriate data
á employ the plan to collect the data
Analyze Data
á select appropriate graphical or numerical methods
á use these methods to analyze the data
Interpret Results
á interpret the analysis
á relate the interpretation to the original question.
The Role of Variability in the Problem Solving Process
Formulate Question
Anticipating Variability Making the statistics question distinction
The formulation of a statistics question requires an understanding of the difference between a question that anticipates a deterministic answer and a question that anticipates an answer based on data that vary.
The question "How tall am I?" will be answered
with a single height. It is not a statistics question. The question "How
tall are adult men in the
The poser of the question "How does sunlight affect the growth of a plant?" should anticipate that the growth of two plants of the same type exposed to the same sunlight will likely differ. This is a statistics question.
The anticipation of variability is the basis for understanding of the statistics question distinction; these are required for proper question formulation.
Collect Data
Acknowledging Variability Designing for differences
Data collection designs must acknowledge variability in data and frequently are intended to reduce variability. Random sampling is intended to reduce the differences between sample and population, and the sample size influences the effect of sampling variability (error). Experimental designs are chosen to acknowledge the differences between groups subjected to different treatments. Random assignment to the groups is intended to reduce differences between the groups due to factors that are not manipulated in the experiment. Some experimental designs pair subjects so that they are similar. Twins are frequently paired in medical experiments so that observed differences might be more likely attributed to the difference in treatments rather than differences in the subjects.
The understanding of data collection designs that acknowledge differences is required for effective collection of data.
Analyze Data
Accounting of VariabilityUsing
Distributions
The main purpose of statistical analysis is to give an accounting of the variability in the data. When results of an election poll state that "42% of those polled support a particular candidate with margin of error +/ 3% at the 95% confidence levelÓ, the focus is on sampling variability. The poll gives an estimate of the support among all voters. The margin of error indicates how far the sample result (42%+/3%) might differ from the actual percentage of all voters who support the candidate. The confidence level tells us how often estimates produced by the method employed will produce correct results. This analysis is based on the distribution of estimates from repeated random sampling.
When test scores are described as "normally distributed with mean 450 and standard deviation 100" the focus is on how the scores differ from the mean. The normal distribution describes a bellshaped pattern of scores and the standard deviation indicates the level of variation of the scores from the mean.
Accounting for variability with the use of distributions is the key idea in the analysis of data.
Interpret Results
Allowing for VariabilityLooking beyond
the data
Statistical interpretations are made in the presence of variability and must allow for it.
The result of an election poll must be interpreted as an estimate that can vary from sample to sample. The generalization of the poll results to the entire population of voters looks beyond the sample of voters surveyed and must allow for the possibility of variability of results among different samples. The results of a randomized comparative medical experiment must be interpreted in the presence of variability due to the fact that different individuals respond differently to the same treatment as well as the variability due to randomization. The generalization of the results looks beyond the data collected from the subjects who participated in the experiment and must allow for these sources of variability.
Looking beyond the data to make generalizations must allow for variability in the data.
Maturing over Levels
The mature statistician understands the role of variability in the statistical problem solving process. At the point of question formulation, the statistician anticipates the data collection, the nature of the analysis, and the possible interpretations, all of which must consider possible sources of variability. In the end, the mature practitioner reflects upon all aspects of data collection and analysis as well as the question itself when interpreting results. Likewise he links data collection and analysis to each other and the other two components.
The beginning student cannot be expected to make all of these linkages. They require years of experience as well as training. Statistical education should be viewed as a developmental process. To meet the proposed goals, this report will provide a framework for statistical education over three levels. If the goal were to produce a mature practicing statistician, there would certainly be several levels beyond these. There is no attempt to tie these levels to specific grade levels.
The Framework uses three developmental Levels, A, B, and C. Although these three levels may parallel grade levels, they are based on development, not age. Thus, a middle school student who has had no prior experience with statistics will need to begin with Level A concepts and activities before moving to Level B. This holds true for a secondary student as well. If a student hasn't had Level A and B experiences prior to high school, then it is not appropriate to jump into Level C expectations. The learning is more teacherdriven at Level A, but becomes student driven at Levels B and C.
The Framework Model
The conceptual structure for statistics education is provided in the twodimensional model shown in Figure 1. One dimension is defined by the problemsolving process components plus the nature of the variability considered and how we focus on variability. The second dimension is comprised of the three developmental levels.
Each of the first four rows describes a process component as it develops across levels. The fifth row indicates the nature of the variability considered at a given level. It is understood that work at Level B assumes and develops further the concepts from Level A, and likewise Level C assumes and uses concepts from the lower levels.
Figure 1: The Framework
Process Component 
Level A 
Level B 
Level C 
Formulate Question 
Beginning awareness
of the statistics question distinction Teachers pose questions of interest. Questions restricted to classroom 
Increased awareness
of the statistics question distinction. Students begin to pose their own questions of interest. Questions not restricted to classroom 
Students can make
the statistics question distinction. Students pose their own questions of interest. Questions seek generalization 
Collect Data 
Do not yet design for differences Census
of classroom Simple experiment 
Beginning awareness of design for differences Sample surveys Begin to use random selection Comparative experiment Begin to use random allocation 
Students make designs for differences Sampling designs with random selection Experimental designs with randomization 
Process Component 
Level A 
Level B 
Level C 
Analyze Data 
Use particular properties of distributions in context of specific example Display variability within a group Compare individual to individual Compare individual to group 
Learn to use particular properties of distributions as tools of analysis Quantify variability within a group Compare group to group in displays Acknowledge sampling error Some quantification of association Simple models for association 
Understand and use distributions in analysis as a global concept Measure variability within a group Measure variability between groups Compare group to group using displays and measures of variability
Describe and quantify sampling error Quantification of association Fitting of Models for association 
Process Component 
Level A 
Level B 
Level C 
Interpret Results 
Do not look beyond the data No generalization beyond the classroom Note difference between two individuals with different conditions Observe association in displays 
Acknowledge that looking beyond the data is feasible Acknowledge that a sample may or may not be representative of larger population Note difference between two groups with different conditions Aware of distinction between observational study and experiment Note differences in strength of association Basic interpretation of models for association Aware of the distinction between ÒassociationÓ and Òcause and effectÓ 
Are
able to look beyond the data in some contexts Generalize from sample to population Aware of the effect of randomization on the results of experiments Understand the difference between observational studies and experiments Interpret measures of strength of association Interpret models for association Distinguishes between conclusions from association studies and experiments. 
Process Component 
Level A 
Level B 
Level C 

Nature of Variability Focus on Variability 
Measurement variability Natural variability Induced variability Variability within a group 
Sampling variability Variability within a group and variability between groups Covariability 
Chance variability. Variability in model fitting 


Illustrations
All four steps of the problem solving process are used at all three levels, but the depth of understanding and sophistication of methods used increases across the Levels A, B, C. This maturation in understanding the problem solving process and its underlying concepts is paralleled by an increasing complexity in the role of variability. The illustrations of learning activities given here are intended to clarify the differences across the developmental levels for each component of the problem solving process. A later section in this report will give illustrations of the complete problem solving process for learning activities at each level.
Formulate Question
Example 1
A: How long are the words on this page?
B: Are the words in a chapter of a fifth grade book longer than the words in a chapter of a third grade book?
C: Do fifth grade books use longer words than third grade books?
Example 2
A: What type of music is most popular among students in our class?
B: How do the favorite types of music compare among different classes?
C: What type of music is most popular among students in our school?
Example 3
A: In our class, are the heights and arm spans of students approximately the same?
B: Is the relationship between arm span and height for the students in our class the same as the relationship between arm span and height for the students in another class?
C: Is height a useful predictor of arm span for the students in our school?
Example 4
A: Will a plant placed by the window grow taller than a plant placed away from the window?
B: Will five plants placed by the window grow taller than five plants placed away from the window?
C: How does the level of sunlight affect the growth of a plant?
Collect Data
Example 1
A: How long are the words on this page?
The length of every word on the page is determined and recorded.
B: Are the words in a chapter of a fifth grade book longer than the words in a chapter of a third grade book?
A simple random sample of words from each chapter is used.
C: Do fifth grade books use longer words than third grade books?
Other sampling designs are considered, compared and some are used. For example, rather than select words in a simple random sample, a simple random sample of pages from the book is selected and all of the words on the pages chosen are used for the sample.
Note At each level, issues of measurement should be addressed. The length of word depends on the definition of ÒwordÓ. For instance, is a number a word? Consistency of definition is important to reduce measurement variability.
Example 2
A: Will a plant placed by the window grow taller than a plant placed away from the window?
A seedling is planted in a pot that is placed on the window sill. A second seedling of the same type and size is planted in a pot that is placed away from the window sill. After six weeks the change in height for each is measured and recorded.
B: Will five plants of a particular type placed by the window grow taller than five plants of the same type placed away from the window?
Five seedlings of the same type and size are planted in a pan which is placed on the window sill. Five seedlings of the same type and size are planted in a pan which is placed away from the window sill. Random numbers are used to decide which plants go in the window. After six weeks the change in height for each seedling is measured and recorded.
C: How does the level of sunlight affect the growth of plants?
Fifteen seedlings of the same type and size are selected. Three pans are used, with five of these seedlings planted in each. Fifteen seedlings of another variety are selected to determine if the effect of sunlight is the same on different types of plants. Five of these are planted in each of the three pans. The three pans are placed in locations with three different levels of light. Random numbers are used to decide which plants go in which pan. After six weeks the change in height for each seedling is measured and recorded.
Note At each level, issues of measurement should be addressed. The method of measuring change in height must be clearly understood and applied in order to reduce measurement variability.
Analyze Data
Example 1
A: What type of music is most popular among students in our class?
A bar graph is used to display the number of students who choose each music category.
B: How do the favorite types of music compare among different classes?
For each class, a bar graph is used to display the percentage of students who choose each music category. The same scales are used for both graphs so that they can easily be compared.
C: What type of music is most popular among students in our school?
A bar graph is used to display the percentage of students who choose each music category. Because a random sample is used, an estimate of the margin of error is given.
Note At each level, issues of measurement should be addressed. A questionnaire will be used to gather studentsÕ music preferences. The design and wording of the questionnaire must be carefully considered to avoid possible biases in the responses. The choice of music categories could also affect results.
Example 2
A: In our class, are the heights and arm spans of students approximately the same?
The difference between height and arm span is determined for each individual.
An XY plot is constructed with X=height, Y=arm span. The line Y=X is drawn on this graph.
B: Is the relationship between arm span and height for the students in our class the same as the relationship between arm span and height for the students in another class?
For each class, an XY plot is constructed with X=height, Y=arm span. An "eye ball" line is drawn on each graph to describe the relationship between height and arm span. The equation of this line is determined. An elementary measure of association is determined.
C: Is height a useful predictor of arm span for the students in our school?
The least squares regression line is determined and assessed for use as a prediction model.
Note At each level, issues of measurement should be addressed. The methods used to measure height and arm span must be clearly understood and applied in order to reduce measurement variability. For instance, do we measure height with shoes on or off?
Interpret Results
Example 1
A: How long are the words on this page?
The frequency plot of all word lengths is examined and summarized. In particular, students will note the longest and shortest word lengths, the most common lengths and least common lengths, and the length in the middle.
B: Are the words in a chapter of a fifth grade book longer than the words in a chapter of a third grade book?
The students interpret a comparison of the distribution of a sample of word lengths from the fifth grade book with the distribution of word lengths from the third grade book using a boxplot to represent each of these. The students also acknowledge that samples are being used which may or may not be representative of the complete chapters.
The boxplot for a sample of word
lengths from the fifth grade book is placed beside the boxplot of the sample
from the third grade book.
C: Do fifth grade books use longer words than third grade books?
The interpretation at Level C includes the interpretation at Level B, but also must consider generalizing from the books included in the study to a greater population of books.
Example 2
A: Will a plant placed by the window grow taller than a plant placed away from the window?
In this simple experiment, the interpretation is just a matter of comparing one measurement of change in size to another.
B: Will five plants placed by the window grow taller than five plants placed away from the window?
In this experiment, the student must interpret a comparison of one group of five measurements with another group.
If a difference is noted, then the student acknowledges that is likely caused by the differences in light conditions.
C: How does the level of sunlight affect the growth of a plant?
There are several comparisons of groups possible with this design. If a difference is noted, then the student acknowledges that it is likely caused by the differences in light conditions or the differences in types of plants. It is also acknowledged that the randomization used in experiment can possibly cause some of the observed differences.
Nature of Variability
Variability Within a Group
This is the only type considered at Level A. In Example 1, differences among word lengths on a single page are considered; this is variability within a group of word lengths. In Example 2, differences among how many students choose each category of music are considered; this is variability within a group of frequencies.
Variability Within a Group and Variability Between Groups
At Level B, students begin to make comparisons of groups of measurements. In Example 1, a group of word lengths from a fifth grade book are compared to a group from a third grade book. Such a comparison not only notes differences between the two groups such as the difference between median or mean word lengths, but must also take into consideration how much word lengths differ within each group.
Induced Variability
In Example 4, Level B, the experiment is designed to determine if there will be a difference between the growth of plants in sunlight and the growth of those away from sunlight. We want to determine if an imposed difference on the environments will induce a difference in growth.
Sampling Variability
In Example 1, Level B, samples of words from a chapter are used. Students observe that two different samples will produce different groups of word lengths. This is sampling variability.
Covariability
Example 3, Level B or C, investigates the "statistical" relationship between height and arm span. The nature of this statistical relationship is described in terms of how the two variables "covary". For instance, if the heights of two students differ by 2 centimeters then we would like for our model of the relationship to tell us by how much we might expect their arm spans to differ.
.
Random Variability from Sampling
When random selection is used, then differences between samples will be random. Understanding this random variation is what leads to the predictability of results. In Example 2, Level C, this random variation is not only considered but it is also the basis for understanding the concept of margin or error.
Random Variability Resulting from Assignment to Groups in Experiments
In Example 4, Level C, plants are randomly assigned to groups. Students consider how this randomization might produce differences in results, although a formal analysis is not done.
Random Variation in Model Fitting
In Example 3, Level C, students assess how well a regression line will predict arm span from height. This assessment is based on the notion of random differences between actual arm spans and the arm spans predicted by the model.
Detailed Descriptions of Each Level
As this document transitions
into detailed descriptions of each level, it is important to note that the
examples selected for illustrating key concepts and the problem solving process
of statistical reasoning are based on real data and real world context. The stakeholders reading the document will
need to be flexible and adaptable in using these examples to fit their teaching
needs and situation.
Level A
Objectives of Level A
Children are surrounded by data. They may think of data as tallying a
studentÕs favorite object or as measurements, such as arm span and number of
books in their school bag, on other students in their classroom such as arm
span and number of books in their school bag.
¤
It is in Level A
that children need to develop data sense  an understanding that data are more
than just numbers. Statistics changes numbers into information.
¤
Students should
learn that data are generated with respect to particular contexts or situations
and can be used to answer questions about the context or situation.
¤
Students should
have opportunities to generate questions about a particular context (such as
their classroom) and determine what data might be collected to answer these
questions.
¤
Students should
learn how to use basic statistical tools to analyze the data and make informal
or casual inferences in answering the posed questions.
¤
Students should
develop basic ideas of probability in order to support their later use of
probability in drawing inferences at Levels B and C.
Statistics helps us make better decisions. It is preferable that
students actually collect data but not necessary in every case. Teachers should
take advantage of naturally occurring situations in which students notice a
pattern about some data and begin to raise questions. For example, when taking
daily attendance one morning, students might note that many students are
absent. The teacher could capitalize on this opportunity to have the students
formulate questions that could be answered with attendance data.
Specifically, Level A recommendations in the Investigative Process
include:
1. Formulate the Question
Teachers
help pose questions (Questions in contexts of interest to the student)
Students
distinguish between statistical solution and fixed answer
2. Collect Data to Answer the Question
Students
conduct a census of the Classroom
Students
understand individualtoindividual variability
Students
conduct simple experiments with nonrandom assignment of treatment
Students
understand variability attributable to an experimental condition
3. Analyze the Data
Students
compare individual to individual
Students
compare individual to a group
Students
understand the idea of a distribution
Students
describe a distribution
Students
observe association between two variables
Students
use tools for exploring distributions and association, including,
Bar
Graph
Dotplot
Stem
and Leaf Plot
Scatterplot
Tables
(using counts)
Mean,
Median, Mode, Range
Modal
Category
4. Interpret Results
Students
infer to the classroom
Students
acknowledge results may be different in another class or group
Students
recognize the limitation of scope of inference to the classroom
Example 1: Choosing the Band for the End of the Year PartyÐConducting a Survey
Children at Level A may be
interested in the favorite type of music among students at a certain grade
level. An end of the year party is being planned and there is only enough money
to hire one musical group for the party. The class might investigate the
question: What
type of music is most popular among students?
This question attempts to measure a characteristic in the
population of the schoolgrade children that will have the dance. The
characteristic, favorite music type is a categorical variable  each child in
that grade would be placed in a particular nonnumerical category based on his
or her favorite music type. The resulting data are often called Categorical Data.
The Level A class would most likely conduct a census of the students in a particular classroom to gauge what the favorite music type might be for the whole grade.
At Level A, we want students to
recognize there will be individualtoindividual variability.
For example, a survey of 24 students in one of the classrooms of a particular grade is taken with the data summarized below in the frequency count table. This frequency count table is a tabular representation that takes Level A students to a summative level for categorical data. Students might first use tally marks by counting to record the measurements of categorical data before finding frequencies (count) for each category.
Table 1:Frequency count table
Favorite Frequency or Count
Country 8
Rap 12
Rock 4
A Level A student might first use a picture graph to represent the tallies for each category. A picture graph uses a picture of some sort (such as a type of musical band) to represent each element. Thus, each child who favors a particular music type would put their cutout of that type of band directly onto the graph the teacher has created on the board. Instead of a picture of a band, another representation, such as a picture of a guitar, an X or a square, can be used to represent each element (or tally) of the data set. A child who prefers ÔCountryÕ would go to the board and place a guitar, dot or X or color in a square above the column labeled ÒCountry.Ó In both of these cases, there is a deliberate recording of each element (or tally), one at a time.
Figure 2: Picture graph of music preferences
Our Favorite Type of Tunes
Number of People Who Like This Kind of
Music 
12 



