The Introductory Statistics Course:
The Entity-Property-Relationship Approach

Donald B. Macnaughton

 NOTES A clickable table of contents of this paper appears at the end. This paper is available in PDF format at ABSTRACT This paper proposes six concepts for discussion at the beginning of an introductory statistics course for students who are not majoring in statistics or mathematics. The concepts are (1) entities, (2) properties of entities, (3) variables, (4) a major goal of empirical research: to predict and control the values of variables, (5) relationships between variables as a key to prediction and control, and (6) statistical techniques for studying relationships between variables as a means to accurate prediction and control. After students have learned the six concepts they learn standard statistical topics in terms of the concepts. It is recommended that each concept be taught in a bottom-up fashion with emphasis on concrete practical examples. It is suggested that the approach gives students a lasting appreciation of the vital role of the field of statistics in empirical research. KEY WORDS: Statistics education; Teaching; Role of statistics in empirical research. 1. INTRODUCTION Two former presidents of the American Statistical Association have stated that "students frequently view statistics as the worst course taken in college" (Hogg 1991, Iman 1994). A third former president has stated that the field of statistics is in a "crisis" and the subject has become "irrelevant to much of scientific enquiry" (Box 1995). The 2001 president has stated that statistics is "still among the most despised of college courses" (Scheaffer 2001). Many statisticians reluctantly agree with these remarks. In contrast, many statisticians agree that the field of statistics is a fundamental tool of the scientific method, which plays a key role in modern society. Thus rather than being a worst course and possibly irrelevant, the introductory statistics course ought to be a friendly introduction to the simplicity, beauty, and truth of the scientific method. Teachers must therefore reshape the introductory course. Many teachers have already contributed to the reshaping, as noted below. This paper proposes further changes. I focus on the introductory statistics course for students who are not majoring in statistics or mathematics, whom I call "non-statistics-majors". Most students who take an introductory statistics course are members of this group. The introductory course for non-statistics-majors is important because it is a main seedbed of public opinion about the field of statistics. Section 2 defines the concept of ‘empirical research', which appears throughout this paper. Section 3 recommends two goals for the introductory statistics course. Section 4 proposes six concepts for discussion at the beginning of an introductory course. Section 5 illustrates how the six concepts provide a deep and broad foundation on which we can build the field of statistics. Section 6 discusses testing the proposed approach. Section 7 identifies considerations for teachers wishing to use the approach, and Section 8 gives a summary. 2. A DEFINITION OF "EMPIRICAL RESEARCH" At many points in this paper I refer to "empirical research", which thus deserves a definition: Empirical research is any activity in which data are gathered from some area of experience and then conclusions are drawn from the data about the area of experience. Empirical research is a crucial step of the scientific method, which is central to many areas of human endeavor, such as in science, education, business, industry, law, and government. Section 5.5 discusses the scientific method.

3.1 The Value of Emphasizing Goals

Emphasizing the goals of an undertaking helps one to define and focus on what is most important. This helps substantially in structuring work efficiently. Thus, following Hogg (1990, 1992), I recommend that all introductory statistics courses have carefully considered goals. Ask yourself -- what are the goals of the introductory statistics course most familiar to you?

I have observed goal-setting exercises in which the goals were given much attention for a brief period, but then were forgotten or ignored in the subsequent months and years -- a waste of a valuable resource. Because the course goals specify what the teacher believes is most important, I recommend that teachers regularly revisit their goals to ask (a) if the goals are still reasonable and (b) if the day-to-day operations are effectively serving the goals.

3.2 Topic-Based Goals Have a Significant Drawback

Many introductory statistics courses have what can be called "topic-based" goals. A teacher using such goals does not specify general goals, but instead simply specifies a list of statistical topics to be covered in the course, perhaps in the form of a syllabus. For example, a teacher of a traditional course might aim to cover (in specified amounts of detail) the topics of probability theory, distribution theory, point and interval estimation, and hypothesis testing. Similarly, a teacher of an activity-based course might make a list of statistical topics and then assign various activities to the students in order to cover the topics.

Unfortunately, topic-based goals have a significant drawback: By emphasizing lower-level statistical topics, these goals usually fail to emphasize what is essential, which is to help students to appreciate the vital role or use of the field of statistics. Unless students understand and appreciate the general role of statistics, knowledge of statistical topics will be both of little interest to them and of little use.

I recommend that the goals of an introductory statistics course for non-statistics-majors be

1. to give students a lasting appreciation of the vital role of the field of statistics in empirical research and
2. to teach students to understand and use some useful statistical methods in empirical research.

I suggest that it is more important to satisfy these goals than to satisfy goals stated in terms of statistical topics.

How can we best satisfy these goals? First, it seems clear that students can appreciate the role of statistics only if they understand it. This leads one to ask, "What is the role of statistics?" The next section describes a sequence of simple concepts that give students a broad overview of the vital role of the field of statistics in empirical research.

Second, but of equal importance, most students will appreciate the role of statistics only if they see the practical value of statistics. Thus I recommend that introductory teachers place heavy emphasis on practical examples of the use of statistics. Surprisingly, some introductory approaches use examples that are not practical. Practical examples are discussed at several places below, with general remarks in Section 7.5.

Other authors (after Hogg) who discuss goals for the introductory course include Chromiak, Hoefler, Rossman, and Tesman (1992), Cobb (1992, 1993, 2000), Iversen (1992), Watkins, Burrill, Landwehr, and Scheaffer (1992), Hoerl and Snee (1995), Gal and Garfield (1997a, pp. 2 - 5), Moore (1997a), and Garfield (2000).

4. SIX CONCEPTS

This section describes six simple concepts I recommend for discussion at the beginning of an introductory statistics course for non-statistics-majors. These concepts help students to appreciate the role of statistics by highlighting a recurring simple pattern in the use of statistics across almost all empirical research. Most university and college students can learn the concepts in between one and three class sessions.

To make the approach easy for teachers to use, the following six subsections present the six concepts as a condensed version of how they might be presented to students. Teachers using the approach in an introductory course will need to expand the discussion with more examples, as discussed in Sections 7.5 through 7.8.

At a few points in this section I discuss pedagogical and statistical issues that are beyond the interest or understanding of most beginning students. These discussions are for teachers and are identified with initial asterisks. I recommend that this material be omitted in an introductory course for non-statistics-majors.

I begin with what may be the most fundamental concept of human reality.

4.1 Entities

If you study your train of thought, you will probably agree that you think about "things". For example, during the next few seconds you may think about, among other things, a friend, an appointment, today's weather, and an idea. Each of these things is an example of an entity.

Many different types of entities exist. Some common types are

• organisms (e.g., people, trees)
• inanimate physical objects
• physical locations
• actions and events
• ideas and emotions
• societal organizations (e.g., governments, schools, businesses)
• entities in science (e.g., waves in physics, motivations in psychology)
• *entities in mathematics (e.g., elements of sets, sets, numbers, functions, vectors).

Clearly, the things in the list are diverse. Thus it may at first seem absurd to think that they have anything in common. But they all do have one important property in common -- they are all things (entities).

*Appendix A discusses whether some of the things in the list are really things or entities.

Entities are fundamental units of human reality because people unconsciously view everything (every thing) in our reality as being an entity. This dramatically simplifies our thinking because it allows us to view everything (at the most basic level) the same way, as discussed in the next subsection.

The External World. When considering entities it is useful to consider the concept of ‘external world', which can be defined as follows:

The external world is what is "out there" -- what we see when we look out the windows in our heads and what we sense through our other senses.

People usually view entities as existing both in the external world and in our minds. We use the entities in our minds mainly to stand for entities in the external world, much as we use a map to stand for its territory.

People learn to use the concept of ‘entity' when they are infants. We use the concept unconsciously as a way of organizing the multitude of stimuli that enter our minds from the external world moment by moment while we are awake.

People Group Entities into Types (Populations). We learn as infants to group entities into types. For example, as infants we observe that the entities "mother" and "father" and other similar entities in the external world all have heads with two eyes, a mouth, and usually hair on top. We (unconsciously) group these entities into a type -- the type we call "people". Similarly, as infants we learn to group all inanimate physical objects (beginning with small familiar objects in the crib) into a type. Grouping entities into types simplifies things because we recognize that all the entities of a given type have many things (properties) in common, as discussed in the next subsection.

In statistics and empirical research the set of all the entities of a given type is called the "population" of entities of that type. For example, a web site on the Internet is an entity (of an electronic or computer-object sort), and the set of all the web sites on the Internet constitutes the population of web sites.

Entities and Language. In language people use nouns to stand for entities. For example, in the sentence "Peter Smith is in room 302" the proper noun "Peter Smith" identities a particular entity that is a person. Similarly, the noun phrase "room 302" identifies an entity that is a physical location.

People usually think of entities unconsciously. However, we sometimes do need to refer to them in general terms. In these situations statisticians and empirical researchers may refer to entities as members of the population, cases, elements, individuals, instances, items, objects, observations, specimens, subjects, things, or (experimental, observational, or survey) units. *I discuss why I recommend the term "entity" for general discussion in a paper (1998a, app. E.1).

*Why Discuss Entities? Most human thoughts can be expressed in sentences in language. Since most human sentences contain at least one noun, most human sentences refer directly to entities. Thus the concept of ‘entity' pervades human thought. However, the concept is almost always at a deep unconscious level. Thus the concept is rarely directly discussed in everyday conversation, or in statistics, or in empirical research. This raises the question of whether we need to discuss the concept of ‘entity' at the beginning of an introductory statistics course.

If we are dealing with a specific problem in statistics or empirical research, we can usually omit direct discussion of entities. This is because in specific problems we rarely need to drill down all the way to the foundational concept and discuss "things" at such a basic level. Instead, discussions invariably concern one or more specific types of entities, which are best referred to by their type names. For example, medical researchers study two familiar types of entities called "people" and "diseases".

On the other hand, if we are dealing with the general problems of statistics and empirical research, it is reasonable to begin with the concept of ‘entity'. This is because the easy-to-understand concept of ‘entity' can be reasonably viewed as a unifying foundation for many other concepts of statistics and empirical research. This idea is directly illustrated in the following subsections and is extended in Section 5.3.

*Teaching the Concept of 'Entity'. The concept (principle) of 'entity' is basic and general (and abstract). Moore, in an important article about statistics education, notes that students in introductory statistics courses (and people in general) often have difficulty learning from basic general principles down to special cases (2001, sec. 4). I fully agree with Moore's point. Clearly, unless we bypass the "top-down" nature of the concept of 'entity', its inherent abstractness and generality will be fatal stumbling blocks for many less advanced students.

To bypass the need for students to learn in a top-down fashion, and to ensure that students understand the concept of 'entity', I recommend that teachers develop it using a "bottom-up" sequence of ideas, beginning with familiar concrete examples of entities and working up at an appropriate pace for the students to the general concept. I further discuss the bottom-up approach to teaching statistical concepts in a Usenet post (forthcoming).

4.2 Properties of Entities

Every entity has associated with it a set of properties. For example, all people have thousands of different properties, two of which are "height" and "blood group".

Values of Properties. For any particular entity, each of its properties has a value. We usually report the value of a property with a number, with words, or with a symbol. For example, on April 12, 2001, in a number (the value of) my height was 176.3 centimeters, in a word my height was "medium", and in a symbol my blood group was "O pos". We know or experience an entity by knowing or experiencing its properties and their values.

(We know or experience a living entity by [in large part] knowing or experiencing its behavior. Researchers who study behavior usually view it as a complicated set of properties of the living entities they study.)

Variation in Values of Properties. The values of properties of entities generally vary. The values almost always vary from one entity to the next and they usually also vary within individual entities over time. For example, peoples' heights vary from person to person, and a person's height varies over time.

Measures of Values of Properties. A key operation of empirical research is to determine the values of selected properties of the entities under study. To determine the value of a property of an entity, researchers apply an appropriate measuring instrument to the entity. If the instrument is measuring correctly, it will return a measurement that is an estimate of the value of the property of the entity at the time of the measurement.

For example, if we wish to know the (value of the) height (property) of a person, we can apply a height-measuring instrument (e.g., a tape measure) to the person, and the instrument will return a value that is (in the specified units) an estimate of the person's height.

Measuring instruments (which are sometimes called "measures") are often physical devices such as a tape measure, a speedometer in a car, or litmus paper. But they may also be of other types, such as paper-and-pencil tests administered to students or subjective judgments provided by experts.

Measuring instruments are important because all conclusions in empirical research are based directly on the estimates of values of properties obtained from measuring instruments, as discussed in the following subsections.

The "True" Value of a Property of an Entity. Since a measuring instrument can generally only provide an estimate of the value of the property it measures, this leads to the question of what it means to speak of the "true" value of a property of an entity. Empirical researchers usually view the true value of a property in terms of commonly-agreed-upon measurement standards because this facilitates communication and understanding. For example, researchers in the physical sciences usually view the "true" values of the properties of the entities they study in terms of the definitions and standards maintained by the International Bureau of Weights and Measures (BIPM 2001). *I further discuss the idea of the true value of a property in a Usenet post (2001).

Properties and Populations. Section 4.1 says that humans learn as infants to group entities into types (populations) such as "people" and "inanimate physical objects". An important aspect of these types is that we learn to (unconsciously) view all the entities of a given type as having exactly the same properties. However, as noted, the values of the properties generally vary from entity to entity and over time. For example, we unconsciously view all solid physical objects as having the properties "weight", "shape", "size", and "color(s)" although different solid physical objects generally have different values of these properties.

Viewing all the entities of a given type as having (sharing) exactly the same properties is a key unifying principle of human reality.

Properties in Language and Thought. In everyday language, people often use adjectives and adverbs to report the values of properties of entities. For example, if someone says "Peter Smith is tall", the adjective "tall" reports (an estimate of the value of) Peter's height. Similarly, if someone says "Peter Smith is very tall", the adverb "very" refines the report of Peter's height. Similarly, if someone says "The tiger is running quickly", the adverb "quickly" reports the value of the current "speed" property of the tiger.