11 



10 



9 



8 


7 


6 


5 


4 

3 

2 

1 


Country 
Rap 
Rock 

Type of Music

Note that a picture graph refers to a graph where an object such as a construction paper cutout is used to represent one element on the graph. (A cutout of a tooth might be used to record how many teeth were lost by children in a kindergarten class each month.) The term pictograph is often used to refer to a graph in which a picture or symbol is used to represent several items that belong in the same category. For example, on a graph showing the distribution of car riders, walkers, and bus riders in a class, a cutout of a school bus might be used to represent 5 bus riders. Thus, if the class had 13 bus riders, there would be approximately 2.5 busses on the graph. This type of graph requires a basic understanding of proportional or multiplicative reasoning, and for this reason we do not advocate its use at Level A except possibly with students who are nearly ready for Level B. Similarly, circle graphs require an understanding of proportional reasoning, so we do not advocate their use at Level A except possibly at the top of Level A.
A bar graph takes the student to the summative level with the data summarized from some other representation, such as a picture graph or a frequency count table. The bar on a bar graph is drawn as a continuous rectangle reaching up to the desired number on the yaxis.
A bar graph is displayed below for the census taken of the classroom represented in the above frequency count table and the picture graph.
Figure 3: Bar graph of music preferences
Students at Level A should recognize the mode as a way to describe a ÔrepresentativeÕ or ÔtypicalÕ value for the distribution.
The mode is the representative value that students naturally use first. The mode is most useful for categorical data. Students should understand that the mode is the category that contains the most data points, often referred to as the modal category. In our favorite music example, rap music was preferred by more children, thus the mode or modal category of the data set is rap music. Students could use this information to help the teachers in seeking a musical group for the end of the year party that specializes in rap music.
The vertical axes on the bar
graphs constructed above could be scaled in terms of the proportion or
percentage of the sample size for each category. Since this involves proportional
reasoning, converting frequencies to proportions (or percentages) will be
developed in Level B.
Because most of the data collected at Level A will involve a
census of the studentÕs classroom, the first stage is for students to learn to
read and interpret at a simple level what the data show about their own class.
It is important to consider the sort of question, ÒWhat might have caused the data to look like this?Ó
It is important for children to think about if and how their findings would Òscale upÓ to a larger group, such as the entire grade level, the whole school, all children in the school system, all children in the state, or all people in the nation. They should note variables (such as age or geographic location) that might affect the data in the larger set. In the music example above, students might speculate that if they collected data on music preference from their teachers, the teachers might prefer a different type of music. Or what would happen if they collected music preference from middle school students in their school system? Level A students should begin recognizing the limitations of the scope of inference to a specific classroom.
Comparing Groups
Students at Level A may be interested in comparing two
distinct groups with respect to some characteristic of those groups. For
example, is there a difference in two groups, boys and girls, with respect to
student participation in sports? The characteristic Òparticipation in sportsÓ is
categorical (yes or no). The resulting categorical data for each gender may be
analyzed using a table of frequencies or bar graph. Another question Level A
students might ask is whether there is a difference between boys and girls with
respect to the distance they can jump, an example of taking measurements on a numerical variable. Data on numerical variables are obtained from
situations that involve taking measurements such as height, temperatures or
where objects are counted (e.g., determining the number of letters in your
first name, the number of pockets on clothing worn by children in the class, or
the number of siblings each child has). These are often called Numerical Data.
Returning to the question of comparing boys and girls with
respect to jumping distance, students may measure the jumping distance for all
their classmates. Once the numerical data are gathered, the children might
compare the lengths of jumps of girls and boys using a backtoback ordered stem and leaf plot like the one below.
Figure 4: Stem and leaf plot of jumping distances
Girls 

Boys 

8 


7 


6 
1


5 
2
6 9 
9 7 2 
4 
1
3 5 5 5 
5 5 3 3 3 2 1 
3 
1
1 2 5 6 7 
9 8 7 7 6 4 4 3 2 
2 
2
3 4 6 

1 

Inches jumped
in the standing broad jump
From the stem and leaf plot, the students can get a sense of shape (more symmetric for the boys than for the girls) with boys tending toward having longer jumps. Looking ahead, in Level C, the above examples of data collection design will be more formally discussed as examples of observational studies. The researcher has no control over which students go into the boy and girl groups (the preexisting condition of gender defines the groups). The researcher then merely observes and collects measurements on characteristics within each group.
The Simple Experiment
Another type of design for collecting data appropriate at
Level A is a simple experiment, which
consists of taking measurements on a particular condition or group. Level A
students may be interested in timing the swing of a pendulum or seeing how far
a toy car runs off the end of a slope from a fixed starting position (future
Pinewood Derby participants?) Also,
measuring the same thing several times and finding a mean helps to lay the
foundation for the fact that the mean has less variability as an estimate of
the true mean value than does a single reading. This idea will be more fully
developed at Level C.
Example 2: Growing Beans ÐA Simple Comparative Experiment
A simple comparative experiment is like a science experiment in which children compare the results of two or more conditions. For example, children might plant dried beans in soil and let them sprout and then compare which one grows fastestÐthe one in the light or the one in the dark. The children decide which beans will be exposed to a particular type of lighting. The conditions to be compared here are the two types of lighting environmentÐlight or dark. The type of lighting environment is an example of a categorical variable. Measurements of the plantsÕ heights can be taken at the end of a specified time period to answer the question of whether one lighting environment is better for growing beans. The heights collected are an example of numerical data. In Level C, the concept of an experiment (where conditions are imposed by the researcher) will be more fully developed.
Another appropriate graphical
representation for numerical data of one variable (in addition to the stem
and leaf plot) at Level A is a dotplot.
Both the dotplot and stem and leaf plot can be used to easily compare two or
more similar sets of numerical data. In creating a dotplot, the xaxis should be labeled with a range of
values that the numerical variable can assume. The xaxis for any onevariable graph is conventionally the axis
representing the values of Òthe variableÓ under study. For example, in the bean
growth experiment, children might record in a dotplot the height of beans that
were grown in the dark (labeled D) and in the light (labeled L) using a
dotplot.
Figure 5: Dotplot of height vs. environment
It is obvious from the dotplot that the heights of the plants in the light environment tend to have greater heights than the plants in the dark environment.
Looking for clusters and gaps in the distribution helps students to identify the shape of the distribution. Students should develop a sense of why a distribution takes on a particular shape for the context of the variable being considered.
Making Use of Available Data
Most children love to eat hot dogs but are aware that too much sodium is not necessarily healthy. Is there as difference in the sodium content between beef hotdogs (labeled B below) and poultry hotdogs (labeled P below)? To investigate this question, students can make use of available data. Using data from the June 1993 issue of Consumer Reports magazine, parallel dotplots can be constructed.
Figure 6: Parallel dotplot of sodium content
Students will notice that the distribution of the poultry hot dogs has two distinct clusters. What might explain the gap and two clusters? It could be another variable, such as the price of the poultry hog dogs with more expensive hot dogs having less sodium. It can also be observed that the beef sodium amounts are more spread out (or variable) than the poultry hot dogs. In addition, it also appears that the center of the distribution for the poultry hot dogs is higher than the center for the beef.
Describing shape connects the student to properties of geometry. As students advance to Level B, considering shape will lead to an understanding of what measures are appropriate for describing center and spread.
Describing Center and Spread
Students should understand that the median describes the center of a numerical data set in terms of how
many data points are above and below it. Half of the data points lie above the
median and half lie below it. Children can create a human graph to show how
many letters are in their first name. All of the children with 2letter names
can stand in a line with all of the children having 3letter names standing in
a parallel line right next to them, etc. Once all children are assembled, the
teacher can ask one child from each end of the graph to sit down, repeating
this procedure until one child is left standing, representing the median. With
Level A students, we advocate using an odd number of data points so that the
median is clear until students have mastered understanding of a midpoint.
Students should understand the mean as a fair share at Level A. In the name length example above, the mean would be interpreted as ÒHow long would our names be if they were all the same length?Ó This can be illustrated in small groups by having children take one snap cube for each letter in their name. In their small groups, have them put all of the cubes in the center of the table and redistribute them one at a time so that each child has the same number. Depending on the childrenÕs experiences with fractions, they may say that the mean name length is 4 R. 2 or 4 1/2 or 4.5. Another example would be for the teacher to collect 8 pencils of varying lengths from children and lay them endtoend on the chalk rail. Finding the mean will answer the question ÒHow long would each pencil be if they were all the same length?Ó That is, if we could glue all of the pencils together and cut them into 8 equal sections, how long would the sections be? This can be modeled using adding machine tape or string by tearing off a piece of tape that is the same length as all 8 pencils laid endtoend. Then fold the tape in half three times to get eighths, showing the length of one pencil out of eight pencils of equal length. Both of these demonstrations can be mapped directly onto the algorithm for finding the mean: combine all data elements (put all cubes in the middle, lay all pencils endtoend and measure, add all elements) and share fairly (distribute the cubes, fold the tape, and divide by the number of data elements). Level A students should master the computation (by hand or using appropriate technology) of the mean so that more sophisticated definitions of the mean can be developed at Levels B and C.
The mean and median are measures
of location for describing the center of a numerical data set. Determining
the maximum and minimum values of a numerical data set assists children in
describing the position of the smallest and largest value in a data set. These
lead to a measure of spread for the
distribution, the range.
In addition to describing the center of a data set, it is useful to know how the data are spread out. Measures of spread only make sense with numerical (measurement) data.
The range is a single number that tells how far it is from the minimum element to the maximum element. In looking at the stem and leaf plot formed in Example 2 for the jumping distances, the range differs for the jumping distance of boys (range = 39 inches) and girls (range = 27 inches). Girls are more consistent in their jumping distances than boys.
Looking for an Association
Students should be able to look at the possible association of a numerical variable and a
categorical variable by comparing dotplots of a numerical variable
disaggregated by a categorical variable. For example, using the parallel
dotplots showing the growth habits of beans in the light and dark, students
should look for similarities within each category and differences between the
categories. As mentioned earlier, students should readily recognize from the
dotplot that the beans grown in the light environment have grown taller overall
and reason that it is best for beans to have a light environment. Measures of
center and spread can also be compared. For example, students could calculate
or make a visual estimate of the mean height of the beans grown in the light
and the beans grown in the dark to substantiate their claim that light
conditions are better for beans. They might also note that the range for plants
grown in the dark is 4 and for plants grown in the light is 5. Putting that
information together with the mean should enable students to further solidify
their conclusions about the advantages of grown beans in the light. Considering
the hot dog data, general impressions from the dotplots are that there is more
variation in the sodium content for beef hot dogs. For beef hot dogs the sodium contents
are between 250 and 650, while for poultry hot dogs all are between 350 and
600. Neither the centers nor the
shapes for the distributions are obvious from the dotplots. It is interesting to note the two
apparent clusters of data for poultry hot dogs. Nine of the 17 poultry hot dogs have
sodium content between 350 and 450 mg, while 8 of the 17 poultry hot dogs have
sodium content between 500 and 650 mg.
A possible explanation for this division is that some poultry hot dogs
are made from chicken, while others are made from turkey.
Example 3: Purchasing SweatsuitsThe Role of Height and Arm span
What about the association between two numerical variables? Parentteacher organizations at elementary schools have as a popular fund raiser Ôspirit wearÕ such as sweatshirts and sweatpants with the school name and mascot. The organizers need to have some guidelines on how to order sizes. Should they offer the shirt and pants separately or offer the sweatshirt and sweatpants as one outfit? Are the heights and arm spans of elementary students closely related or due to growing patterns of children? Thus, some useful questions to answer are:
Is there an association between height and arm span?
How strong is the
association between height and arm span?
A scatterplot can be used to graphically represent data when values of two numerical variables are obtained from the same individual or object. Can we use arm span to predict a personÕs height? Students can measure each otherÕs arm spans and heights, and then construct a scatterplot to look for a relationship between these two numerical variables. Data on height and arm span are measured in centimeters for 26 students. [The data presented below are for college students and is included for illustrative purposes.]
Figure 7: Scatterplot of height vs. arm span
With the use of a scatterplot, Level A students can visually
look for trends and pattern.
For example, in the arm span vs. height scatterplot above,
students should be able to identify the consistent relationship between the two
variables: as one gets larger, so does the other. Based on this sample, the
organizers might feel comfortable in ordering some complete outfits of
sweatshirt and sweatpants based on sizes. However, some students may need to
order the sweatshirt and sweatpants separately based on sizes. Another
important question the organizers will need to ask is whether this sample is
representative of all the students in the school? How was the sample taken?
Students at Level A can also use a scatterplot to
graphically look at the relationship of a numerical variable over time,
referred to as a time plot.
For example, children might chart the outside temperature at various times during the day by recording the values themselves or by using data from a newspaper or the internet.
Figure 8: Timeplot of time vs. temperature
When the student advances to Level B, these trends and patterns will be quantified with
measures of association.
Understanding Variability
Students should explore possible reasons that data look the
way they do and differentiate between
variation and error. For example, in graphing the colors of candies in a
small packet, children might expect the colors to be evenly distributed (or
they may know from prior experience that they are not). Children could
speculate about why certain colors appear more or less frequently due to
variation (e.g., cost of dyes, market research on peopleÕs preferences, etc.).
Children could also identify possible places where errors could have occurred
in their handling of the data/candies (e.g., dropped candies, candies stuck in
bag, eaten candies, candies given away to others, colors not recorded because
they donÕt match personal preference, miscounting). Teachers should capitalize
on naturallyoccurring ÒerrorsÓ that
happen when collecting data in the classroom and help students speculate about
the impact of these errors on the
final results. For example, when asking students to vote for their favorite
food, it is common for students to vote twice, to forget to vote, to record
their vote in the wrong spot, to misunderstand what is being asked, to change
their minds, or to want to vote for an option that is not listed. Counting
errors are also common among young children and can lead to incorrect tallies
of data points in categories. Teachers can help students think about how these
events might affect the final outcome if only one person did this, if several
people did it, or if many people did it. Students can generate additional
examples of ways that errors might occur in a particular datagathering
situation.
The notions of error and variability should be used to
explain the outliers, clusters, and gaps that students observe in the graphical
representations of the data. An understanding of error versus natural or
expected variability will help students to interpret whether an outlier is a
legitimate data value that is unusual or is the outlier unusual due to a
recording error?
At Level A, it is imperative that students begin to understand this concept of variability. As students move from Level A to Level B, then Level C, it is important to always keep at the forefront that understanding variability is the essence of developing data sense.
The Role of Probability
Level A students need to develop basic ideas of probability in order to
support their later use of probability in drawing inferences at Levels B and C.
At Level A, students should understand that probability is a measure of the chance that something will happen. It
is a measure of certainty or uncertainty. Events should be seen as lying on
a continuum from impossible to certain, with less likely, equally likely, and
more likely lying in between. Students learn to informally assign numbers to
the likelihood that something will occur. An example of assigning numbers on a
number line is given below.
0 ¼ 1/2 ¾ 1
_________________________________________________________________
Impossible Unlikely Equally likely Likely
Certain
Or less likely to
occur and or more likely
not occur
Student should have experiences estimating
probabilities using empirical data. Through experimentation (or
simulation), students should develop an explicit understanding of the notion
that the more times you repeat an experiment, the closer the results will be to
the expected mathematical model. At Level A we are only considering simple models
based on equally likely outcomes or, at the most, something based on this, such
as the sum of the faces on two number cubes. For example, very young children
can state that a penny should land on heads half the time and on tails half of
the time when flipped. The student has given the expected model and probability
for tossing a head or tail, assuming that the coin is ÔfairÕ.
If a child flips a penny 10 times to obtain empirical data, it is quite
possible that he or she will not get 5 heads and 5 tails. However, if the child
flips the coin a hundreds of times, we would expect to see that results will
begin stabilizing to the expected
probabilities of 50% heads and 50% tails. This is known as the Law of Large Numbers. Thus, at Level A,
probability experiments should focus on obtaining empirical data to develop
relative frequency interpretations that children can easily translate to models
with known and understandable ÔmathematicalÕ probabilities. The classic
flipping coins, spinning simple spinners and tossing a number cube are reliable
tools to use in helping Level A students develop an understanding of
probability. The concept of relative frequency interpretations will be
important at Level B when the student works with proportional reasoningÐgoing
from counts or frequencies to proportions or percentages.
As students work with empirical data, such as flipping a coin, they can
develop an understanding for the concept of randomness. They will see that when flipping a coin
10 times, although we would expect 5 heads and 5 tails, the actual results will
vary from one student to the next. They will also see that if a head results on
one toss, that doesnÕt mean that the next flip will result in a tail. With a
random process, there is always uncertainty as to how the coin will land from
one toss to the next. However, at Level A, students can begin to develop the
notion that although we have uncertainty and variability in our results, by
examining what happens to the random process in the long run, we can quantify the uncertainty and variability with
probabilitiesÐgiving a predictive number for the likelihood of an outcome in
the long run. At Level B, students will see the role that probability plays in
the development of the concept of the simple random sample and will see the
role that probability plays with randomness.
Misuses of Statistics
The Level A student should learn that proper use of statistical terminology is important as well as the proper use of statistical tools. In particular, the proper use of the mean and median should be emphasized. These numerical summaries are for summarizing a numerical variable, not a categorical variable. For example, when collecting categorical data on favorite type of music, the number of children in the sample who prefer each type of music is summarized as a frequency. It is easy to confuse categorical and numerical data in this case and try to find the mean or median of the frequencies for favorite type of music. However, one cannot use the frequency counts to compute a mean or median for a categorical variable. The frequency counts are the numerical summary for the categorical variable.
Another common misuse at Level A student is the inappropriate use of a bar graph with numerical data. A bar graph is used to summarize categorical data. If a variable is numerical, the appropriate graphical display with bars is called a histogram, which is introduced in Level B. At Level A, appropriate graphical displays for numerical data are the dotplot and the stem and leaf plot.
Conclusion
If students become comfortable with the ideas and concepts described above, they will be prepared to further develop and enhance their understanding of the key concepts for data sense at Level B.
It is also important to recognize that helping students develop data sense at Level A allows mathematics instruction to be driven by data. The traditional mathematics strands of algebra, functions, geometry, and measurement can all be developed with the use of data. Making sense of data should be an integrated part of the mathematics curriculum starting in kindergarten.
Level B
Objectives of Level B
Instruction at Level B should build on the statistical base developed at Level A and set the stage for statistics at Level C. Instructional activities at Level B should continue to emphasize the four main components in the investigative process and should have the spirit of genuine statistical practice. Students who complete Level B should see statistical reasoning as a process for solving problems through data and quantitative reasoning. At Level B:
á
Students investigate
problems with more emphasis placed on possible associations among two or more
variables and understand how a more sophisticated collection of graphical,
tabular and numerical summaries is used to address these questions.
Specifically, Level B recommendations in the Investigative Process include:
1. Formulate Questions
Students begin to pose their own questions.
Students address questions involving a group larger than their classroom and begin to recognize the distinction among a population, a census, and a sample.
2. Collect Data
Students conduct censuses of two or more classrooms.
Students design and conduct nonrandom sample surveys and begin to use random selection.
Students design and conduct comparative experiments and begin to use random assignment.
3. Analyze Data
Students expand their understanding of a data distribution.
Students quantify variability within a group.
Students compare two or more distributions using graphical displays and using summary measures.
Students use more sophisticated tools for summarizing and comparing distributions including:
Histograms,
The IQR (Interquartile Range) and MAD (Mean Absolute Deviation),
FiveNumber Summaries and Boxplots.
Students acknowledge sampling error.
Students quantify the strength of association between two variables, develop simple models for association between two numerical variables, and use expanded tools for exploring association including:
Contingency Tables for two categorical variables,
Time Series Plots,
The QCR (Quadrant Count Ratio) as a measure of strength of association,
Simple lines for modeling association between two numerical variables.
4.
Interpret Results
Students describe differences between two or more groups with respect to center, spread, and shape.
Students acknowledge that a sample may not be representative of a larger population.
Students understand basic interpretations of measures of association.
Students begin to distinguish between an observational study and a designed experiment.
Students begin to distinguish between ÒassociationÓ and Òcause and effect.Ó
Students recognize sampling variability in summary measures such as the sample mean and the sample proportion..
Example 1, Level A Revisited: Choosing a Band for the School Dance
Many of the graphical, tabular and numerical summaries introduced at Level A can be enhanced and used to investigate more sophisticated problems at Level B. LetÕs revisit the problem of planning for the school dance introduced in Level A, where a Level A class investigated the question: What type of music is most popular among students? by conducting a census of the class. That is, the class was considered to be the entire population and data were collected on every member of the population. A similar investigation at Level B would include recognition that one class may not be representative of the opinions of all students at their school, and Level B students might want to compare the opinions of their class with other classes from their school. A Level B class might investigate the questions:
What type of music is most popular among students at our
school?
How do the favorite types of music differ between different
classes?
Since class sizes may be
different, in order to make comparisons, results should be summarized with
relative frequencies or percentages. Percentages are useful in that they allow
us to think of having comparable results for groups of size 100. Level B
students will see more emphasis in proportional reasoning throughout the
mathematics curriculum, and they should be comfortable summarizing and
interpreting data in terms of percentages or fractions.
The results from two classes are
summarized in the table below using both frequency counts and relative
frequency percentages.
Table 2: Frequency counts and relative
frequency percentages
Class 1 