A property of an entity may also be called an ability, aspect, attribute, capability, capacity, character, characteristic, countenance, dimension, disposition, facet, factor, faculty, feature, finding, indication, indicator, nature, quality, quantity, scalar, trait, or vector. *Appendix B discusses why I recommend the term "property" for general discussion.

*Appendix C discusses the evolution of entities and properties in human thought.

*Teaching the Concept of 'Property'. As with the concept of 'entity', the main ideas associated with properties are basic, general, and abstract. Thus the ideas can be hard for less-advanced students to understand. Thus, to ensure understanding, I recommend that the ideas be introduced in a bottom-up sequence, beginning with familiar concrete examples of properties, values, variation, and measures, and working up at an appropriate pace for the students to the general ideas.

4.3 Variables

As noted, empirical researchers use measuring instruments to determine estimates of the values of properties of entities. When these estimates are studied formally, statisticians and empirical researchers usually refer to them as "variables". A reasonable definition of the statistical concept of 'variable' is

A variable is a formal representation of a property of entities.

*Appendix D compares some dictionary definitions of the concept of 'variable'. Appendix E further discusses the distinction between properties and variables.

Values of Variables. Like properties, variables have values. And like the values of properties, the values of variables generally vary. An important subgoal of all serious empirical research is to make the measured values of the variables as close as reasonably possible (at the time of the measurement) to the true values of the associated properties in the entities under study. Researchers do this by using measuring instruments and procedures that are as accurate as possible. This generally increases the accuracy of the conclusions they draw from empirical research.

Clearly, time plays a key role in the idea of the value of a variable: In statistics and empirical research the value of a variable for an entity is generally viewed as an estimate or "snapshot" (possibly with distortion) of the true value of the associated property of the entity at a particular time (or perhaps over a particular time period).

Data and Data Tables. The concepts of 'entity', 'property', and 'variable' lead directly to the concept of 'data', which can be defined as follows:

Data are the (measured) values of one or more variables (properties) for one or more entities. (A single value of a single variable is called a "datum".)

All empirical research projects generate data. The (raw) data from a research project (or from a logical unit of a larger research project) are invariably organized in a table. Each row in the table is associated with one entity of the type under study. Each column is associated with a different property of the entities (or a property of the entities' environment), as reflected in the values of the variable associated with the column. Each cell (intersection of a row and a column) in the table contains the value (at the time of measurement) of the variable associated with the column for the entity associated with the row.

The data table (with appropriate footnotes) is the complete record of what was observed in an empirical research project. Thus the table is central to drawing reasonable conclusions from the project (as explained in the next three subsections). The table also provides a succinct summary of the design of the project. Thus when considering or planning an empirical research project it is helpful for students to study the data table, or a manageable number of rows if the table is large. To increase understanding I recommend that studied tables have carefully worded column headings, and that they contain realistic made-up data if real data are unavailable. *Realistic data are further discussed in Section 7.8.

*Teaching the Concepts of 'Variable' and 'Data Table'. Discussion of the somewhat abstract material above is almost never enough for students to properly understand the concepts of 'variable' and 'data table'. Furthermore, as discussed in Section 5.10, students often misunderstand the concept of 'variable'. But the concept of 'variable' is central to virtually all statistical discussions. Thus students must understand variables (and data tables) if they are to understand statistics.

As with entities and properties, students can readily understand the concepts of 'variable' and 'data table' if the ideas are developed in a bottom-up fashion, beginning with concrete examples and working up at an appropriate pace for the students to the general concepts.

I recommend that teachers choose variables for discussion that are interesting, easy to understand, and that empirical researchers are seriously interested in studying. For example, automotive engineers are seriously interested in studying the variable "fuel usage per kilometer" in automobiles because appropriate study of this variable enables them to minimize fuel usage and thereby make automobiles less expensive to run. Thus a teacher might show students a data table with different types of automobiles representing the rows and relevant automobile variables (including "fuel usage per kilometer") representing the columns. If the components of such a table are carefully discussed, students attain a concrete sense of the entities, properties, and variables associated with the table.

Definition of Empirical Research. To help in the meta-discussion, Section 2 above proposes a definition of the concept of 'empirical research'. For introductory statistics courses following the approach of this paper, that definition is properly presented at this point in the development of the ideas, after the introduction of the concept of 'data', which is used in the definition of "empirical research".

4.4 A Goal of Empirical Research: To Predict and Control the Values of Variables

A central idea in the definition of empirical research is that researchers "draw conclusions from data". Why do researchers wish to do this -- what are the goals of empirical research?

Prediction and Control. One important goal of empirical research is to discover how to predict and control (with maximum accuracy) the values of properties of entities. In other words, the goal is to discover how to predict and control the values of variables in entities. For example, an important goal of (empirical) medical research is to discover how to accurately predict and control the state of the human body, where the state is reflected in various medical properties or variables, such as blood pressure, white blood count, and other measures of health or disease.

We seek the ability to predict and control the values of variables because it provides many social and commercial benefits. For example, if a medical researcher can discover how to better predict or control people's risk of heart attacks, this discovery provides the social benefit of saving lives. Similarly, if an organization can discover how to better control variables that reflect important properties of its operations (e.g., customer satisfaction, product performance, product usefulness, product reliability), this discovery helps the organization to optimize its operations and thereby become more successful.

Since the ability to predict and control the values of variables is of broad usefulness, many branches of society (in science, business, technology, education, and government) provide substantial support to empirical research aimed at learning how to predict or control the values of key properties or variables.

*Explanation and Understanding. A second goal of empirical research is to explain and understand the area of experience under study in the research. However, examination of the concepts of 'explanation' and 'understanding' suggests that these concepts are subordinate to prediction and control for three reasons:

• A main test of whether any explanation or understanding is valid is whether it leads to accurate prediction or control.
• An important application of any (correct) explanation is that it can be used for accurate prediction or control. After conceptually subtracting this application from explanation, one looks in vain for other applications, save supporting development of further explanations (which can themselves be used for prediction or control).
• When given a choice, most people will readily choose accurate prediction or control of some important phenomenon (without explanation or understanding) over accurate explanation or understanding of the same phenomenon (without prediction or control).

I support these points in two Usenet posts (1996a, 1997b). See also Section 5.5 below.

*Prediction and Control Deserve Emphasis. The preceding discussion notes the substantial social and commercial benefits of accurate prediction and control of the values of variables. The discussion also notes the relationship between prediction and control on the one hand and explanation and understanding on the other. Later discussion (Section 5.4) illustrates how it is useful to characterize many of the standard statistical procedures as procedures for achieving accurate prediction or control of the values of variables. These points suggest it is reasonable to focus the introductory statistics course for non-statistics-majors on the use of statistics in empirical research as a means to accurate prediction and control.

Where Do We Predict and Control? Before considering the main question of how to predict and control the values of variables, it is useful to consider the following preliminary question:

Where do statisticians and empirical researchers predict and control the values of variables?

We predict and control the values of variables in the entities in the population of entities under study. We seek the ability to predict and control the values of variables in entities in populations because this approach enables us to make our knowledge as general as possible. Such generality is desirable because the ability to predict or control the values of a variable in any entity in a broad population is almost always more useful than the ability to predict or control (with the same accuracy) the values of the same variable in some subset of the population.

For example, in medical research the population of entities of interest is often all the people in the world. The goal of the research is to find ways to predict or control the values of important medical variables in any person in the population (ideally including all people living, dead, and unborn). Similarly, in organizational research the population of entities under study might be all the weeks in the life of a particular organization. Here the goal of the research might be to find ways to predict and control the values of important organizational variables in any week (especially later weeks) in the life of the organization.

Thus if we include the concept of 'population', we can say

A fundamental goal of empirical research is to discover how to predict and control (with maximum accuracy) the values of variables (properties) in entities in populations.

4.5 Relationships Between Variables as a Key to Prediction and Control

The Concept of 'Relationship Between Variables'. Given the goal of predicting and controlling the values of variables, a key question is How can we predict and control the values of variables? The main answer is

We can predict and control the values of variables by studying relationships between variables.

In a "relationship between variables" one variable (called the response variable) "depends" on one or more other variables (called the predictor variable[s]). Almost all prediction and control in all areas of empirical research is done on the basis of this simple idea.

For example, medical researchers have discovered that a relationship exists between the amount of saturated fat ingested by a person (predictor variable) and the risk that a person will have a heart attack (response variable). The relationship is that more saturated fat is associated with a higher risk of a heart attack. Knowing this relationship helps doctors and patients to predict and control heart attacks.

(Empirical research about the relationship between saturated fat and heart attacks is summarized by Kromhout 1999, Liebson and Amsterdam 1999, and de Lorgeril and Salen 2000.)

In addition to using the concept of 'dependence' to characterize relationships between variables, we can characterize them as follows: A relationship exists between two variables if we find that when the values of the predictor variable(s) "go up and down" in the entities under study (or in the entities' environment), the values of the response variable also go up and down (or down and up) somewhat "in step" with the values of the predictor variable(s).

For many non-statistics-majors the above two informal characterizations of relationships are sufficient if they are properly illustrated with practical examples. For more advanced students I propose a formal definition of the concept of 'relationship between variables' in a paper for students (1997a, sec. 7.10) and I discuss an important alternative definition in a Usenet post (2002).

Population and Sample. When empirical researchers study a relationship between variables they usually do not attempt to directly study the relationship in every entity in the population of interest because that would be impossible or prohibitively expensive. Instead, researchers study the relationship in a subset of the population, which is called a "sample". Researchers usually design empirical research projects with between 6 and 2000 entities in the sample.

A View of Empirical Research. Examination of empirical research projects suggests that most can be reasonably viewed as attempting to make a correct generalization for a population of entities about a relationship between variables. The generalization is made from studying the relationship between the variables in the data table for the entities in the selected sample. The generalization (if made properly) enables us to accurately predict or control the values of the property associated with the response variable in new situations for any entity in the population.

For example, the medical researchers who discovered the relationship between saturated fat consumption and heart attacks did so by studying data tables for samples of people. These tables have one or more variables for each person that reflect the person's heart attacks and also one or more variables that reflect the person's fat consumption. The researchers used statistical procedures to look for a relationship between fat consumption and heart attacks in the tables, and such a relationship has been reliably found. These findings (and other supporting information) lead doctors to believe that the relationship exists in all the people in the world.

Accuracy of Predictions. Prediction or control that is done on the basis of relationships between variables is generally not perfectly accurate. However, if the prediction or control is done properly, mathematical proofs exist to show that it is of the highest possible accuracy given the available information.

Examples of Relationships. We can show students the pervasiveness and usefulness of relationships between variables by discussing practical examples. For example, teachers and students can discuss (using appropriate graphics) whether a relationship exists between

• the net amount of force applied to a physical object and the rate of acceleration of a physical object (physics)
• the weight of a car and the fuel usage of a car (automotive engineering)
• the concentration of alcohol in the bloodstream of a person driving a car and the probability that a person driving a car will be involved in an accident (physiology)
• the number of hours of homework completed by a student during a course and the grade obtained by a student in a course (educational psychology)
• the education of a person and the income of a person (sociology)
• the style of management of a company and the amount of profit of a company (business)
• having a black cat cross one's path and having bad luck (folk beliefs).

Each of these examples identifies a possible relationship between variables. Each of these relationships (and relationships between any other pairs or larger sets of compatible variables) can be studied in an empirical research project. If the research project finds conclusive evidence of the relationship, we can use the knowledge of the relationship to predict and possibly control the values of the response variable in new entities from the population on the basis of their values of the predictor variable(s).

Nine Questions About an Empirical Research Project. One can usually understand an empirical research project by considering it in terms of nine questions, which are

1. What is the population of entities under study?
2. How many entities are in the sample, and is the sample selected from the population in a way that allows us to correctly generalize from the sample to the population? (For economy, compromises to "ideal" samples are often reasonable, but should be kept firmly in mind when drawing conclusions.)
3. What is the response variable that is measured in each entity in the sample? (*Some research projects have multiple response variables, but usually each response variable can be reasonably viewed as defining a separate research project, with the research projects sharing the same entities and predictor variable[s]. See also Appendix I.2.)
4. What is (are) the predictor variable(s) that is (are) measured in each entity in the sample (or measured in each entity's environment)? Which predictor variables (if any) are manipulated by the researcher and which are merely observed?
5. In plain language or in graphical terms (as opposed to mathematical terms), what is the relationship between the response variable and the predictor variable(s) that is sought, discovered, or studied in the research project?
6. Regardless of whether the research project has been performed, suppose it has been performed and a relationship between the response variable and the predictor variable(s) has been found. Can you think of a reasonable alternative explanation for the (apparent) relationship between the variables? (Such an explanation renders the research finding equivocal, and thus of less value. To avoid deceiving themselves, empirical researchers strive to perform research in ways that eliminate all reasonable alternative explanations.)
7. How can we best use the relationship to predict or control the values of the response variable in new entities from the population?
8. How accurate will the prediction or control be?
9. What are the practical implications of the findings?

By considering sufficient practical examples, students recognize that most empirical (including most scientific) research projects can be reasonably understood by considering them in terms of the nine questions. Thus students recognize that most empirical research projects can be reasonably viewed as studies of relationships between variables in entities in samples, with the aim being to develop the ability to accurately predict or control the values of the property associated with the response variable in new situations for any entity in the population.

*Appendix H discusses some possible counterexamples to the points in the preceding paragraph. Section 5.5 discusses the relationship between relationships between variables and some general concepts of science. Mosteller (1990) and Lipsey (1990) discuss the idea of a reasonable alternative explanation. I briefly discuss some history of the concept of 'relationship between variables' in a Usenet post (2001).

Terminology. More than eighty terms are available to name the concept of 'relationship' in the phrase "relationship between variables". For example, we can speak of an "association" between variables, or a "relation" between variables, or a "dependence" between variables. *Appendix F discusses why I recommend the term "relationship" for general discussion.