Class 2 

Favorite 
Frequency 
Relative Frequency Percentage 

Favorite 
Frequency 
Relative Frequency Percentage 
Country 
8 
33% 

Country 
5 
17% 
Rap 
12 
50% 

Rap 
11 
37% 
Rock 
4 
17% 

Rock 
14 
47% 
Total 
24 
100% 

Total 
30 
101% 
The comparative bar graph below
compares the percentage of each favorite music category for the two classes.
Figure 9: Comparative bar graph for music
preferences
Students at Level B should begin
to recognize that there is not only variability from one individual to another
within a group, but that there is variability in results from one group to
another. This second type of variability is illustrated by the fact that in
Class 1 the most popular music is rap music while in Class 2 it is rock music.
That is, the mode for Class 1 is rap music, while the mode for Class 2 is rock
music.
The results from the two samples might be combined in order to have a larger sample of the entire school. The combined results indicate that rap music was the favorite type of music for 43% of the students, rock music was preferred by 33%, while only 24% of the students selected country music as their favorite. Level B students should recognize that although this is a larger sample, it still may not be representative of the entire population (all students at their school). In statistics, randomness and probability are incorporated into the sample selection procedure in order to provide a method that is fair and to improve the chances of selecting a representative sample. For example, if the class decides to select what is called a simple random sample of 54 students, then each possible sample of 54 students has the same probability of being selected. This application of probability illustrates one of the roles of probability in statistics. Although Level B students may not actually employ a random selection procedure when collecting data, issues related to obtaining representative samples should be discussed at this level.
Connecting Two Categorical Variables
Since rap was the most popular music for the combined two classes, the students might argue for a rap band for the dance. However, more than half of those surveyed preferred either rock or country music. Will these students be unhappy if a rap band is chosen? Not necessarily since many students who like rock music may also like rap music as well. To investigate this problem, students might explore two additional questions.
Do students who like rock music tend to like or dislike rap
music?
Do students who like country music tend to like or dislike
rap music?
To address these questions, the survey should ask students not only their favorite type of music, but also whether or not they like rap, rock, and country music.
The twoway frequency table (or contingency table) below provides a way to investigate possible connections between two categorical variables.
Table 3: Twoway frequency table