Similarly, several general terms are available to name the response variable and the predictor variable(s) in a relationship between variables. For example, a response variable may be called a "predicted" variable or a "dependent" variable, and a predictor variable may be called an "explanatory" variable or an "independent" variable. *Appendix G discusses why I recommend the terms "response" and "predictor" for general discussion.

*Statistical Concepts. In the initial discussion of the above ideas I recommend that teachers not introduce any additional statistical concepts beyond some simple graphics to illustrate relationships. I recommend this approach because I believe it is important for students to develop (through practical examples) a strong understanding of the unifying concept of 'relationship between variables' before they try to understand the complicated statistical ideas behind the study of relationships.

4.6 Statistical Techniques for Studying Relationships Between Variables as a Means to Accurate Prediction and Control

Once students properly understand and appreciate the usefulness of relationships between variables as a means to prediction and control, we can then bring the field of statistics out onto the stage. We can introduce the role of statistics as follows:

Statistics is a set of optimal general techniques to help empirical researchers study variables and relationships between variables in entities in samples, mainly as a means to accurately predict and control the values of variables (properties) in entities in populations.

After developing this idea, we can spend the rest of the course and subsequent courses discussing standard statistical principles and methods in terms of it. This approach enables us to unify most discussion in statistics under the concepts of entities, properties, variables, and relationship between variables. Sections 5.3 through 5.9 further discuss this unification.

4.7 General Comments

The preceding subsections propose six concepts for discussion at the beginning of an introductory statistics course for non-statistics-majors. The concepts are

1. entities
2. properties of entities
3. variables (which are formal representations of properties of entities)
4. an important goal of empirical research: to predict and control the values of variables
5. relationships between variables as a key to prediction and control and
6. statistical techniques for studying relationships between variables in empirical research as a means to accurate prediction and control.

After introducing the six concepts, the teacher spends the rest of the course covering statistical techniques for studying relationships between variables. The course is thus centered on the fundamental statistical concept of 'relationship between variables' as a means to accurate prediction and control.

Depending on the level of the students, my experience suggests that the six concepts can be properly introduced in between one and eight class sessions. (As noted above, most university and college students can learn the high-level concepts in between one and three class sessions.) Study of the details of the sixth concept can last a lifetime.

Teachers can ensure that students understand the concepts by developing them through bottom-up sequences of ideas, beginning with familiar concrete examples of each concept and working up at an appropriate pace for the students to the general concept.

I call the approach to the introductory statistics course described above the "entity-property-relationship" (EPR) approach. Section 5 discusses evaluating the EPR approach, Section 6 discusses testing it, and Section 7 discusses implementing it.

5. EVALUATING THE EPR APPROACH

This section presents material to help readers evaluate the entity-property-relationship approach to the introductory statistics course.

5.1 Main Differences Between the EPR Approach and Other Approaches

The EPR approach differs from other approaches to the introductory course in the following ways:

1. The EPR approach unifies many ideas of statistics and empirical research under the goal of accurate prediction and control of the values of variables. Many other approaches do not directly emphasize this unifying goal.
2. The EPR approach focuses on using information about relationships between variables (as gleaned from empirical research) as a key to accurate prediction and control. Many other approaches do not emphasize the concept of 'relationship between variables' as a unifying statistical concept. Sections 5.2 through 5.9 further discuss the unifying aspects of the EPR approach.
3. The EPR approach introduces the concepts of 'entity', 'property of an entity', and 'variable' at the beginning of the course before discussing relationships between variables. Most other approaches omit (or at least minimize) discussion of these important preliminary concepts. Section 5.10 gives further discussion.
4. Although the EPR approach places heavy reliance on data analysis, the concept of 'data analysis' is not emphasized. Many other approaches emphasize the concept of 'data analysis'. Section 5.12 gives further discussion.
5. The EPR approach uses examples of empirical research projects that are practical. Some other approaches pay less attention to ensuring that examples are practical. Section 7.5 gives further discussion.
6. The EPR approach emphasizes the role of statistics in empirical research, as discussed in Section 4.6. Some other approaches pay less attention to the role of statistics and place more emphasis on the mathematical aspects of statistics. Section 7.9 gives further discussion.
7. The EPR approach emphasizes the use of hypothesis testing as a means to detecting relationships between variables. Some other approaches suggest that hypothesis testing should be de-emphasized. Section 7.10 gives further discussion.
8. The EPR approach omits discussion of univariate distributions at the beginning of the course. Many other approaches spend a substantial amount of time discussing univariate distributions at the beginning. Section 7.11 gives further discussion.

Despite the above differences, the EPR approach is consistent with and thus compatible with most other approaches to the introductory statistics course -- the differences above are merely differences in ordering and emphasis of statistical topics. Section 5.14 illustrates the relationship between the EPR approach and several other popular approaches.

5.2 The Concepts of the Approach Are Easy to Understand

Sections 4.1 and 4.2 imply that the concepts of 'entity' and 'property' pervade students' unconscious thought. Therefore, if we carefully bring these concepts into students' consciousness (through sufficient practical examples), students find the concepts easy to understand. Similarly, Sections 4.3 through 4.5 suggest that if we carefully develop the concepts of 'variable' and 'relationship between variables' for students with practical examples, these concepts are also easy for students to understand.

The ease of understanding leads me to conjecture that the concepts of entities, properties, variables, and relationships can be taught at all levels of teaching statistics from late elementary school up, with only the teaching time and depth of coverage of the concepts varying at different levels.

5.3 The Approach Provides a Deep and Broad Foundation for Statistical Concepts

Section 4.3 introduces the fundamental statistical concept of 'variable' in terms of the concepts of 'entity' and 'property'. Section 4.5 introduces the fundamental statistical concept of 'relationship between properties' (relationship between variables), which is clearly also built atop the concepts of 'entity' and 'property'. The concepts of 'entity', 'property', and 'relationship' can be used as a foundation for other statistical concepts. Here is a sequence of definitions that develop some basic statistical concepts from the three concepts:

• number: a fundamental type of entity in everyday life and mathematics that is learned in childhood through instances of numbers in counting and through fractions; numbers are generally used to represent simple counts of entities and to represent the values of properties of entities
• set: a fundamental type of entity in human reality that consists of zero or more entities of a specified type (i.e., with specified properties)
• population: the complete set of entities of a specified type (see also Appendix A.4)
• sample: a set of entities selected from a population
• event: a fundamental type of entity in human reality that generally involves a particular physical location and a particular continuous period of the property "time" and that has other defining properties that make it of interest
• probability: a particular property of an event that reflects how often (in time or in some other dimension) the event occurs or is thought likely to occur
• variation: changing in the values of a property across entities in a population or sample (changing either between entities or within entities over time)
• distribution: the (often unknown) set of the values of a property (or properties) in the entities in a population or sample
• distribution function: a mathematical entity that mimics in probabilistic terms the distribution of the values of a property (properties) in the entities in a population or sample
• model (equation): a mathematical statement of a relationship between properties of entities
• parameter: a general name for a property of a population, distribution, or model
• estimation: the process of assigning a value to a property or parameter, often including an estimate of the accuracy of the value
• hypothesis: a statement whose truth is unknown about (a) the existence of an entity or type of entity, (b) the existence of a relationship between properties of entities, or (c) the value of a parameter
• statistical test: a technique for providing an objective measure of the weight of a body of evidence in support of a hypothesis
• statistic: any of various well-defined properties of a sample whose values are obtained by performing mathematical operations on the values of one or more properties of the entities in the sample; often used to estimate the values of parameters or in performing statistical tests.

The definitions cover many of the main statistical concepts. Each definition is built atop the concepts of 'entity', 'property', or 'relationship', or is built atop concepts that are themselves built atop the three concepts. Furthermore, the concepts of entities, properties, and relationships appear to be among the most fundamental concepts of human reality. Thus the EPR approach provides a deep and broad foundation for statistical concepts.

5.4 The Approach Unifies Statistical Methods

Statistical methods can perform the following four groups of techniques to help empirical researchers study relationships between variables:

1. techniques for detecting relationships between variables
2. techniques for illustrating relationships between variables
3. techniques for predicting and controlling the values of variables on the basis of relationships between variables, and
4. miscellaneous techniques for the study of variables and relationships between or among variables.

These techniques are of substantial help in answering important questions 5, 7, and 8 in Section 4.5. I discuss these techniques further in the paper for students (1997a, secs 8-13).

(Logically, the four groups of techniques seem best listed in the above order. However, pedagogically, in an introductory statistics course it makes sense to discuss simple techniques for illustrating relationships before discussing techniques for detecting relationships.)

The four groups of techniques raise the question: Which of the currently available statistical methods can actually perform these techniques? The following twenty-one statistical methods can perform one or more of the four groups of techniques:

• general linear model (t-test, analysis of variance, linear regression, multiple comparison methods, hierarchical methods, variance components analysis, multivariate analysis of variance, multivariate linear regression)
• generalized linear model
• response surface methods
• exploratory data analysis
• time series analysis
• survey analysis
• survival analysis
• categorical analysis
• graphical methods
• meta-analysis
• Bayesian methods
• nonlinear regression
• neural networks and statistical learning theory
• discriminant analysis
• nonparametric methods
• probit analysis
• logistic regression
• correlation analysis
• structural and path analysis
• data mining methods
• univariate analysis.

Upon consideration, many statisticians will agree that the above list of twenty-one statistical methods contains almost all of the currently popular methods, including what most statisticians would view as the "main" methods. Many statisticians will also agree that the only techniques that the statistical methods in the list can perform are given in the four groups of techniques that appear in the first paragraph of this subsection. Appendix I provides support for these claims and explains why certain statistical methods are excluded from the list.

Since the list of twenty-one statistical methods contains almost all of the currently popular methods (including the main methods), and since each method in the list is fully explained (at a high level) by the four groups of statistical techniques that are emphasized in the EPR approach, therefore the approach unifies statistical methods. That is, the EPR approach allows us to teach each new statistical method in terms of the same set of simple concepts: entities, properties, variables, and relationships between variables. Emphasizing the simple commonalities that exist among the methods makes the field of statistics substantially easier for students to understand.

5.5 The Approach Links Well with General Concepts of Science

The EPR approach has strong links with three important general concepts of science, as follows:

Scientific Method. The scientific method (also known as the hypothetico-deductive method of science) has apparently existed implicitly among skilled artisans and tradespeople since the dawn of civilization. It was brought into formal consideration by a long line of artisan-scientist-philosophers who have shown us that systematic observation and experimentation are keys to understanding any area of experience. Fowler (1962) gives a concise overview of many of the people and events in the development of the formal scientific method, and Dunbar (1995) discusses the "pan-cultural" and "pan-species" nature of science. Dingle (1952) and Drake (1970) suggest that Galileo introduced fundamental advances to the understanding and practice of the method. Many philosophers and scientists have written about the method.

It is reasonable to view the scientific method as consisting of four steps:

1. A researcher frames (i.e., invents) a new hypothesis about some area of experience. (The researcher frames the hypothesis on the basis of knowledge of earlier research and current theories, often together with intuition, logic, or mathematical modeling. If the hypothesis is multi-faceted or fundamental, it may be called a "theory".)
2. A researcher (perhaps the same researcher) then deduces or infers an empirically testable implication of the hypothesis.
3. A researcher (perhaps the same) then performs an empirical research project to test whether the implication is actually present in the area of experience.
4. If reliable evidence of the implication is found (and in the absence of a reasonable alternative explanation), the community with prime interest in the area of experience will (by informal consensus) accept (or will be more inclined to accept) the hypothesis framed in step 1 as being correct.

The scientific method is central to science because almost all modern scientific research (and most other empirical research) proceeds formally according to the method. The method is repeated over and over, as discussed by Box and Draper (1987, sec. 1.3).

Interestingly, actual scientific research often proceeds quite differently from the first three steps above, with frequent reorderings of the steps, surprising serendipity, and many false starts. However, scientific research is usually formally viewed in terms of the scientific method because experience has shown that this point of view is generally the most efficient.

Examination of instances of the use of the scientific method suggests that the implication in step 2 can usually be usefully viewed as a statement of a relationship between variables in some population of entities. This can be seen by applying the nine questions discussed in Section 4.5 to specific research projects that exemplify the method -- the questions almost always reasonably apply. Thus the key concept of the EPR approach of 'relationship between variables' plays a central (though often implicit) role in the scientific method.

Appendix H discusses some possible counterexamples to the point in the preceding paragraph. I further discuss the scientific method in a Usenet post (2001, app. A).

Scientific Explanation and Understanding. As suggested in Section 4.4, "explanation" and "understanding" play prominent roles in science. What are scientific explanation and understanding? To help understand this question I wrote a brief version of the accepted scientific explanation of a particular physical phenomenon -- the phenomenon of ocean tides. After writing the explanation I disassembled it to see what it consisted of (1997b). The disassembly suggests that the scientific explanation of ocean tides contains seven types of statements, as follows:

1. statements of the existence of entities
2. statements of the existence of properties of entities
3. statements of the values of properties
4. statements of the distributions of the values of properties
5. statements of relationships between entities
6. statements of relationships between properties (relationships between variables)
7. statements that link together the statements of the six other types.

Because the seven types of statements are all quite general, and because experience suggests that many (all?) other scientific explanations can be given in terms of (at most) the seven types of statements, it appears that most (all?) scientific explanations consist of merely (at most) the seven types of statements.

We can view scientific "understanding" as taking place in an individual person. A person has understanding of some state of affairs or phenomenon if they have learned to think and speak in terms of the "correct" explanation of it.

The seven types of statements of a scientific explanation are all important, but the sixth type (about relationships between variables) is perhaps the most important. This is because statements of relationships directly enable accurate prediction and control. Thus the key concept of the EPR approach of 'relationship between variables' plays a central role in scientific explanation and understanding.

Mathematical Equations. Mathematical equations are crucial on the theoretical side of many branches of science. But most (all?) mathematical equations in science (as opposed to equations in pure mathematics) are simply statements of known or hypothesized relationships between variables (relationships between properties of entities). Thus the key concept of the EPR approach of 'relationship between variables' again plays a central role.