Like Rap Music? 
Row Totals 

Yes 
No 

Like Rock Music? 
Yes 
27 
6 
33 
No 
4 
17 
21 

Column Totals 
31 
23 
54 
According to these results, of the 33 students who liked rock music, 27 also liked rap music. That is, 82% (27/33) of the students who like rock music also like rap music. This indicates that students who like rock music tend to like rap music as well. Once again, notice the use of proportional reasoning in interpreting these results. A similar analysis could be performed to determine if students who like country tend to like or dislike rap music. A more detailed discussion of this example and a measure of association between two categorical variables is given in the Level B Appendix.
Questionnaires and Their Difficulties
At Level B, students should begin to learn about surveys and the many pitfalls to avoid when designing and conducting a survey. One issue involves the wording of questions. Questions must be unambiguous and easy to understand. For example, the question:
Are you against the school implementing a nodoor policy on bathroom
stalls?
is worded in a confusing way. An alternative way to pose this question is:
The school is considering implementing a nodoor policy on bathroom
stalls. What is your opinion regarding this policy?
Strongly Oppose Oppose No
Opinion Support Strongly
Support
Questions should avoid leading the respondent to an answer. For example, the question:
Since our football team hasnÕt had a winning season in 20 years and is
costing the school money rather than generating funds, do you feel we should
concentrate more on another sport such as soccer or basketball?
is worded in a way that is biased against the football team.
The responses to questions with coded responses should include all possible answers and should not overlap. For example, for the question:
How much time do you spend studying at home on a typical night?
the responses:
none 1
hour or less
1 hour or more
would confuse a student who spends 1 hour a night studying.
There are many other issues concerning question formulation and conducting sample surveys. One issue that should be discussed at Level B involves how the interviewer asks the questions as well as how accurately the responses are recorded. It is important for students to realize that the conclusions from their study depend on the accuracy of their data.
Measure of LocationÐThe Mean as Balance Point
Another idea developed at Level A that can be expanded at Level B is the mean as a numerical summary for a collection of numerical data. At Level A the mean is interpreted as the Òfair shareÓ value for data. That is, the mean is the value you would get if all the data are combined and then redistributed evenly so that each value is the same. Another interpretation of the mean
is that it is the balance point of the corresponding data distribution. Following is an outline of an activity that illustrates the notion of the mean as a balance point.
Nine students were asked: ÒHow many pets do you have?Ó
The resulting data are: 1, 3, 4, 4, 4, 5, 7, 8, 9. These data are summarized in the dotplot below. Note that in the actual activity, stickon notes are used as ÒdotsÓ instead of XÕs.
Figure 10: Dotplot for pet count
X
X
X
X X X
X X X
+++++++++
1 2 3 4 5 6 7 8 9
If the pets are combined into one group there are a total of 45 pets. If the pets are redistributed evenly among the 9 students, then each student would get 5 pets. That is, the mean number of pets is 5. The dotplot representing the result that all 9 students have exactly 5 pets is shown below:
Figure 11: Dotplot showing pets evenly distributed
X
X
X
X
X
X
X
X
X
+++++++++
1 2 3 4 5 6 7 8 9
It is hopefully obvious that if a pivot is placed at the value 5 then the horizontal axis will ÒbalanceÓ at this pivot point. That is, the Òbalance pointÓ for the horizontal axis for this dotplot is 5. What is the balance point for the dotplot displaying the original data?
We begin by noting what happens if one of the dots over 5 is removed and placed over the value 7 as shown below.
Figure 12: Dotplot with one data point moved
X
X
X
X
X
X
X
X
X
+++++++++
1 2 3 4 5 6 7 8 9
Clearly, if the pivot remains at 5, the horizontal axis will tilt right. What can be done to the remaining dots over 5 to ÒrebalanceÓ the horizontal axis at the pivot point? Since 7 is 2 above 5, one solution is to move a dot 2 below 5 to 3 as shown below.
Figure 13: Dotplot with two data points moved
X
X
X
X
X
X
X
X
X
+++++++++
1 2 3 4 5 6 7 8 9
Clearly, the horizontal axis is now rebalanced at the pivot point. Is this the only way to rebalance the axis at 5? Another way to rebalance the axis at the pivot point would be to move two dots from 5 to 4 as shown below:
Figure 14: Dotplot with two different data points moved
X
X
X
X
X X
X X
X
+++++++++
1 2 3 4 5 6 7 8 9
The horizontal axis is now rebalanced at the pivot point. That is, the Òbalance pointÓ for the horizontal axis for this dotplot is 5. Replacing each ÒXÓ (dot) in this plot with the distance between the value and 5, we have:
Figure 15: Dotplot showing distance from 5
0
0
0
0
1 0
1 0
2
+++++++++
1 2 3 4 5 6 7 8 9
Notice that the total distance for the two values below the 5 (the two 4Õs) is the same as the total distance for the one value above the 5 (the 7). For this reason, the balance point of the horizontal axis is 5. Replacing each value in the dotplot of the original data by its distance from 5 yields the following plot.
Figure 16: Dotplot showing original data and distance from 5
1
1
4
2 1 0
2 3 4
+++++++++
1 2 3 4 5 6 7 8 9
The total distance for the values below 5 is 9, the same as the total distance for the values above 5. For this reason, the mean (5) is the balance point of the horizontal axis.
Both the mean and median are often referred to as measures of central location. At Level A the median was also introduced as the quantity that has the same number of data values on each side of it in the ordered data. This sameness of each side is the reason the median is a measure of central location. The previous activity demonstrates that the total distance for the values below the mean is the same as the total distance for the values above the mean and illustrates why the mean is also considered to be a measure of central location.
A Measure of SpreadÐThe Mean Absolute Deviation
Statistics is concerned with variability in data. One important idea is to quantify how much variability exists in a collection of numerical data. Quantities that measure the degree of variability in data are called measures of spread. At Level A students are introduced to the range as a measure of spread in numerical data. At Level B students should be introduced to the idea of comparing data values to a central value such as the mean or the median, and quantifying how different the data are from this central value.
In the number of pets example, how different are the original data values from the mean? One way to measure the degree of variability from the mean is to determine the total distance for all values from the mean. Using the final dotplot from the previous example, the total distance the 9 data values are from the mean of 5 pets is 18 pets. The magnitude of this quantity depends on several factors, including the number of measurements. To adjust for the number of measurements, the total distance from the mean is divided by the number of measurements. The resulting quantity is called the Mean Absolute Deviation or MAD. The MAD is the average distance of all the data from the mean. That is,
MAD = Total Distance from the Mean for all Values
Number of Data Values
The MAD for the data on number of pets from the previous activity is:
MAD = 18/9 = 2
The MAD indicates that the actual number of pets for the 9 students differs from the mean of 5 pets on average by 2 pets. Kader (1999) gives a thorough discussion of this activity and the MAD.
The MAD is an indicator of spread based on all the data and provides a measure of average variation in the data from the mean. The MAD also serves as a precursor to the standard deviation developed at Level C.
Representing Data DistributionsÐThe Frequency Table, Histogram
At Level B, students should develop additional tabular and graphical devices for representing data distributions for numerical variables. Several of these build upon representations developed at Level A. For example, students at Level B might explore the problem of placing an order for hats. To prepare an order, one needs to know which hat sizes are most common and which occur least often. To obtain information about hat sizes it is necessary to measure head circumferences. European hat size sizes are based on the metric system. For example, a European hat size of 55 is designed to fit a person with a head circumference between 550 mm and 559 mm. In planning an order for adults, students might collect preliminary data on the head circumferences of their parents, guardians, or other adults. Such data would be the result of a nonrandom sample. The data summarized in the following stemplot (also known as stem and leaf plot) are head circumferences measured in millimeters for a sample of 55 adults.
Figure 17: Stemplot of head circumference
51 3
52 5
53 133455
54 2334699
55 12222345
56 0133355588
57 113477
58 02334458
59 1558
60 13
61 28
Based on the stemplot, some head sizes do appear to be more common than others. Head circumferences in the 560Õs are most common. Head circumferences fall off in somewhat of a symmetric manner on both sides of the 560Õs with very few smaller than 530 mm or larger than 600 mm.
In practice, a decision of how many hats to order would be based on a much larger sample, possibly hundreds or even thousands of adults. If a larger sample were available, a stemplot would not be a practical device for summarizing the data distribution. An alternative to the stemplot is to form a distribution based on dividing the data into groups or intervals. This method can be illustrated through a smaller data set such as the 55 head circumferences but is applicable for larger sets as well. The grouped frequency and grouped relative frequency distributions and the relative frequency histogram that correspond to the above stemplot are:
Table 4: Grouped frequency and grouped relative frequency distributions
Limits on
Recorded
Measurements Interval
of
on
Head Actual
Head Relative
Stem Circumference Circumferences Frequency Frequency
(%)
51 510
 519 510
 < 520 1 1.8
52 520
 529 520
 < 530 1 1.8
53 530
 539 530
 < 540 6 10.9
54 540
 549 540
 < 550 7 12.7
55 550
 559 550
 < 560 8 14.5
56 560
 569 560
 < 570 10 18.2
57 570
 579 570
 < 580 6 10.9
58 580
 589 580
 < 590 8 14.5
59 590
 599 590
 < 600 4 7.3
60 600
 609 600
 < 610 2 3.6
61 610
 619 610
 < 620 2 3.6
T0TAL 55 99.8
Figure 18: Relative frequency histogram
Relative Frequency (%)
If the hat manufacturer requires that orders be in multiples of 250 hats then, based on the above results, how many hats of each size should be ordered? Using the relative frequency distribution, the number of each size to order for an order of 250 hats would be:
Table 5: Hat size data
Hat Size Number to Order
51 5
52 5
53 27
54 32
55 36
56 46
57 27
58 36
59 18
60 9
61 9
Once again, notice how students at Level B would utilize proportional reasoning in determining the number to order of each size. Kader and Perry (1994) give a detailed description of The Hat Shop problem.
Comparing DistributionsÐThe Boxplot
Problems that require comparing distributions for two or more groups are common in statistics. For example, at Level A students compared the amount of sodium in ÒbeefÓ and ÒpoultryÓ hotdogs by examining parallel dotplots. At Level B more sophisticated representations should be developed for comparing distributions. One of the most useful graphical devices for comparing distributions of numerical data is the boxplot. The boxplot (also called a boxandwhiskers plot) is a graph based on a division of the ordered data into four groups with the same number of data values in each group (approximately onefourth). The four groups are determined from the FiveNumber Summary (the minimum data value, the first quartile, the median, the third quartile, and the maximum data value). The FiveNumber Summaries and comparative boxplots for the data on sodium content for beef (labeled B) and poultry (labeled P) hot dogs introduced in Level A are given below.
Table 6: Five number summaries for sodium content
Beef Hot Dogs (n = 20) Poultry Hot Dogs (n = 17)
Minimum 253 357
First Quartile 320.5 379
Median 380.5 430
Third Quartile 478 535
Maximum 645 588
Figure 19: Boxplot for sodium content
Interpreting results based on a boxplot analysis requires comparisons based on global characteristics of each distribution (center, spread, and shape). For example, the median sodium content for poultry hot dogs is 430 mg, almost 50 mg more than the median sodium content for beef hot dogs. The medians indicate that a typical value for the sodium content of poultry hot dogs is greater than a typical value for beef hot dogs. The range for the beef hot dogs is 392 mg versus 231 mg for the poultry hot dogs. The ranges indicate that overall, there is more spread (variation) in the sodium content of beef hot dogs compared to poultry hot dogs. Another measure of spread that should be introduced at Level B is the interquartile range or IQR. The IQR is the difference between the third and first quartiles and indicates the range of the middle 50% of data. The IQRÕs for sodium content are 157.5 mg for beef and 156 mg for poultry hot dogs. The IQRÕs suggest that the spread within the middle half of data for beef hot dogs is similar to the spread within the middle half of data for poultry hot dogs. The boxplots also suggest that each distribution is slightly skewed right. That is, each distribution appears to have somewhat more variation in the upper half. Considering the degree of variation in the data and the amount of overlap in the boxplots, a difference of 50 mg between the medians is not really that large. Finally, it is interesting to note that more than 25% of beef hot dogs have less sodium than all poultry hot dogs. On the other hand, the highest sodium levels are for beef hot dogs.
Note that there are several variations of boxplots. At Level C construction of a boxplot analysis might include an analysis for outliers (values that are extremely large or small when compared to the variation in the majority of the data). If outliers are identified, these are often detached from the ÒwhiskersÓ of the plot. Outlier analysis is not recommended at Level B, so whiskers are attached to the minimum and maximum data values. However, Level B students may encounter outliers when using statistical software or graphic calculators.
Measuring the Strength of Association between Two Quantitative Variables
At Level B, more sophisticated data representations should be developed for the investigation of problems that involve the study of the relationship between two numeric variables. At Level A, the problem of packaging sweatsuits (shirt and pants together or separate) was examined through a study of the relationship between height and arm span. There are several statistical questions related to this problem that can be addressed at Level B that require a more indepth analysis of the heightarm span data. For example,
How strong is the association between height and arm span?
Is height a useful predictor of arm span?
Following are data on height and arm span measured in centimeters for 26 students. For convenience, the data on height have been ordered.
Table 7: Height and arm span data
Height Arm span Height Arm span
155 151 173 170
162 162 175 166
162 161 176 171
163 172 176 173
164 167 178 173
164 155 178 166
165 163 181 183
165 165 183 181
166 167 183 178
166 164 183 174
168 165 183 180
171 164 185 177
171 168 188 185
The height and arm span data are displayed in the
following scatterplot. The scatterplot suggests a fairly strong increasing
relationship between height and arm span, and the
relationship between height and arm span appears to be quite linear.
Figure 20: Scatterplot of height vs. arm span
Measuring the strength of association between two variables is an important statistical concept that should be introduced at Level B. The scatterplot below for the Height/Arm Span data includes a vertical line drawn through the mean height (_{} = 172.5) and a horizontal line drawn through the mean arm span (_{} = 169.3).
Figure 21: Scatterplot showing means
The two lines divide the scatterplot into four regions (or quadrants). The upper right region (Quadrant 1) contains points that correspond to individuals with above average height and above average arm span. The upper left region (Quadrant 2) contains points that correspond to individuals with below average height and above average arm span. The lower left region (Quadrant 3) contains points that correspond to individuals with below average height and below average arm span. The lower right region (Quadrant 4) contains points that correspond to individuals with above average height and below average arm span.
Notice that most points in the scatterplot are in either Quadrant 1 or Quadrant 3. That is, most people with above average height also have above average arm span (Quadrant 1) and most people with below average height also have below average arm span (Quadrant 3). One person has below average height with above average arm span (Quadrant 2) and two people have above average height with below average arm span (Quadrant 4). These results indicate that there is a positive association between the variables height and arm span. Generally stated, two numeric variables are positively associated when above average values of one variable tend to occur with above average values of the other and when below average values of one variable tend to occur with below average values of the other. Negative association between two numeric variables occurs when below average values of one variable tend to occur with above average values of the other and when above average values of one variable tend to occur with below average values of the other.
A correlation coefficient is a quantity that measures the direction and strength of an association between two variables. Note that in the previous example, points in Quadrants 1 and 3 contribute to the positive association between height and arm span, and there are a total of 23 points in these two quadrants. Points in Quadrants 2 and 4 do not contribute to the positive association between height and arm span, and there are a total of 3 points in these two quadrants. One correlation coefficient between height and arm span is given by the QCR (Quadrant Count Ratio):
QCR = 23 Ð3 = .77
26
A QCR of .77 indicates that there is a fairly strong positive association between the two variables height and arm span. This indicates that a personÕs height is a useful predictor of his/her arm span.
In general, the QCR is defined as:
(Number of Points in Quadrants
1 and 3) Ð (Number of Points in Quadrants 2 and 4)
Number of Points in all Four Quadrants
The QCR has the following properties:
The QCR is unitless.
The QCR is always between Ð1 and +1 inclusive.
Holmes (2001) gives a detailed discussion of the QCR. A similar correlation coefficient for 2x2 contingency tables is described in Conover (1999) and is discussed in the Appendix for Level B. The QCR is a measure of the strength of association based only on the number of points in each quadrant and, like most summary measures, has its shortcomings. At Level C the shortcomings of the QCR can be addressed and used as foundation for developing PearsonÕs correlation coefficient.
Modeling Linear Association
The height/arm span data were collected at Level A in order to study the problem of packaging sweat suits. Should a shirt and pants be packaged separately or together? A QCR of .77 suggests a fairly strong positive association between height and arm span, which indicates that height is a useful predictor of arm span and that a shirt and pants could be packaged together. If packaged together, how can a person decide which size sweatsuit to buy? Certainly, the pantsize of a sweatsuit depends on a personÕs height and the shirtsize depends on a personÕs arm span. Since many people know their height, but may not know their arm span, can height be used to help people decide which size sweatsuit they wear? Specifically:
Can the relationship between height and arm span be described using a
linear function?
Students at Level B will study linear relationships in other areas of their mathematics curriculum. The degree to which these ideas have been developed will determine how we might proceed at this point. For example, if students have not yet been introduced to the equation of a line, then they might simply draw a line through the Òcenter of the dataÓ as shown below.
Figure 22: Sscatterplot showing center of data
This line can be used to predict a personÕs arm span if his or her height is known. For example, to predict the arm span for a person who is 170 cm tall, a vertical line is drawn up from the Xaxis at Height = 170. At the point this vertical line intersects the line, a horizontal line is drawn to the Yaxis. The value where this horizontal line intersects the Yaxis is the predicted arm span. Based on the graph above, it appears that we would predict an arm span of approximately 168 cm for a person who is 170 cm tall.
If students are familiar with the equation for a line and know how to find the equation from two points, then they might use the Mean Ð Mean line, which is determined as follows. Order the data according to the Xcoordinates and divide the data into two ÒhalvesÓ based on this ordering. In the case that there is an odd number of measurements, remove the middle point from the analysis. Determine the means for the Xcoordinates and Ycoordinates in each half and determine the equation of the line that passes through these two points. Using the previous data:
Lower Half (13 Points) Upper Half (13 Points)
Mean Height = 164.8 Mean Height = 180.2
Mean Arm Span = 163.4 Mean Arm Span = 175.2
The equation of the line that goes through the points (164.8, 163.4) and (180.2, 175.2) is: Predicted Arm Span È 37.1 + .766(Height). This equation can be used to predict a personÕs height more accurately than an eyeball line For example, if a person is 170 cm tall, then we would predict his/her height to be approximately : 37.1 + .766(170) = 167.3 cm. A more sophisticated approach (least squares) to determine a ÒbestfittingÓ line through the data will be seen in Level C.
The Importance of Random Selection
In statistics, we often want to
extend results beyond a particular group studied to a larger group, the population. We are trying to gain
information about the population by examining a portion of the population,
called a sample. Such generalizations
are only valid if the data are representative of that larger group. A
representative sample is one in which the relevant characteristics of the
sample members are generally the same as those of the population. Improper or
biased sample selection tends to systematically favor certain outcomes and can
produce misleading results and erroneous conclusions.
Random sampling is a way to remove
bias in sample selection and tends to produce representative samples. Random
sampling attempts to reduce bias in sample selection by being fair to each
member of the population. At Level B, students should experience the
consequences of nonrandom selection and develop a basic understanding of the
principles involved in random selection procedures. Following
is a description of an activity that allows students to compare sample results that are based on personal (nonrandom)
selection versus sample results that are based on random selection.
Consider the 80 circles on the next page. What is the average diameter for these 80 circles? Each student should take about 15 seconds and select five circles that he/she thinks best represents the sizes of the 80 circles. After selecting the sample, each student should find the average diameter for the circles in her/his personal sample. Note that the diameter for the small circles is 1 cm, for the medium sized circles is 2 cm, and for the large circles is 3 cm.
Next, each student should number the circles from 1 to 80 and use a random digit generator to select a random sample of size 5. Each student should find the average diameter for the circles in her/his random sample. The sample mean diameters for the entire class can be summarized for the two selection procedures with backtoback stemplots.
How do the means for the two sample selection procedures compare with the true mean diameter of 1.25 cm? Personal selection will usually tend to yield sample means that are larger than 1.25. That is, personal selection tends to be biased with a systematic favoring toward the larger circles and an overestimation of the population mean. Random selection tends to produce some sample means that underestimate the population mean and some that overestimate the population mean such that the sample means cluster somewhat evenly around the population mean value, i.e., random selection tends to be unbiased.
In the previous example, the fact that the sample means vary
from one sample to another illustrates an idea that was
introduced earlier in the favorite music type survey. This is the notion
that results vary from one sample to another. Imposing randomness into the sampling procedure allows us
to use probability to describe the longrun behavior in
the variability in the sample means resulting from random sampling. The
variation in results from repeated sampling is described through what is called
the sampling distribution. Sampling
distributions will be explored more in depth at Level C.
Eighty Circles
Comparative Experiments
Another important statistical idea that
should be introduced at Level B is that of comparative experimental studies. Comparative experimental studies involve
comparisons of the effects of two or more treatments
(experimental conditions) on some response variable. At Level B, studies
comparing two treatments are adequate. For example, students might want to
study the effects of listening to rock music on oneÕs ability to memorize. Before undertaking a study such as this, it is important
that students have the opportunity to identify and, as much as possible, to
control for as many potential extraneous sources that may interfere with our
ability to interpret the results. To address these issues, the class needs to develop
a design strategy for collecting appropriate experimental data.
One simple experiment would be to randomly
divide the class into two equal sized (or near equal sized) groups. Random
assignment provides a fair way to assign students to the two groups because it
tends to average out differences in student ability and other characteristics
that might affect the response. For example suppose a class has 28
students. The 28 students are
randomly assigned into one of two groups of size 14. One way to accomplish this
is to place 28 pieces of paper in a box Ð14 labeled ÒMÓ and 14 labeled
ÒS.Ó Stir and shake the contents in
the box well and have each student randomly choose a piece of paper. The 14 MÕs will listen to music and the
14 SÕs will have silence.
Each student will be shown a list
of words. Rules for how long
students have to study the words and how long they have to reproduce the words
must be determined. For example,
students may have two minutes to study the words, a oneminute pause, and then
two minutes to reproduce (write down) as many words as possible. The number of
words remembered under each condition (listening to music or silence) is the
response variable of interest.
The FiveNumber Summaries and
comparative boxplots for a hypothetical set of data are shown below. These
results suggest that students generally memorize fewer words when listening to
music compared to when there is silence.
With the exception of the maximum value in the Music Group (which is
classified as an outlier), all summary measures for the Music Group (labeled M)
are lower than the corresponding summary measures for the Silence Group
(labeled S). Without the outlier, the degree of variation in the scores appears
to be similar for both groups, and both distributions appear to be reasonably
symmetric. Considering the degree of variation in the scores and the separation
in the boxplots, a difference of 3 between the medians is quite large.
Table
8: Five number summaries
Music Silence
Minimum 3 6
First
Quartile 6 8
Median 7 10
Third
Quartile 9 12
Maximum 15 14
Figure 23: Boxplot for memory data
Time Series
Another important statistical idea that should be introduced at Level B is that of time series. Problems that explore trends in data over time are quite common. For example, the population of the United States and the world continue to grow, and there are several factors that affect the size of a population such as the number of births and the number of deaths per year. One question we ask is:
How has the number of live births changed
over the past 30 years?
The United States Census Bureau publishes vital statistics in its annual Statistical Abstract of the United States. The data below is from The Statistical Abstract of the United States (20042005) and represent the number of live births per year (in thousands) for residents of the United States since 1970. Note that in 1970, the value 3,731 represents 3,731,000 live births.
Table 9: Live birth data
Year Births(x1,000) Year Births(x1,000)
1970 3731 1985 3761
1971 3556 1986 3757
1972 3258 1987 3809
1973 3137 1988 3910
1974 3160 1989 4041
1975 3144 1990 4158
1976 3168 1991 4111
1977 3327 1992 4065
1978 3333 1993 4000
1979 3494 1994 3979
1980 3612 1995 3900
1981 3629 1996 3891
1982 3681 1997 3881
1983 3639 1998 3942
1984 3669 1999 3959
The time series plot below shows the number of live births over time. This graph indicates that:
from 1970 to 1975, the number of live births generally declined,
from 1976 to 1990, the number of live births generally increased,
from 1991 to 1997, the number live births generally declined,
and it appears that the number of live birth may have started to increase since 1997.
Figure 24: Time series plot of live births
Misuses of Statistics
The introduction of this document points out that data (numbers in context) govern our lives. Because of this, every high school graduate deserves to have a solid foundation in statistical reasoning. Along with identifying proper uses of statistics in questionnaires and graphs, the Level B student should become aware of common misuses of statistics. The purpose of investigating misuses of statistics is so that our citizens of tomorrow are aware of past errors and will guard against similar occurrences in the future in their own personal, professional, or corporate lives.
Proportional reasoning allows the Level B student to
interpret data summarized in a variety of ways. One graph that is often misused
for representing data by using incorrect proportions is the pictograph. For example, suppose that the buying power of a dollar
today is 50% of what it was twenty years ago. How would one represent that in a
pictograph? Let the buying power of a dollar twenty years ago be represented as
the following.
If the buying power today is half
of what it was twenty years ago, one might think of reducing both the width and
height of this dollar by onehalf as illustrated in the pictograph below.
TodayÕs dollar should look half
the size of the dollar of twenty years ago. Does it? Since both the length and
the width were cut in half, the area of todayÕs dollar shown above is
onefourth the original area, not onehalf. The two pictographs shown below
show the correct reduction in area.
The one on the left changes only one dimension, while the other changes
both dimensions but in correct proportion so that the area is onehalf the area
of the original representation. This exercise provides the Level B student with
an excellent exercise in proportional reasoning.
Poorly designed statistical graphs are commonly found in newspapers and other popular media. Several examples of bad graphs including the use of an unwarranted third dimension in bar graphs and pie graphs can be found at the following www.stat.sfu.ca/~cschwarz/Stat301/Handouts/, a website managed by Carl Schwarz. Students at Level B should be given opportunities to identify graphs that incorrectly represent data and then draw, through the aid of statistical computer software, the correct versions. This gives them excellent practice in calculating areas and volumes.
There are many famous misuses of data analysis in the
literature, and three are mentioned here. The magazine Literary Digest erred in 1936 when it projected that Alf Landon
would beat Franklin Delano Roosevelt by a 57 to 43 percent margin based on
responses to its survey. Each
survey included a subscription form to the magazine and more than 2.3 million
were returned. Unfortunately, even large voluntary response surveys are
generally not representative of the entire population and Roosevelt won with
62% of the vote. George Gallup
correctly projected the winner and thereby began a very successful career in
using random sampling techniques for conducting surveys. Learning what
The 1970 Draft Lottery provides an example of incorrectly applying randomness. In the procedure that was used, capsules containing birthdates were placed in a large box. Although there was an effort to mix the capsules, this was insufficient to overcome the fact that the capsules were placed in the box in order from January to December. This resulted in young men with birthdates in the latter months to be more likely to have their dates selected sooner than birthdates elsewhere in the year. Hollander and Proschan (1984) give an excellent discussion of this problem.