5.6 The Approach Unifies Empirical Research

As discussed in Section 4.5 and Appendix H, most empirical research projects can be usefully viewed as studying relationships between variables. Thus by focusing on the concept of 'relationship between variables' the EPR approach unifies most empirical research.

5.7 The Approach Links Well with General Concepts of Commerce

A commercial product is an entity, as are instances of a product, as is a commodity, as is a financial instrument (e.g., a stock or a bond), as is a loan repayment or dividend, and as is an interaction with a customer. These and all other entities that are used in commerce are efficiently handled by the EPR approach.

To achieve a general understanding of the logical constructs used in commerce it is helpful to study how commercial organizations store information. Almost all progressive commercial organizations use a computer "database" as their main repository for information. This is because databases have easy-to-use, versatile, reliable, and secure features that allow one to easily assemble information to generate reports, invoices, charts, and other graphical, statistical, and textual information as a broad and fundamental aid to operating an organization.

A database consists of a set of one or more "tables". Each table holds information about entities of a particular type. For example, a manufacturing company might have one database table that holds information about its products, another that holds information about its customers, another that holds information about its invoices, and so on. The database (or databases) of a larger progressive organization may contain hundreds (or even thousands) of tables holding information about all the types of entities in which the organization has a serious interest.

A database table is conceptually identical to a statistical data table, as described in Section 4.3 -- a rectangular array that contains one or more "rows" associated with the entities the table is tracking and one or more "columns" associated with properties of the entities. Each cell in the table contains the value of the property associated with the column for the entity associated with the row.

The database of a progressive commercial organization will hold a substantial proportion of the organization's information because even "documents" (which are entities) can be stored in a database table to facilitate ready access. (A cell in a modern database table can hold an entire document.)

As noted, database tables are the main repositories for information in commerce, and the rows in a database table are associated with entities and the columns are associated with properties of the entities. Thus the basic concepts of the EPR approach of 'entity' and 'property' play fundamental (implicit) roles throughout commerce.

(Data mining is reasonably viewed as the study of relationships between variables reflected in the columns of database tables.)

5.8 The Approach Links Well with Language

Section 4.1 notes that nouns are used in language to denote entities and Section 4.2 notes that adjectives and adverbs are often used in language to denote the values of properties of entities. Since language is intimately tied to human thought, it is of interest to consider how the concepts of the EPR approach relate to other parts of speech.

A verb usually express one of the following ideas:

• an act or action by an entity or entities
• an occurrence (i.e., an event)
• a state or mode of being of an entity or entities (possibly in terms of the value or values of one or more properties)
• a relationships between entities (including a relationship between properties and a relationship between variables).

When one views entities broadly, the acts, actions, occurrences, events, states, modes, values, properties, variables, and relationships in the list are themselves all entities. (Appendix A further discusses this idea.)

Furthermore, most sentences that contain verbs also contain one or more nouns. (All verbs in coherent sentences have a subject, perhaps implicit, which is represented by a noun. Transitive verbs have an object, perhaps implicit, which is also represented by a noun.) The verbs describe various "things" about the entities denoted by the nouns including things about the properties of the entities.

Thus when verbs are used in language, entities are invariably present and of central interest.

(Verbs are also often linked to the concept of 'time'. We can view time [both duration and point in time] as a property of the entity that contains all other entities -- the entity we call "experience" or "reality". Alternatively, we can view time as a property of events.)

The remaining parts of speech function as or support nouns, verbs, adjectives, and adverbs as follows:

• Articles ("a", "the") are a form of adjective -- they modify a common noun to indicate whether it is being used to denote an entity in a general sense or in a particular sense.
• Prepositions delineate phrases that function as adjectives or adverbs. The choice of a preposition denotes some general aspect (e.g., "in", "of", "beside") of the relationship between the word or phrase (i.e., noun, verb, adjective, or adverb) being modified by the prepositional phrase and the entity named by the noun that is the object of the phrase.
• Conjunctions (e.g., "and", "but", "if", "because") join together words, phrases, or clauses to make more complicated phrases, clauses, sentences, or pairs of sentences. The choice of conjunction indicates the relationship between the ideas (which are a type of entity) it joins.
• Interjections express simple complete thoughts -- often expressions of emotion. Both emotions and expressions of emotion are reasonably viewed as being entities, as discussed in Appendix A.

Thus the concepts of the EPR approach link well with the various parts of speech. Thus the approach links well with language at a fundamental level.

5.9 The Concepts of the Approach Are Fundamental

Subsections 5.2 through 5.8 suggest that the concepts of entities, properties, variables, and relationships between variables

• are easy to understand
• are a foundation for many of the fundamental statistical concepts
• unify statistical methods
• link well with general concepts of science
• unify empirical research
• link well with general concepts of commerce and
• link well with language at a fundamental level.

These points together with consideration of the fundamental statistical concepts discussed in Section 5.3 suggest that the concepts of the EPR approach are more fundamental than many (all?) of the other concepts that are traditionally discussed in statistics courses.

5.10 Easy-to-Understand Fundamental Concepts Should Be Taught First

What Should Come First? Concepts in a body of knowledge are generally easiest to understand and use if they are developed in a logical order beginning with the most fundamental. This is especially true if the fundamental concepts are themselves easy to understand.

As suggested in Section 5.2, the concepts of 'entity' and 'property' and (to a lesser degree) 'variable' and 'relationship' are easy to understand. Furthermore, these simple concepts are (by virtue of their logical priority) substantially easier to understand than the various traditional statistical concepts that depend on them. In view of the ease of understanding, and in view of the logical priority, it is reasonable to carefully cover the concepts of 'entity', 'property', 'variable', and 'relationship' first, before introducing even the most rudimentary of the other traditional statistical topics. As suggested in Sections 5.3 through 5.9, this unifies and simplifies discussion of the traditional topics.

Must We Emphasize Entities and Properties? An associate editor (assumed male for expository convenience) noted that his students have little difficulty distinguishing the concepts of 'entity' and 'property'. (He refers to the concepts respectively as "case" and "variable" -- see my 1998a paper, App. E.1 and Appendices D, and E below.) Thus he is noting that these concepts are easy for students to understand. In view of this he wonders whether it is necessary to emphasize entities and properties at the beginning of the introductory statistics course. Perhaps the concepts are simple enough that we need not discuss them at all.

To address this issue, let us consider the concept of 'variable'. This concept is arguably the most ubiquitous concept in statistics. Clearly, students must understand the concept of 'variable' before they can understand the concept of 'relationship between variables'. How do students usually learn this concept?

Students usually first learn the concept of 'variable' in their first algebra class in grade 6 or later. The introductory algebra teacher usually does not teach the concept directly in terms of entities and properties. Instead, the teacher teaches the concept in terms of examples of entities and properties. For example, in the first discussion of the concept of 'variable' the algebra teacher may say that the variable x will represent the height of some person. The teacher will typically say (in a carefully worded discussion) that we do not know the value of x at the present time, but we do know the value of x plus a known constant. The teacher will then show the students how to use algebra to determine what the value of x must be.

During the rest of the introductory algebra course, and during subsequent mathematics courses, students will encounter many other examples of variables that represent unknown values (of properties of entities), and students will learn various ways to manipulate and solve for these values. But they will usually not consciously understand the unifying concept that a variable is reasonably viewed as a formal representation of a property of entities.

Consider some differences between the use of the concept of 'variable' in mathematics and the use in statistics:

• A variable in a mathematical problem generally has only a single value. (This value is often abstract -- known only in the sense of an algebraic symbol, such as x -- but the value is generally viewed as a single value.) On the other hand, a variable in a statistical problem usually has multiple values -- typically one value for each entity in the sample (and population) under study.
• In mathematics the values of important variables are often unknown. Often the goal of a mathematical analysis is to solve for or prove a theorem about the unknown value(s). On the other hand, in the typical application of statistics to a real-world problem, the values of the relevant variables are (for the sample, after the data collection) generally all known. Generally the goal of a statistical analysis is to enable drawing conclusions (predictions) about the unknown values of the response variable for new entities in the population on the basis of the known values of the predictor variable(s) for these entities and on the basis of the known values of the response and predictor variables for the entities in the sample.

Perhaps due to the above differences between the mathematical and statistical concepts of 'variable', and perhaps due to the mathematical (algebraic) genesis of the concept of 'variable' in students' minds, many non-statistics-majors have difficulty understanding the fundamental statistical concept of 'variable'. This can be seen by asking students to define the concept -- many students have difficulty giving a reasonable definition. Some students may say that a statistical variable is a "measurement of something", which (although vague) is certainly correct. But they are often unable to say, without prompting, what the "something" is -- both in the specific sense of voluntarily identifying the relevant entities in a given situation and in the general sense of linking the concept of 'variable' to the more fundamental concept of 'property of an entity'.

The preceding five paragraphs suggest that many students entering the introductory statistics course lack a clear understanding of the statistical concept of 'variable'. But examination of currently popular textbooks for the introductory course suggests that most approaches assume entering students have a clear understanding of this concept.

(Some books do introduce entities, properties, and variables, but spend only a page or two on these topics at the beginning and then never return to focused discussion of them. This approach forgoes the substantial unifying power of the concepts. For example, using the same concepts but different terminology Moore briefly discusses "individuals", "characteristics", and "variables" at the beginning of two of his introductory texts [1997b, 2000]. I discuss Moore's use of the concepts in a Usenet post [1997c].)

Sections 4.1 and 4.2 above suggest that all students unconsciously learn the concepts of 'entity' and 'property' (the concepts not the words) when they are infants or young children. Thus if we carefully bring these concepts into students' consciousness, and if we then carefully build the important statistical concept of 'variable' atop these foundational concepts, we ensure that students have a proper understanding of the concept of 'variable'. This understanding helps students to understand the central statistical concept of 'relationship between variables'.

Therefore, it is useful to spend time at the beginning of an introductory statistics course discussing the fundamental concepts of 'entity' and 'property'.

5.11 The Approach Gives Students a Lasting Appreciation of Statistics

Section 3.3 recommends that the first goal of an introductory statistics course be to give students a lasting appreciation of the vital role of the field of statistics in empirical research. Does the EPR approach satisfy this goal?

Consider:

• The EPR approach is aimed specifically at satisfying a main goal of empirical research -- the goal of accurate prediction and control.
• Most students are directly interested in prediction and control (of variables of interest to them).
• The idea of prediction and control on the basis of relationships between variables is easy to understand.
• The EPR approach directly shows how the field of statistics can play a broad role across almost all empirical research.
• The EPR approach is developed in a logical sequence from intuitive fundamental concepts (with numerous practical examples).

These points suggest that the EPR approach gives students a lasting appreciation of the field of statistics and its vital role in empirical research.

5.12 The Approach Links Well with the Concept of 'Data Analysis'

Many approaches to introductory statistics emphasize the concepts of 'data' and 'data analysis'. One can see this by noting the frequent occurrence of the word "data" in the preface and early chapters of many textbooks and other discussions. In contrast, the EPR approach does not emphasize the concepts of 'data' or 'data analysis' and instead emphasizes the concept of 'relationship between variables as a means to accurate prediction and control'.

The EPR approach links well with the concepts of 'data' and 'data analysis'. This is because the exact operation of what is called "data analysis" is an essential step of the EPR approach. Data analysis is the step in which we actually study the relevant data to look for information about relationships between the variables.

Tukey initiated emphasis on the concept of 'data analysis' in statistics education with his seminal book Exploratory Data Analysis (1977). Tukey and later reformers emphasize this concept to emphasize the important practical side of the field of statistics, which lies in helping researchers to analyze research data. Emphasis on data analysis has helped to eliminate the detrimental emphasis on mathematical statistics in teaching statistics to non-statistics-majors. As further discussed in Section 7.9, de-emphasizing the mathematical side of statistics makes the material easier for non-statistics-majors to understand.

Because emphasis on mathematical statistics is now greatly diminished in introductory statistics courses for non-statistics-majors, and because these courses now generally focus on analyzing data, it is useful to ask whether central emphasis on the concept of 'data analysis' is still necessary or whether central emphasis on another concept is more effective. This question is important because the concept of 'data analysis', although fully correct, is functionally vague -- to a beginner doing "data analysis" sounds like something medieval monks might do with great rigor in a remote monastery in the mountains, but with no known practical value.

Instead of emphasizing data and data analysis, the EPR approach sharpens the focus by emphasizing the function of data analysis which, as discussed above, can be usefully viewed as being (mainly) to give us the ability to predict and control the values of variables through study of relationships between variables. This emphasis on function or usefulness (when coupled with sufficient practical examples) substantially increases beginning students' appreciation of the role of statistics.

A teacher can show students the link between relationships between variables on the one hand and data analysis on the other by noting that relationships between variables are generally studied in terms of relationships between (the values of the variables in) the columns of data in a data table.

5.13 The Concepts Are Old But the Approach Is New

The concepts of entities, properties, and relationships are not new. Indeed, all statisticians and empirical researchers use these concepts implicitly throughout their thinking and discussion. However, as discussed in Section 5.10, the fundamental concepts of 'entity', 'property', 'variable', and 'relationship' are almost never carefully discussed in a unified approach in introductory statistics courses. I believe that the unfortunate omission of unified discussion of these concepts is the main reason why the field of statistics is so widely misunderstood.

(Some leaders in statistics education have already independently adopted an important aspect of the EPR approach in that they emphasize relationships between variables in their introductory courses. For example, using an idea developed by Gudmund Iversen, George Cobb teaches two introductory courses, both of which start with relationships -- one devoted to experimental design and applied analysis of variance and the other devoted to applied regression [G. Cobb, personal communication, August 21, 1996]. Similarly, Robin Lock teaches an introductory course devoted to time series analysis -- i.e., methods for studying relationships between variables when an important predictor variable is "time" [Cobb 1993, sec. 3.1].)

5.14 The Approach Links Well with Other Approaches to the Introductory Course

Many helpful new approaches to teaching the introductory statistics course have recently been proposed. As suggested by Moore (1997a), these approaches fall neatly into two distinct groups: conceptual approaches and pedagogical approaches.

Each of the conceptual approaches emphasizes a particular set of statistical concepts and de-emphasizes other statistical concepts. The EPR approach is a conceptual approach. Other (sometimes overlapping) conceptual approaches include

• emphasis on data analysis and de-emphasis of theoretical concepts (especially de-emphasis of probability theory, distribution theory, and the theory of statistical tests) (Tukey 1977; Cobb 1992; Moore 1992a, 1992b)
• emphasis on exploratory data analysis and de-emphasis of confirmatory data analysis (Tukey 1977; Velleman and Hoaglin 1992)
• emphasis on statistical reasoning and de-emphasis of statistical methods (computations) (Ruberg 1990; Bradstreet 1996)
• emphasis on the design and analysis of experiments (Bisgaard 1991)
• emphasis on the Bayesian approach to statistics (usually restricted to more mathematically literate students) (Blackwell 1969; DeGroot 1986; Albert 1996, 1997; Berry 1996, 1997; Berry and Lindgren 1996; Antelman 1997; Moore 1997c, 1997d)
• emphasis on (a) process (which is a particular type of entity) and (b) the minimization of the variation in selected properties of a process, mainly through relationships between variables, sometimes with less emphasis on formal statistical methods (Snee 1993; Hoerl and Snee 1995; Britz, Emerling, Hare, Hoerl, and Shade 1997, Hoerl and Snee 2002)
• emphasis on nonparametric statistical methods (Iman and Conover 1983; Noether 1992)
• emphasis on resampling (Boomsma and Molenaar 1991; Simon and Bruce 1991; Smith and Gelfand 1992; Wonnacott 1992; Albert 1993; Simon 1993, 1994; Willemain 1994; Pollack, Fireworker, and Borenstein 1995; Hesterberg 1998)
• emphasis on aspects of a practicing statistician's work that are not directly related to drawing conclusions from data, such as team design of research projects, ensuring data quality, and managing data (Higgins 1999)
• emphasis on probability (Falk and Konold 1992).

In contrast to the conceptual approaches, the pedagogical approaches to teaching the introductory statistics course emphasize new ways of teaching any set of statistical concepts. The pedagogical approaches include

• more interaction between the teacher and the students instead of straight lectures (Mosteller 1988; Zahn 1994)
• application of the principles of Total Quality Management or Continuous Quality Improvement to the design and management of the course (Hogg and Hogg 1995; Wild 1995, Hogg 1999)
• use of multimedia, film, or video, to teach the concepts (Moore 1993; Ottaviani 1996; Velleman and Moore 1996; Cobb 1997; Cryer and Cobb 1997; Doane, Mathieson, and Tracy 1997; Newton and Harvill 1997; Macnaughton 1998b; Velleman 1998)
• use of demonstrations, activities, or projects to teach the concepts (Jowett and Davies 1960; Scott 1976, Hunter 1977; Hansen 1980; Carlson 1989; Bisgaard 1991; Halvorsen and Moore 1991; Bryce 1992; McKenzie 1992; Roberts 1992; Sylwester and Mee 1992; Zahn 1992; Gunter 1993, 1996; Fillebrown 1994; Garfield 1994, 1995; Mackisack 1994; Sevin 1995; Magel 1996; Rossman 1996; Scheaffer, Gnanadesikan, Watkins, and Witmer 1996; Holcomb and Ruffer 2000)
• use of students working in groups instead of working individually (Dietz 1993; Garfield 1993, 1995)
• replacement of lectures by assigned readings, which are motivated by individual and group "Readiness Assessment Tests" (Michaelsen 1999, Simon, Harkness, Buchanan, Chow, Heckard, Lane, and Zimmaro 2000, Teaching Effectiveness Program 2000)
• emphasis on examples that are clearly practical as opposed to examples that have no obvious practical value (see Section 7.5 below)
• use of real or realistic data in examples (see Section 7.8 below)
• use of cases (i.e., detailed examples or problems based on practical situations) to teach the concepts (Chatterjee, Handcock, and Simonoff 1995; Czitrom and Spagon 1997; Parr and Smith 1998; Peck, Haugh, and Goodman 1998; Nolan and Speed 1999)
• use of news stories to teach the concepts (Snell and Finn 1992; Snell 1999)
• use of improved methods for assessing students (Eltinge 1992; Cobb 1993; Garfield 1994; Chance 1997; Gal and Garfield 1997b, 1999; Goldman, McKenzie, and Sevin 1997; Garfield 2000)
• use of concept maps to teach the concepts (Schau and Mattern 1997a, 1997b)
• use of community service projects to teach the concepts (Anderson and Sungur 1999)
• use of sports statistics to teach the concepts
• use of a computerized library of examples that students can analyze to aid in teaching the concepts (Andrews and Herzberg 1985; Hand, Daly, Lunn, McConway, and Ostrowski 1994; Velleman, Hutcheson, Meyer, and Walker 1996; Pearl, Notz, and Stasny 1996; OzData 1999; StatLib 1999, Rumsey 2001)
• emphasis on improving students' writing ability (Radke-Sharpe 1991; Samsa and Oddone 1994; Stromberg and Ramanathan 1996; Holcomb and Ruffer 2000)
• use of computers or calculators to generate data displays, to illustrate and simulate statistical ideas, and to perform statistical analyses (Dixon and Massey 1983, Macnaughton 1998a, sec. 8).

I discuss some criteria for evaluating pedagogical approaches in a paper (1998a, sec. 7).

Most introductory statistics teachers now use some combination of the above conceptual and pedagogical approaches. The main disagreement among teachers is only about the relative emphasis that each approach deserves.

(It is possible to classify the use of multimedia, film, video, computers, and calculators as "technological" approaches to the introductory course, rather than as "pedagogical" approaches. However, it seems more reasonable to view technology as a means to better pedagogy rather than as an end in itself.)

A simple relationship exists between the EPR conceptual approach to the introductory statistics course and the other approaches -- the EPR approach can be effectively used in conjunction with any (or any group) of them.

Moore (1997a) reviews several of the new approaches to statistics education. Cox (1998) comments on some general aspects of statistics education. Gordon and Gordon (1992), Hoaglin and Moore (1992), and T. Moore (2000) give papers by leading statistics educators about teaching statistics. Hawkins, Jolliffe, and Glickman (1992) give a general discussion of teaching statistical concepts.

5.15 Responses to Criticisms of the EPR Approach

The EPR approach has been criticized as being too "abstract" for students to understand. I discuss this important criticism in a Usenet post (forthcoming). I discuss some other insightful criticisms of the approach in a series of Usenet posts (1996-2001).

6. TESTING THE EPR APPROACH

It is interesting that statisticians, who are the keepers of the keys to empirical (scientific) research, perform almost no serious empirical research in statistics education. Instead, much of what is reported as "testing" of approaches in statistics education is anecdotal. That is, the author or proponents of a new approach use the approach one or more times in courses and then report that the approach was successful.

Unfortunately, no matter how "successful" a course might appear to be, anecdotal reports do not reflect valid empirical research about the approach used in the course. This is because reasonable alternative explanations invariably exist that could explain why the course was as successful as it was. Some possible reasons why a course might be successful are

• The teacher may be a very good teacher, and the success of the course may merely reflect the teacher's skill and may not reflect anything notable about the approach used in the course.
• The lessons, assignments, tests, and examinations in the course may have been easy (relative to other courses) and the success of the course may merely reflect the easy course, and may not reflect anything notable about the approach used in the course.
• The teacher may have invested a substantial amount of time and effort in the approach, which may dispose him or her to see the approach from an overly positive point of view.

These alternative explanations (and other situation-specific alternative explanations) imply that anecdotal testing of approaches to teaching statistics is invariably equivocal.

We can eliminate the equivocation of anecdotal testing by testing approaches with randomized experiments. Such experiments (when properly performed) provide clear comparative evidence of the effectiveness of different approaches to teaching statistics.

Some readers may feel that experimentation in statistics education is not possible because too many confounding variables are present. For example, "instructor teaching ability" must be properly accounted for before unequivocal conclusions can be drawn. However, confounding variables can usually be accounted for in experimental research, albeit at the expense of increased cost and complexity. Furthermore, accounting for confounding variables in experimental research in statistics education would appear to be no more complicated than accounting for them in multicenter clinical trials, where accounting for confounding variables is standard practice.

Some readers may feel it would be difficult to ensure protocol adherence by the multiple statistics teachers that are needed in a proper experimental trial of different approaches to statistics education. Clearly, this is a challenging problem, although perhaps no more difficult than ensuring protocol adherence in multicenter clinical trials, where various monitoring systems are used to ensure adherence.

Some readers may feel that experimentation in statistics education may be ineffective because no reasonable response variable can be found that is sensitive enough to discriminate between different treatments. However, this is an empirical question that awaits serious attempts to address it.

I further discuss methods and problems of experimentally testing approaches to the introductory statistics course, including a recommendation that one use "attitude toward statistics" as a response variable, in a paper (1998a, app. A and B).

6.2 Testing of the EPR Approach

Despite my preceding comments, I regret that I cannot report proper experimental testing of the EPR approach -- such testing is beyond my resources.

Although I cannot report proper testing of the EPR approach, I can report some enthusiastic remarks from three teachers who used a draft textbook for the approach (Macnaughton 1986) in their courses. They commented that

... students found the book enjoyable and easy to understand.

Using a unique approach, Macnaughton has provided a comprehensive first-rate introduction to the material. I would highly recommend the book for use in introductory statistics courses ....

- Professor Alexander Even, The Ontario Institute for Studies in Education

... students obtained a good understanding of the basic principles of statistical analysis.

... [the approach] substantially simplifies the material without sacrificing important concepts.

- Professor John Flowers, School of Physical and Health Education, University of Toronto

The absence of overt mathematics enables the underlying principles of scientific research ... to be more directly apprehended by persons who have ... weak grounding in mathematics. ...

... Students' comments have been uniformly favorable ....

... the book is to be commended to the instructor.

- Professor Donald F. Burrill, The Ontario Institute for Studies in Education

These remarks are encouraging, but are far from being definitive about the effectiveness of the EPR approach. I hope that publication of this paper will facilitate proper testing of the approach.

7. IMPLEMENTING THE EPR APPROACH

Until textbooks based on the EPR approach are available, a teacher wishing to use the approach in an introductory course can use the paper for students (1997a) to reinforce class discussion of the six introductory concepts. The following twelve subsections discuss implementation considerations.

7.1 Motivating Students on the First Day of Class

The first day of class is important because if the lesson is properly designed, it will establish a positive attitude about the course in students' minds. What should be the first statistical idea we introduce to students on the first day of class?

I recommend that the first idea be the promise that the course will teach students how to make accurate predictions. For example, we can promise students they will learn how to accurately (but generally not perfectly) predict

• the mark they will get on the final
• their average annual income over the next several years
• their longevity
• whether it will rain tomorrow
• just about anything else of interest (if it can be reliably measured).

(Along with prediction methods, I recommend that the introductory course devote substantial attention to the methods of exercising accurate control through formal experimentation. However, for simplicity, I recommend that discussion of control and experimentation be omitted at the very beginning -- the promise of accurate predictions seems quite enough to engage students. Section 7.3 further discusses experimentation.)

If we promise students on the first day of class that they will learn how to make accurate predictions, we arouse their curiosity and set the stage for development of the six concepts discussed in Section 4. The promise also sets a practical tone for the course, which is more likely to impress most students than if we begin with mathematical discussion.

If we promise students on the first day of class that they will learn how to make accurate predictions, we must later fulfill our promise. In particular, the thoughtful student will be interested in whether we can demonstrate practical methods for making accurate predictions. Fortunately, statistical methods (broadly viewed) are universally accepted among statistically experienced researchers as the most practical and most accurate methods available for making predictions (and exercising control).

7.2 What Topics Should Follow the Six Concepts?

Section 4.6 recommends that after introducing the six concepts of the EPR approach the teacher spend the rest of the course expanding the sixth concept by covering standard statistical topics. The present and the next subsections discuss ways of covering the standard topics.

As a general principle for choosing topics, I recommend that teachers cover topics that are used more frequently in empirical research first.

One easy way to implement the EPR approach is to follow discussion of the six concepts with material selected from an already existing introductory statistics course. The teacher can use the six concepts to introduce and unify the material. This enables the teacher to use the EPR approach in an already existing course with only a minimum amount of modification to the course.

7.3 A Syllabus

A more unified way of implementing the EPR approach is to break the course into five phases: an introductory phase, a practical-experience phase, a generalization phase, a specific-methods phase (optional), and a mathematics phase (also optional).

Introductory Phase. In this first phase the teacher introduces the six concepts discussed in Section 4.

Practical-Experience Phase. To solidify students' understanding of the six concepts, the teacher follows the introductory phase with a practical-experience phase in which students obtain hands-on experience with specific statistical methods.

I recommend that the teacher begin the practical-experience phase with discussion of a commonly occurring simple type research project -- the observational research project that studies the relationship between two continuous variables (2000 Hayden response, red tab 6). Possibly using the material in the paper for students (1997a) as an introduction, the teacher can discuss how to design an observational research project to study the relationship between two continuous variables, how to use a scatterplot to illustrate such a relationship, how to use statistical techniques to analyze data from such a research project to determine if a relationship is present between the variables, and how to use the model equation derived from such a relationship to make predictions. To reinforce the discussion, I recommend that students be given computer assignments to detect and study (practical) relationships between pairs of continuous variables in various sets of data.

If time permits, the bivariate case can be extended to the multiple regression case.