The 25^{th} flight of NASAÕs space shuttle program
took off on January 20, 1986. Just after liftoff a puff of gray smoke could be
seen coming from the right solid rocket booster. Seventythree seconds into the
flight, the Challenger exploded killing all seven astronauts aboard. The cause
of the explosion was determined to be an Oring failure due to cold weather.
The disaster could possibly have been avoided had available data been displayed
in a simple scatterplot and correctly interpreted. The Challenger
disaster has become a case study in the possible catastrophic consequences of
poor data analysis. An excellent
description of the Challenger accident can be found at: http://wps.aw.com/wps/media/objects/15/15719/projects/ch5_challenger/
Summary of Level B
Understanding the statistical concepts of Level B enables a student to begin to appreciate that data analysis is an investigative process consisting of formulating their own questions, collecting appropriate data through various sources (censuses, nonrandom and random sample surveys, and comparative experiments with random assignment), analyzing data through graphs and simple summary measures, and interpreting results with an eye toward inference to a population based on a sample. As they begin to formulate their own questions, students become aware that the world around them is filled with data that affect their own lives, and they begin to appreciate that statistics can help them make decisions based on data, investigation, and logic.
Level C
Objectives of Level C
Level C is designed to build on the foundation developed in Levels A and B. In particular, Levels A and B introduced students to
á Statistics as an investigatory process
á The importance of using data to answer appropriately framed questions
á Types of variables (categorical versus numerical)
á Graphical displays (including bar graph, histogram, boxplot, scatterplot)
á Tabular Displays (including twoway frequency tables for categorical data and both ungrouped and grouped frequency/relative frequency tables for numerical data)
á Numerical summaries (including counts, proportions, mean, median, range, quartiles, interquartile range, measures of correlation)
á Common study designs (including census, simple random sample, randomized designs for experiments)
á The process of drawing conclusions from data
á The role of probability in statistical investigations
At Level C, all of these ideas are revisited, but the types of studies emphasized here are of a deeper statistical nature. Statistical studies at this level require students to draw on basic concepts from earlier work, to extend the concepts to cover a wider scope of investigatory issues, and to develop a deeper understanding of inferential reasoning and increased ability to explain that reasoning to others.
At Level C, additional concepts are developed for the interpretation and the use of statistical methods to answer questions. In general,
á Students should be able to formulate questions that can be answered with data.
á Students should be able to devise a reasonable plan for collecting appropriate data through observation, sampling or experimentation.
á Students should be able to draw conclusions and use data to support these conclusions.
á
Students should be able to understand the role
that variability plays in the decision making process.
Specifically, Level C recommendations include:
1. Formulate Questions
Students should be able to formulate questions and determine how data can be collected and analyzed to provide an answer.
2. Collect Data
Students should understand what constitutes good practice in conducting a sample survey.
Students should understand what constitutes good practice in conducting an experiment.
Students should understand what constitutes good practice in conducting an observational study.
Students should be able to design and implement a data collection plan for statistical studies, including observational studies, sample surveys, and simple comparative experiments.
3. Analyze Data
Students should be able to summarize numerical and categorical data using tables, graphical displays, and numerical summary statistics such as the mean and standard deviation.
Students should understand how sampling distributions (developed through simulation) are used to describe sampletosample variability.
Students should be able to recognize association between two categorical variables.
Students should be able to describe relationships between two numerical variables using linear regression and the correlation coefficient.
4. Interpret Results
Students should understand the meaning of statistical significance and the difference between statistical significance and practical significance.
Students should understand the role of Pvalues in determining statistical significance.
Students should be able to interpret the margin of error associated with an estimate of a population characteristic.
An Introductory ExampleÐObesity in America
Data and the stories that surround the data must be of interest to students! It is important to remember this when teaching data analysis. It is also important to choose data and stories that have enough depth to demonstrate the need for statistical thinking. The following example illustrates this.
Students are interested in issues that affect their lives, and issues of health often fall into that category. News items are an excellent place to look for stories of current interest, including items on health, and one healthrelated topic making lots of news lately is obesity. The following relates to a news story that is rich enough to provide a context for many of the statistical topics to be covered at Level C.
A recent newspaper article begins with the following lines. ÒAsk anyone: Americans are getting fatter and fatter. Advertising campaigns say they are. So do federal officials and the scientists they rely on. É In 1991, 23 percent of Americans fell into the obese category; now 31 percent do, a more than 30 percent increase. But Dr. Jeffrey Friedman, an obesity researcher, at Rockefeller University, argues that contrary to popular opinion, national data do not show Americans growing uniformly fatter. Instead, he says, the statistics demonstrate clearly that while the very fat are getting fatter, thinner people have remained pretty much the same. É The average weight of the population has increased by just 7 to 10 pounds.Ó The discussion in the article refers to adults.
The following are suggested questions to explore with students who have
a Level B background in statistics, but are moving on to Level C.
a. Sketch what you think a distribution of weights of American adults
might have looked like in 1991.
Adjust the sketch to show what the distribution of weights might look
like today (actually, 2002). Before
making your sketches, think about the shape, center and spread of your
distributions. Will the
distribution be skewed or symmetric?
Will the median be smaller, larger or about the same size as the
mean? Will the spread increase as
you move from the 1991 distribution to the 2002 distribution?
b. Which sounds more newsworthy, ÒObesity has increased by over 30%Ó or
ÒOn the average, the weight of Americans has increased by less than 10
pounds?Ó Explain your
reasoning.
c. The title of the article is The Fat Epidemic: He Says ItÕs an Illusion. [See New York Times, June 8, 2004 or Chance, Vol. 17. No. 4, Fall 2004, p. 3 for the complete article.] Do you think this is a fair title? Explain your reasoning.
d. The data on which the percentages are based come from the National Center for Health Statistics, National Health and Nutrition Examination Survey (NHANES) 2002. This is survey of approximately 5800 residents of the U.S. Although the survey design is more complicated than a simple random sample, the margin of error calculated as if it were a simple random sample is a reasonable approximation. What is an approximate margin of error associated with the 31% estimate of obesity for 2004? Interpret this margin of error for a newspaper reader who never studied statistics.
For the curious, information on how obesity is defined can be found at http://www.obesity.org/
In answering these questions, students at Level C should realize that a distribution of weights is going to be skewed toward the larger values (many are overweight but few are underweight). This generally produces a situation in which the mean is larger than the median. Because 8% shifted over the obesity line between 1991 and 2002, but the average weight (or center) did not shift very much, the upper tail of the distribution must have gotten Òfatter,Ó indicating a larger spread for the 2002 data. Students will have a variety of interesting answers for parts (b) and (c). The role of the teacher is help students understand whether or not their answers are supported by the facts. Part (d) gets students thinking about a concept studied at Level C.
The Investigatory Process at Level C
Because Level C revisits many of the same topics as Levels A and B but at a deeper and more sophisticated level, we begin by describing how the investigatory process looks at Level C. This general discussion is then followed by several examples.
Formulating Questions
As stated at the beginning of Level A, data are more than just numbers. Students need to understand the types of questions that can be answered with data. ÒIs the overall health of high school students declining in this country?Ó That is too big a question to answer by a statistical investigation (or even many statistical investigations). Certain aspects of the health of students, however, can be investigated by formulating more specific questions like ÒWhat is the rate of obesity among high school students?Ó; ÒWhat is the average daily caloric intake for high school seniors?Ó; ÒIs a threedayaweek exercise regimen enough to maintain heart rate and weight within acceptable limits?Ó Question formulation, then, becomes the starting point for a statistical investigation.
Collecting DataÑTypes of Statistical Studies
Most questions that can be answered through data collection and interpretation require data from a designed study, either a sample survey or an experiment. These two types of statistical investigations have some common elements, as each requires randomization both for purposes of reducing bias and building a foundation for statistical inference, and each makes use of the common inference mechanisms of margin of error in estimation and pvalue in hypothesis testing (both to be explained later). But these two types of investigations have very different objectives and requirements. Sample surveys are used to estimate or make decisions about characteristics (parameters) of populations and a welldefined, fixed population is the main ingredient of such a study. Experiments are used to estimate or compare the effects of different experimental conditions, called treatments, and require welldefined treatments and experimental units on which to study those treatments.
Estimating the proportion of residents of a city who would support an increase in taxes for education requires a sample survey. If the selection of residents is random, then the results from the sample can be extended to represent the population from which the sample was selected. A measure of sampling error (margin of error) can be calculated to ascertain how far the estimate is likely to be from the true value. Testing to see if a new medication to improve breathing for asthma patients produces greater lung capacity than a standard medication requires an experiment in which a group of patients who have consented to participate in the study are randomly assigned to either the new or the standard medication. With this type of randomized comparative design, an investigator can determine, with a measured degree of uncertainty, whether or not the new medication caused an improvement in lung capacity. Randomized experiments are, in fact, the only type of statistical study capable of establishing cause and effect. Any generalization extends only to the types of units used in the experiment, however, as the experimental units are not usually sampled randomly from a larger population. To generalize to a larger class of experimental units, more experiments would have to be conducted. That is one reason why replication is a hallmark of good science.
Studies that have no random selection of sampling units or random assignment of treatments to experimental units are called observational studies in this document. A study of how many students in your high school have asthma and how this breaks down among gender and age groups would be of this type. Observational studies are not amenable to statistical inference in the usual sense of that term, but they can provide valuable insights on the distribution of measured values and the types of associations among variables that might be expected.
At Level C, Students should understand the key features of both sample surveys and experimental designs, including how to set up simple versions of both types of investigations, how to analyze the data appropriately (as the correct analysis is related to the design), and how to clearly and precisely state conclusions for these designed studies. Key elements of the design and implementation of data collection plans for these types of studies follow.
Sample Surveys
Students should understand that obtaining good results from a sample survey depends on four basic features  the population, the sample, the randomization process that connects the two, and the accuracy of the measurements made on the sampled elements. For example, to investigate a question on health of students, a survey might be planned for a high school. What is the population to be investigated? Is it all the students in the school (which changes on a daily basis)? Perhaps the questions of interest involve only juniors and seniors. Once the population is defined as precisely as possible, what is an appropriate sample size and how can a random sample be selected? Is there, for example, a list of student who can then be numbered for random selection? Once the sampled students are found, what questions are to be asked of them? Are the questions fair and unbiased (as far as possible) and can or will the students actually answer them accurately?
Two samples of size 50 from the same population of students will not give the same result on, say, the proportion of students who eat a healthy breakfast. This variation from sample to sample is called sampling variability or sampling error and this is the type of error that we can measure statistically. Other errors (or biases) that slip in because the sample was not a random sample to begin with, because the sample was selected from the wrong population, because a number of the students selected in the sample refused to participate, or because the questions were poorly written and the responses ambiguous, are not easily measured. These types of errors should be considered carefully before the study begins so that plans can be made to reduce them as far as possible.
Experiments
At Level C, Students should understand that obtaining good results from an experiment depends upon four basic features  welldefined treatments, appropriate experimental units to which these treatments can be assigned, the randomization process that is used to assign treatments to experimental units, and accurate measurements of the results of the experiment. Experimental units generally are not randomly selected from a population of possible units. Rather, they are the ones that happen to be available for the study. In experiments with human subjects, the people involved have to sign an agreement stating that they are willing to participate in an experimental study. In experiments with agricultural crops, the experimental units are the field plots that happen to be available. In an industrial experiment on process improvement the units may be the production lines in operation in a given week.
As in the sample survey, variability due to the random assignment of treatments to experimental units can be measured statistically, but variability resulting from a poor design cannot. Suppose treatment A gets assigned only to patients over the age of 60 and treatment B only to patients under the age of 50. If the treatment responses differ, it is now impossible to tell whether the difference is due to the treatments themselves or the ages of the patients. (This kind of bias in experiments is called confounding.) The randomization process, if properly done, will usually balance treatment groups so that this type of bias is minimized.
Observational Studies
At Level C, Students should understand that observational studies are useful for suggesting patterns in data and relationships between variables, but do not provide a strong basis for estimating population parameters or establishing differences among treatments. Asking the students in one classroom whether or not they eat a healthy breakfast is not going to help you establish the proportion of healthy breakfast eaters in the school because the students in one particular classroom may not be representative of the students in the school. Random sampling is the only way to be confident of a representative sample for statistical purposes. Similarly, feeding your cats diet A and your neighbors cats diet B is not going to allow you to claim that one diet is better than the other in terms of weight control because there was no random assignment of experimental units (cats) to treatments (diets), and as a consequence there may be many biasing or confounding variables. Studies of the type suggested above are merely observational; they may suggest patterns and relationships but they are not a reliable basis for statistical inference.
Analyzing Data
When analyzing data from welldesigned sample surveys, students at Level C should understand that an appropriate analysis is one that can lead to justifiable inferential statements based on estimates of population parameters. The ability to draw conclusions about a population using information from a sample depends on information provided by the distribution (called the sampling distribution) of the sample statistic that is being used to summarize the sample data. The two most common statistics used in applications are the sample proportion for categorical data and the sample mean for measurement data.
It is at Level C that sampletosample variability is addressed in a bit more depth. Exploring how the information provided by a sampling distribution is used for generalizing from a sample to the larger population enables students at Level C to draw more sophisticated conclusions from the statistical studies that they conduct. At Level C, it is recommended that the sampling distribution of a sample proportion and of a sample mean be developed through simulation. More formal treatment of sampling distributions can be left to AP Statistics and collegelevel introductory statistics courses.
Because the sampling distribution of a sample statistic is a topic with which many teachers may not be familiar, the next sections show how simulation can be used to obtain an approximate sampling distribution for a sample proportion and for a sample mean.
Example 1: The Sampling Distribution of a Sample Proportion
Properties of the sampling distribution for a sample proportion can be illustrated quite simply by using random digits as a device to model various populations. Suppose a population is assumed to have 60% ÒsuccessesÓ (p = .6) and we are to take a random sample of n=40 cases from this population. How far can we expect the sample proportion of successes to deviate from the population value of .60? This can be answered by determining what the sampling distribution of sample proportions looks like through repeated selection of samples of 40 random digits (with replacement) from a population in which 6 of the ten digits from 0 to 9 are labeled ÒsuccessÓ and 4 are not. A simulated distribution constructed using sample proportions from 200 different random samples of size 40 from a population with 60% successes is shown in Figure 25. This is an approximation to the sampling distribution of the sample proportion for samples of size 40 from a population where the actual proportion is .60.
Getting into the habit of using shape, center and spread to describe distributions, one can state that this simulated sampling distribution of sample proportions has a mound shape (approximately normal). Because the mean and standard deviation of the 200 sample proportions were .59 and .08 respectively, the simulated distribution shown in Figure 25 has a mean of .59 (very close to p = .6) and a standard deviation of .08. By studying this approximate sampling distribution and others that can be generated the same way, students will see patterns emerge and can verify that the sampling distributions for sample proportions center at p, the population proportion of ÒsuccessesÓ, and that the standard deviations for the sampling distribution is approximately
_{}
where _{}is the observed sample proportion in a single sample. In addition, if the sample size is reasonably large, the shape of the sampling distribution is approximately normal.
A followup analysis of these simulated sampling distributions can show students that about 95% of the sample proportions lie with a distance of
_{}
from the true value of p. This distance, sometimes called the margin of error, is useful in estimating where a population proportion might lie in relation to a sample proportion for a new study in which the true population proportion is not known.
Example 2: The Sampling Distribution of a Sample Mean
Properties of the sampling distribution for a sample mean can be illustrated in way similar to that used for proportions in Example 1. Figure 26 shows the distribution of the sample mean when samples of 10 random digits are selected (with replacement) and the sample mean is computed. This models sampling from a population that has a uniform distribution with equal numbers of 0Õs, 1Õs, 2Õs, and so on.
The approximate sampling distribution shown in Figure 26 can be described as approximately normal with a mean of 4.53 (the mean of the 200 sample means from the simulation) and a standard deviation of 0.87 (the standard deviation of the 200 sample means). The population described has a mean of 4.5 and a standard deviation of 2.9. By studying this approximate sampling distribution and others produced similarly, students will see patterns and can verify that the standard deviation of the sampling distribution for a sampling mean is approximately the population standard deviation divided by the square root of the sample size, in this case 2.9/Ã10 = 0.92.
The margin of error in estimating a population mean from a single random sample is approximately
_{}
where s denotes the sample standard deviation for the observed sample. The sample mean should be within this distance of the true population mean about 95% of the time in repeated random sampling.
Interpreting Results
Generalizing from Samples
The key to statistical inference is the sampling distribution of the sample statistic, which provides information on the population parameters being estimated or the treatment effects being tested. As described in the previous section, knowledge of the sampling distribution for a statistic, like a sample proportion or sample mean, leads to a margin of error that provides information about the maximum likely distance between a sample estimate and the population parameter being estimated. Another way to state this key concept of inference is that an estimator plus and minus the margin of error produces an interval of plausible values for the population parameter, any one of which could have produced the observed sample result as a reasonably likely outcome.
Generalizing from Experiments
Do the treatments differ? In analyzing experimental data this is one of the first questions asked. This question of difference is generally posed in terms of differences between the centers of the data distributions (although it could be posed as a difference between 90^{th} percentiles or any other measure of an aspect of a distribution). Because the mean is the most commonly used statistic for measuring center of a distribution, this question of differences is generally posed as a question about the differences in means. The analysis of experimental data, then, usually involves a comparison of means.
Unlike sample surveys, experiments do not depend on random samples from a fixed population. Instead, they require random assignment of treatments to preselected experimental units. The key question, then, is of the following form: ÒCould the observed difference in treatment means be due to the random assignment (chance) alone, or can it be attributed to the treatments administered?Ó
Examples Illustrating Important Concepts of Level C
The
following examples are designed to illustrate and further illuminate the
important concepts of Level C by carefully considering the four phases of a
statistical analysis (question, design, analysis, interpretation) in a variety
of real contexts.
Example 3: A Survey of Music Preferences
A survey of student music preferences was introduced at Level A, where the analysis consisted of making counts of student responses and displaying the data in a bar graph. At Level B the analysis was expanded to consider relative frequencies of preferences and crossclassified responses for two types of music displayed on a twoway table. Suppose the survey included the following questions:
1. What kinds of music do you like?
a. Do
you like country music? Yes
or No
b. Do
you like rap music? Yes
or No
c. Do
you like rock music? Yes
or No
2. Which of the following types of music do you like to
most? Select only one.
Country Rap/Hip
Hop Rock
In order to be able to generalize to all students at the school, a representative sample of students from a school is needed. This could be accomplished by selecting a simple random sample of 50 students from the school. The results can then be generalized to the school (but not beyond) and the Level C discussion will center on basic principles of generalization, or statistical inference.
A Level C analysis begins with a twoway table of counts that summarizes the data on two of the questions, ÒDo you like rock music?Ó and ÒDo you like rap music?Ó The table provides a way to examine the responses to each question separately as well as a way to explore possible connections (association) between the two categorical variables. Suppose the survey data resulted in the twoway table shown in Table 10.
Table 10: Two way frequency table
Like Rock Music?
Yes No Row Totals
Yes 25 4 29
Like Rap Music?
No 6 15 21
Column Totals 31 19 50 = Grand Total
As demonstrated at Level B, there are a variety of ways to interpret data summarized in a twoway table such as Table 10. Some examples based on all 50 students in the survey include:
25 of the 50 students (50%) liked both rap and rock music.
29 of the 50 students (58%) liked rap music.
19 of the 50 students (38 %) did not like rock music.
One type of statistical inference relates to conjectures (hypotheses) made before the data were collected. Suppose a student says ÒI think more than 50% of the students in the school like rap music.Ó The statistical question then is whether the sample data support this claim or not. One way to arrive at an answer is to set up a hypothetical population that has 50% successes (like even and odd digits produced by a random number generator) and repeatedly take samples of size 50 from it, each time recording the proportion of even digits. The sampling distribution of proportions so generated will be similar to the one in Figure 27.
A sample proportion greater than or equal to the observed .58 occurs 12 times out of 100 just by chance when the actual population proportion is only .50.. This suggests that the result of .58 is not a very unusual occurrence when sampling from a population with .50 as the ÒtrueÓ proportion of students who like rap music, so the evidence in support of the studentÕs claim is not strong enough. A population value of .50 (or maybe even something smaller) is plausible based on what was observed in the sample. The fraction of times the observed result is matched or exceeded (.12 in this investigation) is called the approximate pvalue. The pvalue represents the chance of observing a sample result as extreme as the one observed in the sample when the hypothesized value is in fact correct. A small pvalue would have supported the studentÕs claim, because this would have indicated that if the population proportion was .50, it would have been very unlikely that a sample proportion of .58 would have been observed.
Suppose another student hypothesized that at most 40% of the students in the school like rap music. Now, the samples of size 50 must be repeatedly selected from a population that has 40% successes (like random digits with 1 through 4 representing success and the other digits representing failure). Figure 28 shows the results of one such simulation. The observed result of .58 was reached only 1 time out of 100 (approximate pvalue = .01). It is not likely that a population in which 40% of the students like rap music would have produced a sample proportion of 58% in a random sample of size 50. This studentÕs claim can be rejected!
Another way of stating the above is that .5 is a plausible value for the true
population proportion, based on the sample evidence, but .4 is not. A set of plausible values can be
found by using the margin of error.
As explained previously, the margin of error for a sample proportion is
approximately
_{}
Thus, any population proportion between .58  .14 = .44 and .58+.14=.72 can be considered a plausible value for the true proportion of students who like rap music. Notice that .5 is well within this interval but .4 is not.
Another type of question that could be asked about the student preferences on music is of the form ÒDo those who like rock music also tend to like rap music?Ó In other words, is there an association between liking rock music and liking rap music? The same data from the random sample of 50 students can be used to answer this question.
According to Table 10 a total of 31 students in survey liked rock music. Among those students, the proportion who also like rap music is (25/31) = .81. Among the 19 students who do not like rock music, 4/19 = .21 is the proportion who like rap music. The large difference between these two proportions (.60) suggests that most students who like rock also like rap. There appears to be a strong association between liking rock music and liking rap music.
But could this association simply be due to chance (a consequence only of the random sampling)? If there were no association between the two groups the 31 students who like rock would behave as a random selection from the 50 in the sample. To simulate this situation we make up a population of 29 1Õs (those who like rap) and 21 0Õs (those who do not like rap) and mix them together. Then, we select 31 (representing those who like rock) at random and see how many 1Õs (those who like rap) we get. It is this entry that goes into the (yes, yes) cell of the table, and from that data the difference in proportions can be calculated. Repeating the process many times produces a simulated sampling distribution for the difference between two proportions, as shown in Figure 29.
The observed difference of .6 was never reached in 100 trials, indicating that the observed difference cannot be attributed to chance alone. There is evidence of a real association between liking rock music and liking rap music.
Example 4: An Experiment on the Effects of Light on the Growth of Radish Seedlings
What is the effect of different lengths of light and dark on the growth of radish seedlings? This question was posed for a class of biology students who then set about designing and carrying out an experiment to investigate the question. All possible relative lengths of light to dark cannot possibly be investigated in one experiment, so the students decided to focus the question on three treatments: 24 hours of light, 12 hours of light and 12 hours of darkness, and 24 hours of darkness. This covers the extreme cases and one in the middle.
With the help of a teacher, the class decided to use plastic bags as "growth chambers." The plastic bags would permit the students to observe and measure the germination of the seeds without disturbing them. Two layers of moist paper towel were put into a disposable plastic bag, with a line stapled about 1/3 of the way from the bottom of the bag (side opposite opening) to hold the paper towel in place and provide a seam to hold the radish seeds in place for germination.
Figure 30: Seed experiment
Although three growth chambers would be sufficient to examine the three treatments, this class made four growth chambers with one designated for the 24 hours of light treatment, one for the12 hours of light and 12 hours of darkness treatment, and two for the 24 hours of darkness treatment. One hundred twenty seeds were available for the study. Thirty of the seeds were chosen at random and placed along the stapled seam of the 24 hours of light bag. Thirty seeds were then chosen at random from the remaining 90 seeds and placed in the bag the 12 hours of light and 12 hours of darkness bag. Finally, 30 of the remaining 60 seeds were chosen at random and placed in one of the 24 hours of darkness bags and the final 30 seeds placed in the final bag. After three days, the lengths of radish seedlings for the germinating seeds were measured and recorded. These data are provided in Table 11; the measurements are in millimeters.
Treatment 1 24 light 
Treatment 2 12 light, 12
dark 
Treatment 3 24 dark 