Next, the teacher can discuss "experiments" and the associated statistical methods as a powerful tool for studying causal relationships between variables. This topic is important because the study of causal relationships through experiments is the soul of empirical research. Discussion can be in terms of designing simple experiments to study causal relationships, illustrating causal relationships, detecting causal relationships in experimental data, and predicting and controlling on the basis of causal relationships. To reinforce the discussion, I recommend that students be given computer assignments to detect and study simple (practical) causal relationships between variables.

If time permits, the fully randomized one-way case can be extended to the multi-way case, repeated measurements, blocking, analysis of covariance, and so on.

The length of the practical-experience phase should be adjusted to allow enough time at the end of the course to properly cover the material in the important next phase.

Generalization Phase. The teacher follows the practical-experience phase with a generalization phase that introduces important topics that span almost the entire field of statistics. The recommended topics are

1. Types of Variables. This topic introduces students to a standard typology standard typology for variables, with four mutually exclusive and exhaustive categories: continuous, discrete-ordinal, discrete-nominal, and binary. This topic also introduces the idea of the univariate distribution of the values of a variable, with coverage of ways of illustrating the distribution of the values of a variable and measures of the center and spread of a distribution. (This topic is sometimes taught at the beginning of an introductory statistics course -- see Section 7.11.)
2. Overview of Statistical Methods. This topic provides a brief high-level overview of statistical methods (such as those listed in Section 5.4 and Appendix I.2), with brief discussion of the conditions under which each method is applicable. Most of the twenty-one methods listed in Section 5.4 are best summarized in a table, with rows of the table indexing the four possible types of the response variable and with columns indexing the four possible types of the predictor variable(s) in the research. Each cell in the body of the table contains the names of the statistical methods that are typically used to study relationships between variables when the response variable and predictor variables are of the indicated types. (As emphasized by Velleman and Wilkinson [1993], who discuss another typology for variables, valid exceptions to the categorization can occur. Nevertheless, the table of methods is useful because it categorizes the standard approaches to the study of relationships between variables and thereby provides a unifying overview of many standard statistical methods.) I recommend that students not be required to memorize the table, but rather it be provided as a key to understanding and choosing statistical methods.
3. Underlying Assumptions of Statistical Methods. This topic discusses how every statistical method is based on certain assumptions (which are different for different methods), and why therefore (to avoid later possible embarrassment) a researcher should always verify that the underlying assumptions are adequately satisfied before attempting to draw conclusions from the use of a statistical method. I recommend that teachers characterize the common assumptions using dot plots or graphs that illustrate situations in which assumptions are both satisfied and not. But, except when students are mathematically mature, I recommend that teachers not discuss the mathematical details of the assumptions -- the important point is the existence of the assumptions, not their complicated mathematical details, which we can leave to the computer and later courses.
4. Statistical Thinking. This topic gives students practice in understanding, criticizing, and writing reports of empirical research, generally involving detailed consideration of the nine questions introduced in Section 4.5, and generally observing the principles of the scientific method, as summarized in Section 5.5.

A reasonable ending-point for an introductory statistics course is at the end of the generalization phase.

Specific-Methods Phase. For students who are likely to perform empirical or statistical research in their careers, I recommend that the preceding three phases be extended (in courses following the introductory course) with detailed discussion of selected statistical methods, such as those listed in Section 5.4 and Appendix I.2.

For each method in Section 5.4, I recommend that the following topics be covered (when applicable):

• Design: How to design a research project to study relationships between variables using the method.
• Power: How to use statistical software to compute the power of statistical tests for detecting relationships between variables using the method.
• Data Checking: How to use statistical software to examine the data from a research project using the method in order to identify and (when appropriate) correct anomalies in the data prior to analysis (using methods for studying univariate and possibly bivariate distributions of the values of variables).
• Assumptions: How to analyze the description and results of a research project in order to determine whether the underlying assumptions of the method are sufficiently satisfied to permit drawing conclusions from the output of the method.
• Detection: How to use statistical software to analyze the results of a research project in order to detect relationships between variables using the method.
• Illustration: How to use statistical software to illustrate relationships between variables using the method.
• Prediction and Control: How to use statistical software to analyze the results of a research project in order to derive model equations for relationships between variables using the method, as an aid to prediction or control.
• Accuracy: How to determine the accuracy of the prediction or control of a model equation.
• Reporting: How to interpret and write reports of empirical research projects that use the method.

Except for statistics or mathematics majors, I recommend that the use of mathematics be avoided in the specific-methods phase. Instead, I recommend that attention be focused on designing research projects and on correctly interpreting the relevant output from statistical software.

Mathematics Phases. For students pursuing careers in statistics or mathematics, I recommend that discussion in the earlier phases be interwoven and extended with discussion of the underlying mathematics. I illustrate an approach to such discussion in a paper about computing sums of squares in unbalanced analysis of variance (1998d). I recommend that statistics majors be made (at a high level) as aware of the role of statistics in empirical research as students in other disciplines. This facilitates interaction between the theoretical and practical sides of statistics.

7.4 "Basis for Action" Versus "Decision Procedure"

In choosing topics for the introductory statistics course it is helpful to distinguish between a basis for action and a decision procedure. A basis for action is a statement that a relationship exists between a response variable that we wish to predict or control and one or more predictor variables. If the variables are appropriately chosen, such a relationship will often suggest that some action be taken. For example, if social researchers find that a relationship exists between secondary school practices with difficult students and the amount of subsequent delinquent behavior by these students, this suggests a basis for action in the choice of secondary school practices.

On the other hand, a decision procedure is a procedure that assists us to make some form of decision. Some possible forms of decision are

1. a decision in some substantive area (e.g., in secondary school practices, in medical care, or in business) whether to take the action suggested by a basis for action, which I call an "action" decision
2. a decision whether a relationship exists between given variables, which is discussed further in Section 7.10
3. a decision about the likely value of a population parameter (possibly a parameter in a model)
4. a decision about the next step to take in conducting an empirical research project.

The second through fourth types of decision play important roles in statistics, but I focus here on the first.

Procedures for making action decisions are studied in a branch of statistics called "decision theory", which was founded by Wald (1950). Such procedures must take account of many diverse inputs. Some typical inputs are one or more bases for action (i.e., relationships between variables), various social or commercial values (or goals, perhaps stated as objective or utility functions), alternative explanations, error sizes, costs, and side-effects. A procedure for making an action decision takes appropriate account of all these inputs and provides as output an "optimal" recommendation whether (and possibly how) to perform various actions.

In view of the multiple diverse inputs, useful procedures for making action decisions are much more complex than relationships between variables. Perhaps due in part to this complexity, most action decisions are still made (in all areas) on the basis of informal and intuitive criteria rather than by formal decision procedures. Swets, Dawes, and Monahan (2000) and Edwards and Fasolo (2001) discuss some current work in decision procedures.

(A formal procedure for making action decisions can be efficiently characterized as a set of relationships between variables in which the response variables are indicators of whether [or how] actions should be taken and the predictor variables reflect the various inputs to the decisions. Decision procedures for assisting with the second through fourth types of decisions above can be similarly characterized.)

The question arises whether the introductory statistics course should discuss procedures for making action decisions. Because such procedures are complicated and infrequently used, I recommend that the introductory course omit this topic. And although the formal procedures for making optimal action decisions are an interesting and important area of study, it seems more in keeping with the standard use of statistics in empirical research to focus the introductory course on the study of relationships between variables. A relationship can suggest a basis for action in a substantive area although (except indirectly) relationships do not provide the final decision whether to act.

Barnett discusses the distinction between statistical inference (which can often be reasonably viewed as inference about relationships between variables) and decision procedures (1982). Bordley (2001) discusses teaching decision theory in applied statistics courses.

7.5 Practical Examples

As noted at several points above, and following Jowett and Davies (1960), Scott (1976), Hunter (1977, pp. 16-17), Cobb (1987, sec. 4.2), and Willett and Singer (1992, p. 91), I recommend that any implementation of the EPR approach discuss each main concept in terms of numerous practical examples. Practical examples can appear in lectures, textbooks, multimedia courseware, class discussions, exercises, activities, and projects.

A Criterion for Practicality of Research Projects. An important type of example in an introductory statistics course is an example of an empirical research project that studies a relationship between variables. To determine whether such an example is "practical", I recommend (after Deming [Wallis 1980, p. 321] and Scheaffer [1992, p. 69]) that the teacher consider the following question:

Does an understanding of the relationship between variables in the example have an obvious potential social, scientific, or commercial benefit? That is, does knowledge of the relationship suggest some clear basis for action?

If we choose examples that suggest a clear basis for action, and if we ensure that students see the practical benefits provided in the examples, we help students to appreciate the practical value of statistics.

Finding Practical Examples. Examples of empirical research projects that suggest a clear basis for action are easy to find in most fields of empirical research. For example, consider research in medicine to study the relationship between AIDS symptoms (as reflected in a response variable) and a new treatment for AIDS (as reflected in a predictor variable, typically a measure of the dose of the new treatment). This research is very practical according to the criterion. That is, if AIDS research finds a new relationship between relevant variables, it suggests a clear basis for action in the treatment of AIDS. Similarly, research to study relationships between variables that help to make computer hardware or software more efficient or less expensive is also (in a commercial sense) very practical because if this research finds a new relevant relationship, it suggests a clear basis for action in designing computer hardware or software.

Examples that fail to satisfy the practicality criterion seriously detract from the field of statistics because they associate the field with problems that appear to be frivolous (or at best inconsequential). For example, a research project that studies the relationship between people's forearm lengths and their foot lengths is a "frivolous" research project because students can see no obvious practical use of knowledge of this relationship. Study of frivolous research projects trivializes the field of statistics.

(Interestingly, if one looks hard enough, one can find practical uses of most relationships between variables. For example, the relationship between forearm length and foot length is occasionally used in orthopedics, paleontology, and physical anthropology. If a particular group of students is likely to be impressed by an obscure example, this is a reasonable example for them. But if a complicated explanation is needed before students can see the practicality of an example, most students are unimpressed.)

If the students in a particular introductory statistics course are all specializing in the same discipline, and if that discipline performs empirical research, we can almost certainly make the greatest impression on these students by discussing examples from among the milestone empirical research projects in the discipline.

We can also impress students if we discuss practical examples of research projects that use response variables that students themselves are directly interested in predicting and controlling, such as variables reflecting student grades, student health, student skills, student happiness, student expenses, and student income.

Practical Research in Pure Science. How can we judge whether an empirical research project in pure science is practical? Here it seems less reasonable to view the word "practical" as meaning that the research must suggest a basis for direct physical action because pure science is not done with direct physical applications in mind (although such applications often arise). Instead, it seems more reasonable to say that empirical research in pure science is "practical" if and only if it suggests a basis for intellectual action -- if it has the potential to advance a scientific theory or to otherwise usefully advance scientific knowledge.

Practical Examples in Statistics Textbooks. Surprisingly, many examples of empirical research projects in some statistics textbooks are not practical. And when one studies such examples and asks "Would an enlightened empirical researcher every actually do the research project discussed in the example?" the answer is (for various reasons) often a clear "No".

On the other hand, some introductory textbook writers provide an abundance of excellent practical examples (e.g., Moore [2000], Freedman, Pisani, and Purves [1998]).

(The frequent use of impractical examples in some statistics textbooks is one reason why some writers insist that teachers and textbook writers use real data in examples. Real data guarantee that at least one empirical researcher has judged the research to be in some sense practical. Unfortunately, insisting on real data does not ensure practicality according to the criterion given in the indented paragraph above. Section 7.8 contrasts real data with easier-to-obtain realistic data.)

Practical Student Projects. Many student projects in some statistics courses are impractical. However, such projects need not be impractical, as illustrated by some fascinating examples of practical projects discussed by Wardrop (2000, examples 1 - 24).

I further discuss the use of practical examples in the introductory statistics course in a paper (1998a, sec. 6) and in a Usenet post (forthcoming).

7.6 Generalization and Instantiation

Once students have studied a concept through a sufficient number of practical examples, I recommend that the teacher cement the appropriate generalizations about the concept in students' minds. This helps students to use the concept in new situations. For example, once students understand the concept of 'relationship between variables', the teacher can make the generalization that most empirical research projects can be usefully viewed as studying relationships between variables.

After stating a generalization, I recommend that the teacher assign exercises in which students identify details of the generalization in specific new instances. In particular, after discussing the idea that most empirical research projects can be usefully viewed as studying relationships between variables, I recommend that the teacher assign exercises in which students answer the nine questions given in Section 4.5 for various empirical research projects, including research projects of the students' own choosing. Answering these questions shows students that the questions almost always usefully apply.

Appendix H supports the point that the questions almost always usefully apply. Appendix I.2 discusses some infrequently occurring types of empirical research projects that lack a response variable.

7.7 Feedback Systems

How many explanations, examples, exercises, or activities should a teacher provide or assign to ensure that students understand a particular generalization? This depends, of course, on the generalization and on the nature of the students and is often difficult to determine at the front line of teaching -- especially if a teacher is using a new approach. To reduce this difficulty, I recommend that teachers use feedback systems to assess whether students understand each main concept and generalization.

Some effective feedback systems for assessing students' understanding are

• minute papers (which are ungraded and may be anonymous) in which students briefly report their understanding, their questions, and the muddiest point to help the teacher evaluate a lecture or lesson (Mosteller 1988)
• graded quizzes, exercises, and essay questions that focus on testing students' understanding of the ideas
• two-way discussions between the teacher and students about the ideas.

Garfield (2000) discusses approaches to assessing students as an aid to improving their learning and understanding. Gal and Garfield (1997b, pt. 2) give four interesting essays by statistics educators about assessing students' understanding of statistical ideas.

7.8 Realistic Data Versus Real Data

Some statistics educators recommend that the introductory statistics course rely heavily on real data and not merely realistic data (Cobb 1987, 1992; Moore and Roberts 1989; Singer and Willett 1990; Willett and Singer 1992; Witmer 1997; Moore 1997, 2000b; American Statistical Association 2000; Ballman 2000, Hayden 2000). I give a detailed argument in a Usenet post why I believe real data are unnecessary and why realistic data should be broadly allowed in introductory courses (forthcoming). The main ideas of the argument are

• Good realistic data are much easier to obtain than good real data.
• If realistic data are properly described and properly supported by ancillary material, students either cannot tell or do not appreciate the difference between the data they are studying and real data.
• Insistence on real data may lead teachers to use impractical examples because (despite the available Internet data libraries) appropriate practical real data are often hard to find.

In view of these points, I recommend the following criteria for data in examples (including exercises) in an introductory statistics course:

• Use real data in examples if good real data are available. But if a better example can be made by adapting real data or by completely making up the data, it is quite reasonable to use realistic made-up data.
• Regardless of whether an example uses realistic or real data, the teacher should strive to make the example practical, as discussed in Section 7.5.

Generating Realistic Data. Perhaps the easiest way to obtain realistic data for a research project is to generate the data with statistical software using a model equation. That is, one uses random number generators or fixed values (as necessary) to generate values of predictor variables and one uses a properly parameterized model equation (with a random number generator for the error term) to generate values of the response variable from the values of the predictor variables. For realism, a teacher may wish to hand-adjust the generated data, possibly adding an outlier or two or including some missing values. Most general statistical software can be easily programmed to generate realistic research data using this method.

Realistic Data in Assignments. Permitting realistic data allows teachers to assign extended exercises in which students are asked to pose a research hypothesis of their choosing and then provide a complete written proposal for an empirical research project that is capable of efficiently confirming the hypothesis (if the hypothesis is correct). To help students see value, I recommend that teachers stipulate that students' research hypotheses must be practical in the sense described in Section 7.5. I also recommend that the teacher describe numerous examples of earlier work by other students as an aid to students in choosing their own research hypotheses, as illustrated by Wardrop (2000). It is useful to have students present interim versions of their research proposals to the class, where the teacher and class may suggest improvements, as discussed by Chance (1997).

After students have finished planning a research project, I recommend (following Hunter 1977) that the teacher provide them with appropriate realistic made-up data that the students might have obtained if they had actually performed their project. The students can then analyze these data and report the results.

Providing students with realistic made-up data enables them to proceed in sequence through research planning and research analysis using an example that is of direct interest to them, even though (for reasons of cost, accessibility, or time) performing such a research project in real life would generally be impossible for students. Also, providing students with made-up data ensures that students obtain interesting results because in generating the data the teacher can tailor the results to contain interesting features.

I recommend that students be required to write a "codebook" describing the expected data table for their planned research. This aids student understanding and aids the teacher in generating the data.

I recommend that teachers provide students with data with relationships between variables that are only moderately significant, that is, with p-values between .05 and .001, or sometimes not significant because that is the way results usually appear in empirical research. (When generating data for a continuous response variable one can easily control the p-value by controlling the standard deviation of the error term in the model.)

In addition to the above approach, I recommend that students be assigned a small number of activities or projects at the beginning in which they actually conduct empirical research. However, in later work, I recommend that this time-consuming physical activity (which is generally non-statistical) be omitted through the use of realistic made-up data, thereby providing students with more statistical experience per unit of time spent on the course.

Realistic Data in Self-Study. For a teacher or student engaged in self-study of statistics, it is helpful to make up realistic data from an interesting empirical research project (perhaps one described in a newspaper, magazine, or journal article) and then to perform a detailed analysis of the data with one (or more) of the available statistical software products. It is usually helpful to study the description of the relevant computer output in the software manual because the manuals for the better products are well written and generally provide functional descriptions of statistical analysis as opposed to mathematical descriptions. (Functional descriptions best satisfy the needs of most statistical software users.) By analyzing realistic made-up data containing known relationships between the variables, one learns how the relevant statistical procedures identify and describe the relationships, which is a substantial aid to understanding the study of relationships between variables.

7.9 The Discussion of Mathematics in the Introductory Statistics Course

As suggested in Section 7.3, except in courses aimed at statistics or mathematics majors, I recommend that discussion of the underlying mathematics of statistics (e.g., probability theory, distribution theory, theory of statistical models, theory of statistical tests) be omitted from the introductory statistics course. This recommendation is motivated by the needs of the typical user of statistical methods. This person is interested in the field of statistics only to the extent that it can help them to detect and study relationships between variables (or possibly perform equivalent functions under another name). And like the typical automobile driver who needs transportation, but who cares little about the mechanical details of the engine, the typical user of statistical methods needs help studying relationships between variables, but cares little about the mathematical details of the help. Instead, the user's attention is directed toward a substantive area of empirical research (e.g., toward a particular branch of medicine). Thus the less we engage (and confuse) potential users with the complicated mathematical details of statistical methods, and the more we teach them how to properly use the methods in empirical research, the greater the chance we have of turning potential users into users.

The paper for students (1997a) illustrates one approach to showing due deference to the underlying mathematics without becoming immersed in complicated details.

Moore (1997a, sec. 4) and the American Statistical Association (2000) also recommend de-emphasizing mathematics in statistics education.

7.10 Hypothesis Testing

Section 5.4 introduces four groups of techniques that statistical methods can perform to help empirical researchers study relationships between variables. The first group of techniques are techniques for detecting relationships between variables. I recommend that teachers discuss detecting relationships between variables in the introductory statistics course in terms of modern Fisher-Neyman-Pearson hypothesis testing (Lehmann 1993).

Hypothesis testing is theoretically broader than testing for evidence of the existence of relationships between variables. However, examination of the use of statistics in empirical research suggests that most practical instances of hypothesis testing can be usefully viewed as testing for evidence of the existence of a relationship between variables (or testing for evidence of the existence of an extension to an already known relationship between variables). That is, the instances can be viewed as testing the hypothesis that a relationship exists between a response variable and one or more predictor variables. Appendix H.4 expands this point.

Cobb (1992, 2000) notes that some statistics teachers feel that (general) hypothesis testing should not be taught or should be de-emphasized in the introductory course. This view reflects the controversy that presently exists about the use of hypothesis testing (sometimes called "inference" or "significance testing") in empirical research (Wilkinson and Task Force on Statistical Inference 1999).

A central idea of hypothesis testing about relationships between variables is that we can (tentatively) conclude that a relationship exists if the relevant p-value is less than .05 and if no reasonable alternative explanation of the finding is available.

Testing hypotheses about the existence of relationships between variables is important because (to avoid embarrassing and costly errors) we must first verify that we have proper evidence that a relationship exists between the relevant variables before we attempt to use information about a putative relationship for prediction or control. Testing hypotheses about the existence of relationships between variables by computing relevant p-values and checking whether they are low enough is an objective and standardized aid in verifying that we have proper evidence of the existence of a relationship between variables.

(Tukey has suggested that a relationship likely exists between any given response variable and all what might be called "compatible" predictor variables [1989]. This point may seem to make hypothesis testing about relationships between variables less important. I discuss this point in a Usenet post [forthcoming].)

In some empirical research projects -- especially in the physical sciences -- the evidence of the existence of a relationship between variables is so strong that the use of hypothesis testing, although not incorrect, is superfluous. However, data are generally noisy. If we are analyzing noisy data, hypothesis testing provides an objective procedure to help us decide whether we have reasonable evidence of the existence of a relationship. Also, hypothesis testing can reliably detect subtle phenomena in data that may otherwise go unnoticed.

The rationale behind hypothesis testing is not easy for beginning students to understand. However, hypothesis testing will likely remain a pillar of empirical research because no easier general way to objectively check for evidence of the existence of a relationship between variables seems possible.

(Another approach to objectively checking for evidence of the existence of a relationship is to use confidence intervals for parameters. However, this approach is harder to understand than the hypothesis-testing approach because the confidence-interval approach requires that students understand statistical model equations and the distributions of the parameters of these models. The hypothesis-testing approach allows us to keep these technical matters behind the scenes, focusing instead on the clear-cut simple criterion discussed in the fourth paragraph of this subsection.)

To facilitate student understanding of hypothesis testing, I recommend that teachers emphasize the idea that the p-value provides a way of detecting relationships between variables. I recommend that teachers refer to the lower-level ideas and caveats in passing, but the complicated details of these ideas can be discussed later, after students understand the use of p-values to detect relationships.

In teaching students about hypothesis testing it is important to distinguish between statistical significance and practical significance, which are mutually independent. A relationship between variables is statistically significant if the associated p-value is properly computed and is low enough (i.e., less than .05 or, better, less than .01). If a relationship is statistically significant (and in the absence of a reasonable alternative explanation), we can (tentatively) conclude that the associated relationship between variables exists in the population of entities under study. On the other hand, as discussed in Section 7.5, a relationship between variables is practically significant if it provides us with a practical benefit -- that is, if it provides a basis for action. Clearly, both statistical significance and practical significance are necessary for an empirical research project studying a relationship between variables to be successful.

I illustrate an approach to discussing hypothesis testing in the introductory course in the paper for students (1997a, sec. 9) and I discuss these ideas further in a paper (1998a, sec. 5).

7.11 Univariate Distributions

Traditionally, the introductory statistics teacher spends a substantial amount of time near the beginning of the course covering univariate distributions of the values of variables. (The coverage generally includes ways of summarizing and illustrating univariate distributions and may also include the mathematics of univariate distributions.) Because many statistical concepts depend on the idea of a univariate distribution, it is clearly mandatory to cover this topic at some point in students' statistical careers -- but where?

Except possibly in courses for statistics majors, I recommend that discussion of univariate distributions be omitted at the beginning of the introductory course. This is because students find the topic to be boring and of little obvious use.

Nor is the topic of univariate distributions necessary at the beginning. That is, almost all practical examples of the use of statistics in empirical research do not study univariate distributions but instead study relationships between variables. As suggested by the discussion in Section 4, one does not need to understand univariate distributions to attain a proper basic understanding of relationships between variables.

If we omit univariate distributions at the beginning of the introductory course, where should we discuss them? For teachers using the phased approach described in Section 7.3, I recommend discussing univariate distributions as part of the discussion of types of variables in the Generalization Phase of the course. Placing univariate distributions at this point makes them available for discussion during the later steps in the Generalization Phase and for discussion of specific statistical methods in the Specific Methods phase.

(The concepts of univariate distributions are especially helpful in the mathematical derivation of p-values, in understanding statistical power, in checking data for anomalies, in examining data to determine whether the underlying assumptions of a statistical method are sufficiently satisfied to justify the use of the method, and in specifying the estimated accuracy of predictions or control based on a model derived from empirical research.)

Appendix H.2 discusses how univariate distributions are rigorously a degenerate case of relationships between variables. I further discuss univariate distributions in a paper (1998a, sec. 9.1 and app. G) and in some Usenet posts (1998c).

7.12 Implementation With Software Support

Appendix J discusses future software systems for studying relationships between variables. Such systems will make it substantially easier for teachers to convey statistical concepts to students.

8. SUMMARY

Under the entity-property-relationship approach, students learn the following six concepts at the beginning of an introductory statistics course:

1. entities
2. properties of entities
3. variables
4. an important goal of empirical research: to predict and control the values of variables
5. relationships between variables as a key to prediction and control
6. statistical techniques for studying relationships between variables as a means to accurate prediction and control.

To facilitate understanding, students learn the concepts in terms of numerous practical examples.

After students have learned the six concepts, they learn standard statistical principles and methods in terms of the concepts, again with emphasis on practical examples.

The EPR approach is broad, and the concepts of the approach are fundamental. The approach gives students a lasting appreciation the vital role of the field of statistics in empirical research.

Section 4.1 notes that people use nouns in language to denote entities. This implies that certain unusual things that we might not initially think of as entities are, under the EPR approach, entities. For example, the words "event", "behavior", "emotion", "set", "statistical distribution", "property", "color", "trial", and "flying saucer" are all nouns. Therefore, these words (when used in specific situations) all denote entities. Is it reasonable to view the "things" denoted by these nouns as entities?

To address this question, note that all instances of these "things" have properties. For example, all events have the properties of "location", "duration", and "identities of participants". Similarly, all behaviors have the properties "social appropriateness" and "duration". Similarly, all emotions in people have the properties "type", "intensity", and "duration". Similarly, all sets have the properties "type of elements" and "number of elements". Similarly, all statistical distributions have the properties "type" (i.e., continuous or discrete) and "expected value". Similarly, all properties have the properties "type" (i.e., continuous or discrete), "distribution" (e.g., normal or binomial), and "expected value". Similarly, all colors have the properties of "intensity of the yellow light component of the overall color" and "saturation". Since these things all have properties, and since the only things we know with properties are things or entities, these things are all reasonably viewed as being entities.

An important type of entity studied in some empirical research projects are "trials" (or "cases", "runs", or "instances"). For example, in a simple physics experiment we may operate a piece of apparatus for several "trials" under some condition A and we may also operate it for several trials under some other condition B. These trials typically become the main entities in the data analysis because each row in the statistical data table of the results is associated with a different trial. Trials have the properties of "duration", "time of occurrence", and "outcome", and thus are reasonably viewed as being entities.

Let us define a "flying saucer" as a type of vehicle used by extraterrestrials. All flying saucers have the properties of "size" and "does it exist?". In the case of the "does it exist?" property of flying saucers, many people believe that the value of this property is probably always "no". But the fact that the value of the property may be "no" does not remove entity-hood from flying saucers -- something need not exist in the external world for it to be a valid entity. This approach frees us to think about things that may or may not exist, which is a useful freedom.

Similarly, for any other noun we can find properties of the thing or things denoted by the noun. (For even if a thing denoted by a noun has no other properties, it still has the property of having no other properties.) Since all things denoted by nouns have properties, it makes sense to view whatever a noun names or identifies (in a given context) as being an entity of some type.

(Section 4.2 suggests that a behavior is a property of a living entity, but the first two paragraphs of this appendix suggest that a behavior is an entity. This dual view of behavior applies to other properties as well: As noted above, since all properties [and all variables] are denoted by nouns, all properties [and variables] can be viewed as entities. For example, the word "height", which names a property of physical objects, is a noun. Thus the word "height" names an entity -- an entity that is a property. Thus the root concept is the concept of 'entity' -- everything in human reality is an entity. Thus even properties [of entities] are entities [with properties]. However, I do not recommend discussing this confusing philosophical issue in an introductory statistics course.)