2 
3 
5 
20 
3 
4 
5 
20 
5 
5 
8 
22 
5 
9 
8 
24 
5 
10 
8 
25 
5 
10 
8 
25 
5 
10 
10 
25 
7 
10 
10 
25 
7 
10 
10 
25 
7 
11 
10 
26 
8 
13 
10 
29 
8 
15 
11 
30 
8 
15 
14 
30 
9 
15 
14 
30 
10 
17 
15 
30 
10 
20 
15 
30 
10 
20 
15 
30 
10 
20 
15 
31 
10 
20 
15 
33 
10 
20 
15 
35 
10 
21 
16 
35 
10 
21 
20 
35 
14 
22 
20 
35 
15 
22 
20 
35 
15 
23 
20 
35 
20 
25 
20 
36 
21 
25 
20 
37 
21 
27 
20 
38 


20 
40 
A good first step is to make a plot of the data to look for patterns and any unusual departures from the patterns. Boxplots are ideal for comparing data from more than one treatment, as you can see in Figure 31. Both the centers and the spreads increase as the amount of darkness increases. There are three outliers (one at 20mm and two at 21mm) in the Treatment 1 (24 hours of light) data. Otherwise, the distributions are fairly symmetric, which is good for statistical inference.
Treatment 1is 24 hours of light
Treatment 2 is 12 hours of light and 12 of darkness
Treatment 3 is 24 hours of darkness.
The summary statistics for these data are as follows:
Treatment n MEAN MEDIAN ST DEV
1 28 9.643 9.500 5.027
2 28 15.82 16.00 6.76
3 58 21.86 20.00 9.75
Experiments are designed to compare treatments, usually by comparing means. The original question on the effect of different lengths of light and dark on the growth of radish seedlings might be turned into two questions about treatment means. Is there evidence that 12 hours of light and 12 hours of dark (Treatment 2) has a higher mean than 24 hours of light (Treatment 1)? Is there evidence that 24 hours of dark (Treatment 3) has a higher mean than 12 hours of light and 12 hours of dark (Treatment 2)? Based on the boxplots and the summary statistics, it looks like the means differ, but students should now realize that this casual observation should be verified by ruling out chance as a possible explanation for the observed difference. The Treatment 2 mean is 6.2 mm larger than the Treatment 1 mean. If there is no real difference between the two treatments, then the observed difference must be due to the random assignment of seeds to the bags; one bag was simply lucky enough to get a preponderance of good and lively seeds. But, if a difference this large (6.2 mm) is likely to be the result of randomization alone, then we should see differences of this magnitude quite often if we repeatedly rerandomize the measurements and calculate a new difference in observed means. This, however, is not the case, as is seen in Figure 32. This figure was produced by mixing the measurements from Treatments 1 and 2 together, randomly splitting them into two groups of 28 measurements each, recording the difference in means for the two groups, and repeating the process 200 times.
The observed difference of 6.2 mm was exceeded only one time in 200 trials, for an approximate pvalue of 1/200. This is very small, and gives strong evidence to support the hypothesis that there is a real difference between the means for Treatments 1 and 2. The observed difference of 6.2 mm is very unlikely to be due simply to chance.
In a comparison of the means for Treatments 2 and 3 the same procedure is used, except that the combined measurements are split into groups of 28 and 58 each time. Again, the observed difference of 6 mm was exceeded only 1 time out of 200 trials (see Figure 33), giving strong evidence of a real difference between the means for Treatments 2 and 3. In summary, all three means show real differences that cannot be explained by the random assignment of seeds to the bags; the more hours of darkness, the greater the growth of the seedling, at least for these three light versus darkness times.
Figure
33: Dotplot showing differences of means
Example 5: Estimating the Density of the Earth Ð A Classical Study
What is the density of the earth? This is a question the intrigued the great scientist Henry Cavendish, who attempted to answer the question in 1798. Cavendish estimated the density of the earth by using the crude tools that were available to him at the time. He did not literally take a random sample; he measured on different days and at different times, as he was able. But, the density of the earth does not change over time, so his measurements can be thought of as a random sample of all the measurements he could have taken on this constant. The variation in the measurements is due to his measurement error, not to changes in the earthÕs density. The earthÕs density is the constant that is being estimated.
This is a typical example of an estimation problem that occurs in science. There is no real ÒpopulationÓ of measurements that can be sampled; rather, the sample data is assumed to be a random selection from the conceptual population of all measurements that could have been selected. At this point there may be some confusion between an ÒexperimentÓ and a Òsample surveyÓ because Cavendish actually conducted a scientific investigation to get his measurements. The key, however, is that he conducted essential the same investigation many times with a goal of estimating a constant, much like interviewing many people to estimate the proportion who favor a certain candidate for office. He did not randomly assign treatments to experimental units for the purpose of comparing treatment effects.
A famous Cavendish dataset contains his 29 measurements of the density of the earth, in grams per cubic centimeter. The data are shown below. [Source: http://lib.stat.cmu.edu/DASL/]
Density
5.50 5.57 5.42 5.61 5.53 5.47 4.88 5.62 5.63 4.07 5.29 5.34
5.26 5.44 5.46 5.55 5.34 5.30 5.36 5.79 5.75 5.29 5.10 5.86
5.58 5.27 5.85 5.65 5.39
A look at the data is appropriate before proceeding with an analysis. The histogram below shows the data to be skewed toward the smaller values, with one unusually small value. If Cavendish were alive, you might check to see if that measurement might have been a mistake (and that is certainly what you should do for a current data set).
The mean of the 29 measurements in 5.420 and the standard deviation is 0.339. That makes the margin of error
_{}
The analysis shows that any value between 5.420  0.126 and 5.420 + 0.126, or in the interval (5.294, 5.546), is a plausible value of the density of the earth. That is, any value in the interval is consistent with the data obtained by Cavendish. Now, the questionable low observation should be taken into account as it will lower the mean and increase the standard deviation. If that measurement is regarded as a mistake and removed from the data set, the mean of the 28 remaining observations is 5.468 and the standard deviation is 0.222, producing a margin of error of 0.084 and an interval of plausible values of (5.384, 5.552).
Students can now check on how well Cavendish did; modern methods pretty much agree that the average density of the earth is about 5.515 grams per cubic centimeter. The great 18^{th} century scientist did well!
Example 6: Linear Regression AnalysisÑHeight v. Forearm Length
Regression analysis refers to the study of relationships between variables. If the ÒcloudÓ of points in a scatterplot of paired numerical data has a linear shape, a straight line may be a realistic model of the relationship between the variables under study. The least squares line runs through the center (in some sense) of the cloud of points. Residuals are defined to be the deviations in the y direction between the points in the scatterplot and the least squares line; spread is now the variation around the least squares line, as measured by the standard deviation of the residuals. When using a fitted model to predict a value of y from x, the associated margin of error depends on the standard deviation of the residuals.
Relationships among various physical features, such as height versus arm span and neck size versus shoe size, can be the basis of many interesting questions for student investigation. If I were painting a picture of a person, how could I get the relative sizes of the body parts correct? This question prompted students to carry out an investigation of one of the possible relationships, that between forearm length and height.
The students responsible for the study sampled other students on which to make forearm and height measurements. Although the details are not clear on how the sample was actually selected, we will suppose that it is representative of students at the school and has the characteristics of a random sample. An important consideration here is to agree on the definition of ÒforearmÓ before beginning to take measurements. The data obtained by the students (in centimeters) are provided in Table 13.
Forearm (cm) 
Height (cm) 
45.0 
180.0 
44.5 
173.2 
39.5 
155.0 
43.9 
168.0 
47.0 
170.0 
49.1 
185.2 
48.0 
181.1 
47.9 
181.9 
40.6 
156.8 
45.5 
171.0 
46.5 
175.5 
43.0 
158.5 
41.0 
163.0 
39.5 
155.0 
43.5 
166.0 
41.0 
158.0 
42.0 
165.0 
45.5 
167.0 
46.0 
162.0 
42.0 
161.0 
46.0 
181.0 
45.6 
156.0 
43.9 
172.0 
44.1 
167.0 
Regression Analysis: Height
versus Forearm
The regression equation is
Height = 45.8 + 2.76 Forearm
Predictor Coef SE
Coef T P
Constant 45.78 19.74
2.32 0.030
Forearm 2.7631 0.4459 6.20 0.000
S = 5.76625 RSq = 63.6% RSq(adj) = 61.9%
Is the slope of 2.8 ÒrealÓ or simply a result of the random selection process? This question can be investigated using simulation. A description of this simulation is included in the Appendix to Level C.
Example 7: Comparing Mathematics Scores: An Observational Study
Data are often presented to us in a form that does not call for much analysis, but does require some insight into statistical principles for correct interpretation. Test scores often fall into this category. Table 14 gives data on the state mean scores for the National Assessment of Educational Progress (NAEP) 2000 grade 4 mathematics scores for Louisiana and Kentucky. Even though these scores are based on a sample of students, these are the scores assigned to the states and consequently they can be considered observational data from that point of view.

Overall Mean 
Mean for Whites 
Mean for Nonwhites 
Percent White 
Louisiana 
217.96 
229.51 
204.94 

Kentucky 
220.99 
224.17 

87 
To see if students understand the
table, it is often informative to ask them to fill in a few omitted
entries.
a. Fill in the two missing entries in the
table. (53% and 199.71)
More substantive questions
involve the seeming contradictions that often occur in data of this type. The might be phrased as follows.
b. For the two states, compare the overall
means. Compare the means for
whites. Compare the means of
nonwhites. What do you
observe?
c. Explain why the reversals in direction
take place once the means are separated into racial groups.
It is genuinely surprising to students
that data summaries (means in this case) can go in one direction in the
aggregate but can go in the opposite direction for each subcategory when
disaggregated. This
phenomenon is called SimpsonÕs paradox.
Example 8: Observational Study Ð Toward Establishing Cause
Observational studies are the only option for situations in which it is impossible or unethical to randomly assign treatments to subjects. Such situations are a common occurrence in the study of causes of diseases. A classical example from this field is the study of the relationship between smoking and lung cancer, a debate that was raging during the 1950Õs and 1960Õs. Society will not condone the notion of assigning some people to be smokers and others to be nonsmokers in an experiment to see if smoking causes lung cancer. So, the evidence has to be gathered from observing the world as it is. But, the data collection process can still be designed in clever ways so as to obtain as much information as possible.
Here is an example from the smoking versus lung cancer debates. A group of 649 men with lung cancer were identified from a certain population in England. A control group of the same size was established by matching these patients with other men from the same population who did not have lung cancer. The matching was on background variables such as ethnicity, age and socioeconomic status. (This is called a casecontrol study.) The objective, then, is to compare the rate of smoking among those with lung cancer to the rate for those without cancer.
Table 15: Cigarette Smoking and Lung Cancer

Lung cancer cases 
Controls 
Totals 
Smokers 
647 
622 
1269 
Nonsmokers 
2 
27 
29 
[Source: http://wwwphm.umds.ac.uk/teaching/ClinEpid/ObservS.htm]
First, make sure the students understand the nature of the
data in the table. Does this show,
for example, that there was a very high percentage of smokers in
Another step in establishing cause in observational studies is to see if the increase in exposure to the risk factor produces an increase in the disease. This was done with the same casecontrol study by looking at the level of smoking for each person, producing Table 16.
Cigarettes / day 
Lung cancer cases 
Controls 
Probability 
0 
2 
27 
0.07 
114 
283 
346 
0.45 
1524 
196 
190 
0.51 
25+ 
168 
84 
0.67 
The term ÒProbabilityÓ is used in the same sense as above. If a person is randomly selected from the 114 level, the chance that the person falls into the cancer column is .45, and so on for the other rows. The important result is that these ÒprobabilitiesÓ increase with the level of smoking. This is evidence that an increase in exposure produces an increase in the disease rate.
Even with this additional evidence, students should understand that a cause and effect relationship is not established beyond a reasonable doubt. The main reason for this is that these observational studies are subject to bias in the selection of patients and controls. Another study of this type could have produced a different result. (As it turned out, many studies of this type produced remarkably similar results. That, coupled with laboratory experiments on animals that established a biological link between smoking and lung cancer, eventually settled the issue for most people.)
The Appendix to Level C contains more examples of the types discussed in this section.
The Role of Probability in Statistics
Teachers and students must understand that statistics and
probability are not the same thing.
Statistics uses probability, much as physics uses calculus, but only
certain aspects of probability make their way into statistics. The concepts of probability needed for
introductory statistics (with emphasis on data analysis) include relative
frequency interpretations of data, probability distributions as models
of populations of measurements, an introduction to
the normal distribution as a model for sampling distributions, and the basic
ideas of expected value. Counting
rules, most specialized distributions, and development of theorems on the
mathematics of probability should be left to areas of discrete mathematics
and/or calculus.
Understanding the reasoning and logic of statistical
inference requires a basic understanding of some important ideas in
probability. Students should be
able to
In addition, because so many of the standard inferential
procedures are based on the normal distribution, students should be able to
evaluate probabilities using the normal distribution (preferably with the aid
of technology).
Probability is an attempt to quantify uncertainty. The fact that, even though it is not
possible to predict individual outcomes, the longrun behavior of a random
process is predictable leads to the longrun relative frequency interpretation
of probability. Students should be
able to interpret the probability of an outcome as the proportion of the time,
in the long run, that the outcome would occur. This longrun relative frequency
interpretation of probability also provides the justification for using
simulation to estimate probabilities.
After observing a large number of chance outcomes, the observed
proportion of occurrence for the outcome of interest can be used as an estimate
of the relevant probability.
Students also need to understand the concept of
independence. Two outcomes are
independent if our assessment of the chance that one outcome occurs is not
affected by knowledge that the other outcome has occurred. Particularly important to statistical
inference is the notion of independence in sampling settings. Random selection (with replacement) from
a population ensures that the observations in a sample are independentÑfor
example, knowing the value of the third observation does not provide any
information about the value of the fifth (or any other) observation. Many of the methods used to draw
conclusions about a population based on data from a sample require the
observations in a sample to be independent.
Most importantly, the concepts of probability play a
critical role in developing statistical methods that make it possible to draw
conclusions based on data from a sample and to assess the reliability of such
conclusions.
To clarify the connection between data analysis and probability we will return to the key ideas presented in the inference section. Suppose an opinion poll shows 60% of sampled voters in favor of a proposed new law. A basic statistical question is, ÒHow far might this sample proportion be from the true population proportion?Ó That the difference between the estimate and truth is less than the margin of error approximately 95% of the time is based on a probabilistic understanding of the sampling distribution of sample proportions. For large random samples, this relative frequency distribution of sample proportions is approximately normal. Thus, students should be familiar with how to use appropriate technology to find areas under the normal curve.
Suppose an experimenter divides subjects into two groups with one group receiving a new treatment for a disease and the other receiving a placebo. If the treatment group does better than the placebo group, a basic statistical question is, ÒCould the difference have been caused by chance alone?Ó Again, the randomization allows us to determine the probability of a difference being greater than that observed under the assumption of no treatment effect, and this probability allows us to draw a meaningful conclusion from the data. (A proposed model is rejected as implausible not primarily because the probability of an observed outcome is small, but rather because it is in the tail of a distribution.) Adequate answers to the above questions also require knowledge of the context in which the underlying questions were asked and a ÒgoodÓ experimental design. This reliance on context and design is one of the basic differences between statistics and mathematics.
As demonstrated in the Analysis of Data section, it is known that the sampling distribution of a sample mean will be approximately normal under random sampling, as long as the sample size is reasonably large. The mean and standard deviation of this distribution are usually unknown (introducing the need for inference) but sometimes these parameter values can be determined from basic information about the population being sampled. To compute these parameter values students will need some knowledge of expected values, as demonstrated next.
According to the March 2000 Current Population Survey of the U.S. Census Bureau, the distribution of family size is as given by the data Table 17. (A family is defined as two or more related people living together. The number Ò7Ó really is the category Ò7 or moreÓ, but very few families are larger than 7.)
Family Size, x 
Proportion, p(x) 
2 
0.437 
3 
0.223 
4 
0.201 
5 
0.091 
6 
0.031 
7 
0.017 
The first connection between data and probability to note is that these proportions (really estimates from a very large sample survey) can be taken as approximate probabilities for the next survey. In other words, if someone randomly selects a U.S. family for a new survey, the probability that it will have 3 members is about .223.
The second point is that we can now find the mean and standard deviation of a random variable (call it X) defined as the number of people in a randomly selected family. The mean, sometimes called the expected value of X and denoted by E(X), is found by formula
_{}
For the family size data, V(X) = 1.54 and SD(X) = 1.24.
Thirdly, these facts can be assembled to describe the sampling distribution of the mean family size in a random sample of 100 families yet to be taken. That sampling distribution will be approximately normal in shape, centering at 3.11 with a standard deviation of 1.24/Ã100 = 0.124. This would be useful information for the person designing the next survey.
In short, the relative frequency definition of probability, the normal distribution, and the concept of expected value are the keys to understanding sampling distributions and statistical inference.
Summary of Level C
Students at Level C should become adept at using statistical tools as a natural part of the investigative process. Once an appropriate plan for collecting data has been implemented and the resulting data is in hand, the next step is usually to summarize the data using graphical displays and numerical summaries. At Level C, students should be able to select summary techniques appropriate to the type of data available, construct these summaries, and describe in context the important characteristics of the data. Students will use the graphical and numerical summaries learned at Levels A and B, but should be able to provide a more sophisticated interpretation that integrates the context and the objectives of the study.
At Level C, students should also be able to draw conclusions from data and support these conclusions using statistical evidence. Students should see statistics as providing powerful tools that enable them to answer questions and to make informed decisions. But, students should also understand the limitations of conclusions based on data from sample surveys and experiments, and should be able to quantify uncertainty associated with these conclusions using margin of error and related properties of sampling distributions.
References
American Diploma Project http://www.achieve.org/achieve.nsf/AmericanDiplomaProject?openform
Cobb, G. and D. Moore (2000). ÒStatistics and Mathematics: Tension and Cooperation,Ó
American Mathematical Monthly, pp. 615630.
College
Entrance Examination Board (2004). Course
Description: Statistics.
Conference Board of the Mathematical Sciences (2001). The Mathematical Education of Teachers.
Conover, W. J. (1999).
Practical Nonparametric Statistics. John Wiley and Sons, Page 235 (Equation
17).
Consumers Reports (June, 1993) Hot dogs.
51(6), 364367.
Data Ð Driven
Mathematics Series (1998),
Gnanadesikan,
Mrudulla, Richard L. Scheaffer, James M. Landwehr, Ann E. Watkins, Peter
Barbella, James Kepner, Claire M. Newman, Thomas E. Obremski, and Jim Swift
(1995). Quantitative Literacy Series,
Hollander, Miles and Proschan, Frank (1984). The Statistical Exorcist: Dispelling
Statistics Anxiety, Marcel Dekker, Pages 8388 and 121130.
Holmes, Peter (2001).
Correlation: From Picture to
Formula. Teaching Statistics 23(3), 6770.
Kader, Gary (1999).
Means and MADS. Mathematics Teaching in the Middle School
4(6), 398403.
Kader, Gary and Perry, Mike (1984). Learning Statistics with Technology. Mathematics Teaching in the Middle School
1(2), 130136.
Moore, D, and G. Cobb (1997). ÒMathematics, Statistics, and Teaching,Ó American Mathematical Monthly, 104, 801823
National Assessment Governing Board (2004). Mathematics Framework for 2005 National
Assessment of Educational Progress. Available: http://www.nagb.org/pubs/m_framework_05/toc.html.
National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics,
National Council of Teachers of Mathematics (20022004). Navigating through Data Analysis and
Probability Series.
National Council of Teachers of Mathematics (2000). Principles and Standards for School
Mathematics.
Steen,
Utts, Jessica A.
(1999). Seeing Through Statistics.
Appendix for Level A
What are Common Name Lengths?
What They Wonder AboutÐFormulate Questions
During the first week of school, a thirdgrade teacher is trying to help her students learn one anotherÕs names by playing various games. During one of the games, a student named MacKenzie noticed that she and her classmate Zacharius each have 9 letters in their name. MacKenzie conjectured that their names were longer than everyone elseÕs names. The teacher decided that this observation by the student provided an excellent opening for a statistics lesson.
The next school day, the teacher reminds students of MacKenzieÕs comment from the day before and asks the class what they would like to know about their classmatesÕ names. The class generates a list, which the teacher records on the board as follows:
¥ Who has the longest name? The shortest?
¥ Are there more 9 letter names or 6 letter names? How many more?
¥ WhatÕs the most common name length?
¥ How many letters are in all of our names?
¥ If you put all of the 8 and 9 letter names together, will there be as many as the 5 letter names?
Collect Data
The statistics lesson begins with students writing their names on sticky notes and posting them on the white board at the front of the room. This is a census of the classroom because they are gathering data from all students in the class.
Figure 36: Random placement of names
Given no direction about how to organize the notes, the students simply place them randomly on the board.
Analyze Data
In order to help students think about how to use graphical tools to analyze data, the teacher asks the students if they are able to answer any of the posed questions easily now by looking at the sticky notes, and the students say they cannot. The teacher then suggests that they think of ways to better organize the notes. A student suggests grouping the names according to how many letters are in each name.
Figure 37: Names clustered by length
The teacher again asks if they can easily answer the questions that are posed. The students say that they can answer some of the questions, but not easily. The teacher asks what they can do to make it easier to answer the questions. Because the students have been constructing graphs since kindergarten, they readily answer, ÒMake a graph!Ó The teacher then facilitates a discussion of what kind of graph they will make, and the class decides on a dotplot, given the fact that their names are already on sticky notes and given the available space on the board. Note that this display is not a bar graph because bar graphs are made when the data represent a categorical variable (such as favorite color). A dotplot is appropriate for a numerical variable such as the number of letters in a name.
Figure 38: Preliminary dotplot
Figure 39: Computergenerated dotplot
Interpret Results
The teacher then facilitates a discussion of each question
posed by the students, using the data displayed in the graph to answer the
questions. Students also add appropriate labels and titles to the graph. The
teacher helps students use the word ÒmodeÓ to answer the question about the
most common name length. She introduces the term ÒrangeÓ to help students
answer the questions about shortest and longest names. Students visualize from
the dotplot that there is variability in name length from individual to
individual. The range gives a sense of the amount of variability in name length
within the class.
The teacher then tells the students that there is another useful question they
can answer from this data. Sometimes it is helpful to know Òabout how long most
names are.Ó For instance, if you were making place cards for a luncheon, you
might want to know how long the typical name is in order to decide which size
of place cards to buy. The typical or average name length is called the mean.
Another way to think of this is ÒIf all of our names were the same length, how
long would they be?Ó To illustrate this new idea, the teacher has students work
in groups of 4, and each child takes a number of snap cubes so that the number
of cubes is the number of letters in his/her name. Then all four children at
one table put all of their snap cubes in a pile in the middle of the table.
They count how many cubes they have in total. Then they share the cubes fairly
with each child taking one at a time until they are all gone or there are not
enough left to share. They record how many cubes each child received. (Students
at some tables are able to use fractions to show that, for example, when there
are 2 cubes left each person could get 1/2 cube. At other tables the students
simply leave the remaining 2 cubes undistributed.) The teacher then helps the
students symbolize what they have done by using addition to reflect putting all
of the cubes in the middle of the table and using division to reflect sharing
the cubes fairly among everyone at the table. They attach the words ÒmeanÓ and
ÒaverageÓ to this idea.
Finally, the students are asked to transfer the data from the sticky notes on the board to their own graph. The class helps the teacher generate additional questions about the data that can be answered for homework. Because the studentsÕ graphs look different, the next day the teacher will lead a discussion about the features of the various graphs the students have constructed and the pros and cons of each.
Figure 40: Studentdrawn graphs
ValentineÕs Day and Candy Hearts
What They Wonder AboutFormulate Questions
As ValentineÕs Day approaches, a teacher decides to plan a
lesson in which children will analyze the characteristics of a bag of candy
hearts. To begin the lesson, the teacher holds up a large bag of candy hearts
and asks the children what they know about them from prior experience. The
children know that the hearts are different colors and that they have words on
them. The teacher asks the children what they wonder about the bag of hearts
she is holding. The children want to know how many hearts are in the bag, what
they say, and whether there are a lot of pink hearts because most people like
pink ones the best. The teacher tells the children that they will be able to
answer some of those questions about their own bag of candy.
Collect Data
Each child receives a small packet of candy hearts. Students are asked how they can sort their hearts, and the students suggest sorting them by colorÐa categorical variable. The teacher asks students what question this will help them answer, and the students readily recognize that this will tell them which color has the most candies in the bag.
Figure 41: Initial sorting of candies
Analyze Data
After sorting the candies into piles and counting and recording the number of candies in each pile, the teacher guides the students to make a bar graph with their candies on a blank sheet of paper. The children construct individual bar graphs by lining up all of their pink candies, all of their white candies, etc. The teacher then provides a grid with color labels on the xaxis and numerical labels on the yaxis so that the students can transfer their data from the actual candies to a more permanent bar graph.
Figure 42: Bar graph of candy color
Interpret Results
After students construct their individual graphs using the candies as a guide, the teacher distributes a recording sheet on which each student records what color occurred the most frequently (the modal category) and how many of each color they had. This is followed by a class discussion in which the teacher highlights issues of variability. First, the students recognize that the number of each color varies within a package. Students also recognize that their packets of candy were not identical, noting that some students had no green hearts while others had no purple hearts. Some students had more pink hearts than any other color while other students had more white hears than any other color. At Level A, students are acknowledging variability between packages Ð the concept of between group variability that will be explored in more detail at Level B. The students hypothesize that these variations in packages were due to how the candies were packed by machines. The students also noted differences in the total number of candies per packet but found this difference to be small. The student with the fewest candies had 12 while the student with the greatest number of candies had 15. The teacher asked students if they had ever read the phrase Òpacked by weight, not by volumeÓ on the side of a package. The class then discussed what this meant and how it might relate to the number of candies in a bag.
(Note: Images in this example taken from http://www.littlegiraffes.com/valentines.html.)
Appendix for Level B
Many questionnaires ask for a ÒYesÓ or ÒNoÓ response. For example, in the Level B document, we explored connections between whether or not students like rap music and whether or not they liked rock music. To investigate possible connections between these two categorical variables, the data were summarized in the following twoway frequency table (or contingency table) shown below.
Table 18: Twoway frequency table

Like Rap Music? 
Row Totals 

Yes 
No 

Like Rock Music? 
Yes 
27 
6 
33 
No 
4 
17 
21 

Column Totals 
31 
23 
54 
Since 82% (27/33) of the students who like rock music also like rap music, students who like rock music tend to like rap music as well. Since students who like rock music tend to like rap music, there is an association between liking rock music and liking rap music.
In the Level B document, we explored the association between height and arm span by examining the data in a scatterplot and the strength of the association was measured with the Quadrant Count Ratio or QCR. For the Height/Arm Span problem, both variables are numerical. It is also possible to measure the strength and direction of association between certain types of categorical variables. Recall that two numerical variables are positively associated when above average values of one variable tend to occur with above average values of the other and when below average values of one variable tend to occur with below average values of the other. Two numerical variables are negatively associated when below average values of one variable tend to occur with above average values of the other and when above average values of one variable tend to occur with below average values of the other.
The scatterplot below for the Height/Arm Span data includes a vertical line (x = 172.8) drawn through the mean height and a horizontal line (y = 169.3) drawn through the mean arm span.
Figure 43: Scatterplot of height/arm span data
An alternative way to summarize the data would have been to ask each student the following two questions:
Is
your height above average?
Is
your arm span above average?
Note that for these data the response to each question is either ÒYesÓ or ÒNo.Ó
The 12 individuals in the scatterplot with below average height and below average arm span (Quadrant 3) responded ÒNoÓ to both questions. Since their responses to both questions are the same, these 12 responses are in Òagreement.Ó The 11 individuals in the scatterplot with above average height and above average arm span (Quadrant 1) responded ÒYesÓ to both questions. Since their responses to both questions are the same, these 11 responses are in Òagreement.Ó When the responses to two ÒYes/NoÓ questions are the same (No/No) or (Yes/Yes), the responses are in agreement.
The one individual with below average height and above average arm span (Quadrant 2) responded ÒNoÓ to the first question and ÒYesÓ to the second question, (No/Yes). Since her/his responses to the two questions are different, these two responses are in Òdisagreement.Ó The two individuals with above average height and below average arm span (Quadrant 4) responded ÒYesÓ to the first question and ÒNoÓ to the second question (Yes/No). Since their responses to the two questions are different their responses are in Òdisagreement.Ó When the responses to two ÒYes/NoÓ questions are different (No/Yes) or (Yes/No), the responses are in disagreement.
For the data in the above scatterplot, the results to the above two questions can be summarized in the following 2x2 twoway frequency table:
Table 19: 2x2 Twoway frequency table

Height Above Average? 
Row Totals 

No 
Yes 

Arm Span Above Average? 
No 
12 
2 
14 
Yes 
1 
11 
12 

Column Totals 
13 
13 
26 
Notice there are a total of 23 responses in agreement (12 No/No and 11 Yes/Yes to the height/arm span questions), and these correspond to the points in Quadrants 3 and 1, respectively, in the scatterplot. Also, there are a total of 3 responses in disagreement (2 Yes/No and 1 No/Yes), and these correspond to the points in Quadrants 4 and 2, respectively, in the scatterplot. Recall that the QCR is determined as follows:
QCR = (Number of points in quadrants 1 and 3)Ð(Number of points in quadrants 2 and 4)
Number of points in all four quadrants
Restated in terms of the above twoway frequency table,
QCR = (Number of Points in Agreement) Ð (Number of Points in Disagreement)
Total Number of Responses
Based on this, we can say that two ÒYes/NoÓ categorical variables are positively associated when the responses tend to be in agreement  the more observations in agreement, the stronger the positive association. Negative association between two ÒYes/NoÓ categorical variables occurs when the responses tend to be in disagreement  the more observations in disagreement, the stronger the negative association.
The responses to two ÒYes/NoÓ questions can be summarized as follows in a twoway frequency table:
Table 20: Twoway frequency table

Question 1 
Row Totals 

No 
Yes 

Question 2 
No 
a 
b 
r_{1 }= a+b 
Yes 
c 
d 
r_{2 }=c+d 

Column Totals 
c_{1} = a+c 
c_{2} = b+d 
T = a+b+c+d 
Note that a = the number who respond No/No; b = the number who respond Yes/No;
c = the number who respond No/Yes; d = the number who respond Yes/Yes.
Conover (1999) suggests the following measure of association based for a 2x2 table summarized as above:
(a+d) Ð (b+c) .
T
LetÕs call this measure the AgreementDisagreement Ratio (ADR). Note that this measure of association is analogous to the QCR correlation coefficient for two numerical variables.
The ADR for the height/arm span data is:
ADR = (12+11) Ð (2+1) = .77
26
An ADR of .77 indicates a strong association between height and arm span measurements.
Recall the music example data which were summarized as follows:
Table 21: Twoway frequency table

Like Rap Music? 
Row Totals 

Yes 
No 

Like Rock Music? 
Yes 
27 
6 
33 
No 
4 
17 
21 

Column Totals 
31 
23 
54 
The ADR for the rap/rock data presented above is:
ADR = (27 +17) Ð (6+4) = .63
54
An ADR of .63 indicates a fairly strong association between liking rock and liking rap music.
Another question presented in the Level B was:
Do students who like country music tend to like or dislike
rap music?
Data collected on 54 students are summarized in the following twoway frequency table:
Table 22: Twoway frequency table

Like Rap Music? 
Row Totals 

Yes 
No 

Like Country Music? 
Yes 
1 
21 
22 
No 
22 
10 
32 

Column Totals 
23 
31 
54 
For these data,
ADR = (1 +10) Ð (21+22) = .59
54
An ADR of .59 indicates a fairly strong association between liking one type of music, and not caring for the other type.
The QCR and the ADR are additive in nature in that they are based on Òhow manyÓ data value are in each quadrant or cell. Conover (1999) suggests the phi coefficient below as another possible measure of association for data summarized in a 2x2 table.
_{}
Conover points out that Phi is analogous to PearsonÕs correlation coefficient for numerical data. Both Phi and PearsonÕs correlation coefficient are multiplicative and PearsonÕs correlation coefficient is based on Òhow farÓ the points in each quadrant are from the center point.
Appendix for Level C
Example 1 (Level C Example 6 continued)
Recall that in Example 6 of Level C, students investigated the relationship between height and forearm length. The observed data is shown again here as Table 23 and the resulting plots and regression analysis are given in Figure 44.
Forearm (cm) 
Height (cm) 
45.0 
180.0 
44.5 
173.2 
39.5 
155.0 
43.9 
168.0 
47.0 
170.0 
49.1 
185.2 
48.0 
181.1 
47.9 
181.9 
40.6 
156.8 
45.5 
171.0 
46.5 
175.5 
43.0 
158.5 
41.0 
163.0 
39.5 
155.0 
43.5 
166.0 
41.0 
158.0 
42.0 
165.0 
45.5 
167.0 
46.0 
162.0 
42.0 
161.0 
46.0 
181.0 
45.6 
156.0 
43.9 
172.0 
44.1 
167.0 
Regression Analysis: Height
versus Forearm
The
regression equation is
Height
= 45.8 + 2.76 Forearm
Predictor Coef SE Coef T P
Constant 45.78 19.74 2.32 0.030
Forearm 2.7631 0.4459 6.20 0.000
S
= 5.76625 RSq = 63.6% RSq(adj) = 61.9%
Is the slope of 2.8 ÒrealÓ or simply a result of the random selection process? This question can be investigated using simulation.
If there were no real relationship between height and forearm length, then any of the height values could be paired with any of the forearm values with no loss of information. In the spirit of the comparison of means in the radish experiment, you could then randomly mix up the heights (while leaving the forearm lengths as is), calculate a new slope, and repeat this process many times to see if the observed slope could be generated simply by randomization. The results of 200 such randomizations are shown in Figure 45. A slope as large as 2.8 is never reached by randomization, which provides strong evidence that the observed slope is not due simply to chance. An appropriate conclusion is that there is a positive association between forearm length and height.
Figure 45: Dotplot showing association
Example 2: A Survey of Healthy Lifestyles
A high school class interested in healthy lifestyles carried out a survey to investigate various questions they thought were related to that issue. A random sample of 50 students selected from those attending a high school on a particular day were asked a variety of health related questions, including these two:
1. Do you think you have a healthy lifestyle?
2. Do you eat breakfast at least three times a week?
The data are given in Table 24.

Eat Breakfast


Healthy
Lifestyle

Yes 
No 
Total 
Yes 
19 
15 
34 
No 
5 
11 
16 
Total 
24 
26 
50 
From these data, collected in welldesigned sample survey, it is possible to estimate the proportion of students in the school who think they have a healthy lifestyle and the proportion who eat breakfast at least three times a week. It is also possible to assess the degree of association between these two categorical variables.
For example, in the lifestyle survey previously described, 24 students in a random sample of 50 students attending a particular high school reported that they eat breakfast at least three times per week. Based on this sample survey, it is estimated that the proportion of students at this school who eat breakfast at least three times per week to be 24/50 = .48 with a margin of error of
_{}
Using the margin of error result from above (.14), the interval of plausible values for the population proportion of students who eat breakfast at least three times a week is (0.34, 0.62). Any population proportion in this interval is consistent with the sample data in the sense that the sample result could reasonably have come from a population having this proportion of students eating breakfast.
Figure 46: Dotplot showing difference in sample proportions
Example 3: An Experimental Investigation of Pulse Rates
On another healthrelated issue, a student decided to answer the question of whether or not simply standing for a few minutes increases the pulse (heart rate) by an appreciable amount. Subjects available for the study were the 15 students in a particular class. The ÒsitÓ treatment was randomly assigned to eight of the students; the remaining seven were assigned the ÒstandÓ treatment. The measurement recorded was a pulse count for 30 seconds, which was then doubled to approximate a oneminute count. The data, arranged by treatment, are in Table 25. From these data it is possible either to test the hypothesis that standing does not increase pulse rate, on the average, or to estimate the difference in mean pulse between those who stand and those who sit. The random assignment to treatments is intended to balance out the unmeasured and uncontrolled variables that could affect the results, such as gender and health conditions. This is called a completely randomized design.
Table 25: Pulse data
But, randomly assigning 15 students to two groups may not be the best way to balance background information that could affect results. It may be better to block on a variable related to pulse. Since people have different resting pulse rates, the students in the experiment were paired on resting pulse rate by pairing the two students with the lowest resting pulse rate, then the two next lowest, and so on. One person in each pair was randomly assigned to sit and the other to stand. The matched pairs data are in Table 26. As in the completely randomized design, the mean difference between sitting and standing pulse rate can be estimated. The main advantage of the blocking is that the variation in the differences (which now form the basis of the analysis) is much less than the variation among the pulse measurements that form the basis of analysis for the completely randomized design.
Table 26: Pulse data in matched pairs
In the first pulse rate experiment (Table 25) the treatments of ÒsitÓ or ÒstandÓ were randomly assigned to students. If there is no real difference in pulse rates for these two treatments, then the observed difference in means (4.1 beats per minute) is due to the randomization process itself. To check this out the data resulting from the experiment can be rerandomized (reassigned to sit or stand after the fact) and a new difference in means recorded. Doing the rerandomization many times will generate a distribution of differences in sample means due to chance alone and one can assess the likelihood of the original observed difference. Figure 47 shows the results of 200 such rerandomizations. The observed difference of 4.1 was matched or exceeded 48 times, which gives a probability of 0.24 of seeing a result of 4.1 or greater by chance alone. Because this is a fairly large probability, it can be concluded that there is little evidence of any real difference in means pulse rates between the sitting and the standing positions based on the observed data.
In the matched pairs design the randomization occurs within each pair, one person randomly assigned to sit while the other stands. To assess whether the observed difference could be due to chance alone and not due to treatment differences, the rerandomization must occur within the pairs. This implies that the rerandomization is merely a matter of randomly assigning a plus or minus sign to the numerical values of the observed differences. Figure 48 shows the distribution of the mean differences for 200 such rerandomizations; the observed mean difference of 5.14 was matched or exceeded 8 times. Thus, the estimated probability of getting a mean difference of 5.1 or larger by chance alone is 0.04. This very small probability provides evidence that the mean difference must be caused by something other than chance (induced by the initial randomization process) alone. A better explanation is that standing increases mean pulse rate over the sitting rate. The mean difference shows up as significant here, while it did not for the completely randomized design, because the matching reduced the variability. The differences in the matched pairs design have less variability than the individual measurements in the completely randomized design, making it easier to detect a difference in mean pulse for the two treatments.
Example 4: Observational StudyÐRates over Time
Vital statistics are a good example of observational data that are used every day be people in various walks of life. Most of these statistics are reported as rates, so an understanding of rates is a critical skill for high school graduates. The following table shows the death rates for sections of the population of the United States over a period of twelve years. Such sequences over time are often referred to as time series data.
Table 27: United States Death Rates (deaths per 100,000 of population)
Year 
All Races 
White 
Black 


Male 
Female 
Male 
Female 
Male 
Female 
1990 
1,202.8 
750.9 
1,165.9 
728.8 
1,644.5 
975.1 
1991 
1,180.5 
738.2 
1,143.1 
716.1 
1,626.1 
963.3 
1992 
1,158.3 
725.5 
1,122.4 
704.1 
1,587.8 
942.5 
1993 
1,177.3 
745.9 
1,138.9 
724.1 
1,632.2 
969.5 
1994 
1,155.5 
738.6 
1,118.7 
717.5 
1,592.8 
954.6 
1995 
1,143.9 
739.4 
1,107.5 
718.7 
1,585.7 
955.9 
1996 
1,115.7 
733.0 
1,082.9 
713.6 
1,524.2 
940.3 
1997 
1,088.1 
725.6 
1,059.1 
707.8 
1,458.8 
922.1 
1998 
1,069.4 
724.7 
1,042.0 
707.3 
1,430.5 
921.6 
1999 
1,067.0 
734.0 
1,040.0 
716.6 
1,432.6 
933.6 
2000 
1,053.8 
731.4 
1,029.4 
715.3 
1,403.5 
927.6 
2001 
1,029.1 
721.8 
1,006.1 
706.7 
1,375.0 
912.5 
StudentsÕ understanding of the rates can be established by posing problems such as:
a. Carefully explain the meaning of the number 1,029.1 in the lower lefthand data cell.
b. Give at least two reasons why the White Male and Black Male entries do not add up to the All Races male entry.
c. Can you tell how many people died in 2001 based on the above table alone?
Hopefully, students will quickly realize that they cannot change from rates of death to frequencies of deaths without knowledge of the population sizes. Table 28 provides the population sizes overall as well as for the male and female categories.
Table 28: Population of the
Year 
Total persons 
Male 
Female 
1990 
249,623 
121,714 
127,909 
1991 
252,981 
123,416 
129,565 
1992 
256,514 
125,247 
131,267 
1993 
259,919 
126,971 
132,948 
1994 
263,126 
128,597 
134,528 
1995 
266,278 
130,215 
136,063 
1996 
269,394 
131,807 
137,587 
1997 
272,647 
133,474 
139,173 
1998 
275,854 
135,130 
140,724 
1999 
279,040 
136,803 
142,237 
2000 
282,224 
138,470 
143,755 
2001 
285,318 
140,076 
145,242 
Noting that the population figures are in thousands but the rates are per 100,000, it takes a little thinking on a studentÕs part to realize that she can go from rates to counts for females by making the computation shown in the formula:
Female Deaths = _{}
Now, time series questions can be explored. For example, how does the pattern of female death rates over time compare to the pattern of actual female deaths? The plots of Figure 49 provide a visual impression. The death rates are trending down hill over time, with considerable variation, but the actual deaths are going up.
Figure 49: Scatterplot of death rates
Figure 50: Scatterplot of actual deaths
Students will discover that the picture for males is quite different, and this can lead to interesting discussions.
Example 5: Modeling the Dissolve Time of Alka Seltzer: Transformations of Data
Some experiments lead to modeling the relationship between two variables, instead of testing hypotheses or estimating population or treatment means. Such an experiment is a study of the dissolve time of Alka Seltzer tablets carried out by a group of science students. Their question was ÒWhat is the relationship between the temperature of the water and the dissolve time of an Alka Seltzer tablet?Ó The students heated cups of water to various temperatures (centigrade) and then randomly choose a tablet to drop into the water. The measurement of interest was the time (seconds) until the tablet was completely dissolved. The data from the experiment is in Table 29 and the scatterplot of the data is in Figure 51. It is obvious that the dissolve time decreases with increasing temperature of the water, but the curvature in the pattern makes the actual relationship between time and temperature somewhat challenging to model. A transformation is in order if techniques of simple linear regression analysis are to be used.
Time (sec) 
Temperature
(C) 
23 
43 
116 
3 
122 
5 
66 
23 
30 
42 
25 
35 
60 
14 
45 
20 
24 
38 
22 
42 
121 
7 
60 
17 
40 
22 
26 
43 
90 
10 
28 
32 
24 
87 
25 
42 
65 
17 
15 
70 
53 
23 
23 
45 
64 
18 
25 
42 
45 
21 
18 
74 
10 
72 
13 
47 
15 
18 
20 
45 
49 
16 
57 
17 
28 
33 
55 
20 
Figure 51: Scatterplot showing time for Alka Seltzer to dissolve
In modeling the relationship between the time for an Alka Seltzer tablet to dissolve and the temperature of the water (see Figure 51) it is observed that the relationship is more complicated than a simple straight line. Straightening out this relationship requires a transformation of one or both variables; either the extreme temperatures need to be reduced or the extreme times need to be reduced, or both. It turns out that a logarithmic transformation on both variables works well, as shown in Figure 52. The equation of the least squares line is given under the figure. The slope of 0.76 indicates that the ln(time) will decrease, on the average, 0.76 units for every one unit increase in ln(temp). The residual plot shows no strong pattern (although there may still be some curvature here) and the correlation of 0.88 (negative square root of 0.77) shows a fairly strong relationship between the two variables. The standard deviation of the residuals is 0.32, which indicates that a prediction of ln(time) for a new situation with known ln(temp) would have a margin of error of approximately 2(0.32)= 0.64 units.
Example 6: Graphs: Distortions of Reality?
Study the graph pictured below. Do you see any weaknesses in this graphic presentation? If so, describe them and explain how they could be corrected.
Figure 53: Distorted graph
Here are some plausible plots to correct errors of interpretation Ð and to raise other questions. Better presentations begin with a data table, É
Table 30: Enrollment data
Year 
Total Students 
African American 
1996 
29404 
2003 
1997 
29693 
1906 
1998 
30009 
1871 
1999 
30912 
1815 
2000 
31288 
1856 
2001 
32317 
1832 
2002 
32941 
1825 
2003 
33878 
1897 
2004 
33405 
1845 
and then go to more standard graphic displays of such data. The plot below shows total and African American enrollments on the same scale. When viewed this way one can see that the latter is a small part of the former, with little change, by comparison, over the years.
Figure 54: Plot of African American vs. total enrollments
By viewing African American enrollments by themselves, one can see that the marked decrease between 1996 and 2002 may be turning around Ð or leveling off.
Figure 55: Plot of African American enrollments
only
But, the ratio of African American to total enrollment is still on the decrease!
Figure 56: Ratio of African American to total enrollment