A.2 Are Certain Things Really Entities or Properties?

I say that all "things" denoted by nouns can be viewed as entities. But one can reasonably ask whether some of these things that I am calling entities are really entities. Similarly, one can ask whether some of the properties I name above are really properties of the entities I attribute them to. These are question of how entity-hood and property-hood are actually decided. Here, two approaches seem possible -- we can appeal to some authority to define entities and their properties or we can simply let our minds define them as we see fit.

A problem with appealing to an authority to define what are entities and properties is that we have no ready access to a reliable authority who will indicate whether a given candidate for entity-hood or property-hood is a valid entity or property. (We could have people, perhaps experts, vote on these matters, but this would be time-consuming and perhaps without much benefit.)

Because we lack access to an appropriate authority, we generally fall back on simply bestowing entity-hood on any "thing" that is of interest. That is, most people unconsciously view a thing as being an entity as soon as "it" attracts our attention. Similarly, once we have a thing of interest, we may recognize properties of it.

For example, most people unconsciously view a flock of birds as an entity with properties. If the birds suddenly scatter, the flock is no more. But while the birds are together the concept of 'flock' is a useful concept to enable us to view the birds as a group and to act as a container for the various properties of the group, such as "number of birds in the group", "cohesiveness of the group", and "age of the group". Whether the flock and its properties are "real" would appear to be irrelevant. (It is relieving that many entities in the external world are not as ephemeral as a flock of birds, with many entities existing continuously throughout one's lifetime.)

A.3 Alternative Approaches

An alternative approach to using the concept of 'entity' is to use an entity-less fog of unattached properties. However, this approach seems much less viable, and perhaps impossible because it is generally necessary to link together individual values of different variables in an analysis (as reflected, say, in the linkages between the values within a row in a data table). It is the concept of 'entity' that does the linking.

Since we cannot easily abandon the concept of 'entity', and since several other important concepts can be built atop the concept, a reasonable approach to describing human reality is to begin with the concept of 'entity' as a primitive. Another approach is to begin with the concept of 'property' as a primitive and then to define an entity as a cohesive cluster of properties together with their values. However, the property-first approach has the logical problem that it presupposes the concept of 'entity' -- clusters (i.e., sets), properties, and values are all entities.

A.4 Do Populations and Samples Contain Entities, Variables, or Values?

This paper views populations and samples as containing entities. In contrast, some discussions of mathematical statistics view populations and samples as containing only variables or values (e.g., Freund and Walpole 1987, p. 271-2). (Variables and values are entities, but are special limited types, so deserve separate consideration.)

Viewing a population as containing only variables or values reflects the mathematical approach of abstracting only the variables and their values from the situation under study, ignoring the entities and their properties. This approach makes the material substantially more general -- it is designed to be correct (i.e., logically consistent) regardless of the identity of its referents. Therefore, viewing a population as containing only variables or values is clearly a useful approach in mathematical statistics.

However, viewing a population as containing only variables or values diverges from standard human thinking in which we organize the external world into populations of entities that have properties that have values, rather than organizing it directly into populations of variables or values without (at least tacit) acknowledgement of the entities and properties to which the variables and values are "attached".

Therefore, although the abstract mathematical approach is essential for generality in discussions of mathematical statistics, it is reasonable to introduce the ideas to beginners in terms that reflect standard human thinking. Thus in the introductory statistics course for non-statistics-majors it is reasonable to view populations and samples as containing entities. Thus under this approach, statistical data contain estimates (at the time of measurement) of the values of properties of the entities that are in a sample that was drawn from the population.

APPENDIX B: THE TERM "PROPERTY"

Section 4.2 introduces the concept of 'property' and notes that several alternative terms are available to name this concept. Which term is preferred?

A reasonable approach to answering this question is to choose the term that (in the relevant sense) students are most familiar with. Breland and Jenkins report the frequency of occurrence of common English terms in the literature typically encountered by high school and university students (1997). Table B.1, which is extracted from their report, shows the estimated frequency of occurrence of terms that we might use to name the concept of 'property'.

Table B.1
Estimated Frequencies of Selected Terms
Per Million Printed Terms Encountered by
High School and University Students

 Term Estimated Frequency nature 309.1 character 145.6 property 81.6 quality 67.7 ability 48.4 finding 46.5 aspect 42.0 capacity 39.2 characteristic 34.4 quantity 27.2 countenance 25.8 feature 23.3 factor 20.6 faculty 20.1 disposition 20.0 trait 7.1 indication 6.4 attribute 6.2 capability 3.9 dimension 3.0 vector 1.6 indicator 1.4 facet 0.5 scalar -
SOURCE: English Word Frequency Statistics: Analysis of a Selected Corpus of 14 Million Tokens by Hunter M. Breland & Laura M. Jenkins. College Entrance Examination Board, 1997. www.collegeboard.com Used with permission.

Unfortunately, the table is not definitive because several of the terms in the table have more than one sense and these different senses were not distinguished in generating the word frequency counts (which were done by computer). For example, the term "nature" can denote the concept of 'property' discussed in this paper, but it can also denote the "natural" world and the events that take place in it. Similarly, the term "property" can denote the concept of 'property' discussed in this paper, but it can also denote "something owned or possessed". Similarly, some of the other terms (e.g., "character", "quality", and "attribute") have more than one sense.

(Also, the table is less definitive because Breland and Jenkins did not estimate the size of the sampling error of their estimates. However, since their sample contained 14 million words [from texts that students frequently study], intuition suggests that 95% confidence intervals for the frequency estimates in the table will be less (perhaps substantially less) than 20 frequency units wide. See also the discussion by Carroll, Davies, and Richman of error estimates in a statistical model for word frequencies [1971, pp. xxxiii-xxxiv].)

Although we cannot use the table to determine the frequency of occurrence of the terms in just the sense under discussion, we can use the table together with a notion of the popularity of the terms in their different senses to develop a feeling for the appropriateness of the various terms.

The table indicates that the term "nature" is by far the most frequently occurring of the terms. However, when this term occurs the writer or speaker is usually referring to some form of "mother nature" rather than referring to a nature (i.e., property) of some entity. To avoid ambiguity with the popular alternate sense, this suggests that the term "nature" is ruled out from being the chosen term to name the concept of 'property'.

Similarly, the term "character" seems at least as popular in its other senses as it is in the sense of 'property'. Thus use of the term "character" would likely cause ambiguity for beginners, which suggests that this term is also ruled out from being the chosen term to name the concept of 'property'.

The term "quality" is a strong contender. However, it has the disadvantage that it may connote judgment or evaluation and the sense of "superiority" or "excellence". Also, when I read Section 4.2 with the string "propert" replaced everywhere with the string "qualit", the section seems less effective. (It seems more natural to speak of the height property of a person instead of the height quality.) Thus my intuitions prefer the term "property" to the term "quality".

The table suggests that the term "attribute" occurs (in its multiple senses) less than one-tenth of the times that the term "property" occurs (in its multiple senses). Thus if we wish to use a familiar term, this suggests that the term "attribute" is ruled out.

Thus for me the term "property" works best to name the concept of 'property'.

APPENDIX C: THE EVOLUTION OF ENTITIES AND PROPERTIES IN HUMAN THOUGHT

The genesis of the unconscious use of the concepts of 'entity' and 'property' in human thought seems straightforward: The ability to discriminate and conceptually manipulate entities and properties has an evolutionary advantage, and thus the concepts and ability evolved through natural selection. Thus presumably the ability to discriminate the values of properties began as an ability in a simple organism to discriminate and act on differences in a single simple property of its external world (perhaps temperature or light intensity). That ability gave the organism a selective advantage. That ability has evolved into our current human ability to react to and improvise in the external world in many complicated ways as a means to improving our survival and comfort. (We act at various levels of maturity ranging from empathetic philanthropy down to unconscious greed.) The reactions and improvisations are based on our goals or needs coupled with mental models of the external world in terms of entities, properties of entities, relationships between entities, and relationships between properties.

The preceding paragraph suggests that properties preceded entities in the evolution of human thought. That is, natural selection led organisms first to (unconsciously) develop the concept of 'property'. Following (or partly overlapping) the development of the concept of 'property', higher animals developed the ability to (unconsciously) recognize that several associated properties could be "attached" to similar things or entities, which enabled the concept of 'entity' to function as a key simplifying concept for understanding the external world. Entities also provide the necessary conceptual framework for study of relationships between properties (which occur across entities) -- relationships that directly assist us in prediction and control.

Because presumably properties preceded entities in the evolution of human thought, and because presumably the infant recognizes properties in its external world at an earlier age than when it recognizes entities, it is reasonable to view properties as being temporally more basic in human thought than entities. However, it is also reasonable to view entities as being logically more basic than properties, as discussed in Appendix A.3.

APPENDIX D: DEFINING THE TERM
"VARIABLE"

Section 4.3 proposes a definition of the term "variable". This appendix first discusses whether we can define two important concepts that are used in that definition and then discusses and compares some other possible definitions of the term "variable".

D.1 Can We Define the Terms "Entity" and "Property"?

The definition of the term "variable" given in Section 4.3 uses the terms "entity" (actually "entities") and "property". Thus one might reasonably ask for definitions of these more fundamental terms. However, not every term in a body of knowledge can be verbally defined without introducing undesirable circularity. Thus the terms "entity" and "property" are verbally undefined under the EPR approach. However, these terms have ostensive definitions. I discuss this further in a Usenet post (2001).

D.2 Some Definitions of "Variable"

Several different definitions are available for the term "variable" and confusion among these (closely related) definitions can easily arise. We can say that a variable is

1. a formal representation of a property of entities.
2. a symbol representing a property of entities.
3. a placeholder for a value of a property, in effect a pronoun (Rubin 1996).
4. a set of values of a property in a population or sample.
5. a concept in statistical software that represents a column in a data table, which contains the measured values of some property for members of a set of entities.
6. a symbol that stands for some entity (where the concept of 'entity' includes properties, variables, and values).

Consider some dictionary and encyclopedia definitions of the concept of 'variable'. A variable is

1. a feature that is subject to systematic and haphazard variation (Cox 1999).
2. a : a quantity that may assume any one of a set of values b : a symbol representing a variable (Merriam-Webster 1994).
3. something that may or does vary; a variable feature or factor (Random House 1993).
4. a. a quantity or function that may assume any given value or set of values. b. a symbol that represents this (Random House 1993).
5. a quantity or force which, throughout a mathematical calculation or investigation, is assumed to vary or be capable of varying in value (Oxford University Press 1971).
6. generally any quantity which varies. More precisely, a variable in the mathematical sense, i.e. a quantity which may take any one of a specified set of values. ... It is useful, but far from being the general practice, to distinguish between a variable as so defined and a random variable or variate (Marriott 1990).
7. any finding (an attribute or characteristic) that can change, that can vary, or that can be expressed as more than one value or in various values or categories (Vogt 1993).
8. some characteristic that differs from subject to subject or from time to time (Everitt 1998).
9. a symbol which is used to represent some undermined element from a given set, usually the domain of a function (Parker 1994).

Note how definitions 7 through 14 define the term "variable" in terms of the words "feature", "quantity", "factor", "finding", "attribute", and "characteristic". As suggested in Appendix B, these words are all (in the relevant one of their senses) near synonyms for the word "property".

Definitions 8, 10, 11, and 12 define the term "variable" in terms of the word "quantity" which suggests that the values of a variable must be numeric. However, this approach is too limiting because variables in statistics and empirical research sometimes have non-numeric values.

Definitions 10 and 15 use the concept of 'function'. My sense is that this mathematical concept (a mapping between two sets) is not needed to define "variable". Definition 11 uses the concept of 'force'. Although it might be viewed more broadly, this concept is generally viewed as merely a particular type of variable from physics, which does not seem to deserve special mention.

D.3 Which Definition Is Preferred?

As noted, most of the above definitions appeal to the concept of 'property' under one name or another, as discussed in Appendix B. Some of the definitions also refer to the concept of 'representation' or 'symbol'. Thus, to be consistent with common usage, I believe the preferred definition of the term "variable" must refer to the concepts of 'property' and 'representation'.

None of the dictionary or encyclopedia definitions use the word "property". I suspect that this word was avoided in the definitions because it leads to the question "property of what?", and lexicographers felt that entities or "things" are not always present when variables are present. But although it is true that entities are often not present when variables are used in mathematics, it would appear to be difficult or impossible to find a variable that is used in empirical research that cannot be reasonably viewed as reflecting some property of some type of entity.

Since variables in empirical research are invariably associated with entities, and in view of the fundamental role of statistics in empirical research, and in view of the fundamental role of the concept of 'entity' in human thought, I believe the concept of 'entity' should be present in the statistical definition of the concept of 'variable'. These ideas lead to definitions 1 and 2.

To choose between definitions 1 and 2 we must decide whether to view a variable as a "formal representation" or as a "symbol". I prefer the phrase "formal representation" because variables have several notions associated with them (such as 'distribution', and 'relationship between variables'), and these notions seem consistent with the concept of 'formal representation'. On the other hand, the concept of 'symbol' connotes for me the idea of an empty container, thereby diminishing the sense of the associated notions. This leads me to recommend definition 1.

Definition 1 omits four concepts that are closely associated with variables:

• the concept of the 'value' of a variable
• the concept of 'variation' in the values of a variable
• the concept of 'measurement' of the values of a variable
• the concept of the time of the measurement of the value of the variable.

For simplicity, instead of including these concepts in the definition of "variable", I recommend that they be introduced immediately after the definition as concepts that are associated with the definition. This approach is illustrated in Section 4.3.

This approach is consistent with the approach taken with the concept of 'property' in Section 4.2 in which the concept is first introduced and then the other four concepts are introduced.

I further discuss the concept of 'variable' in a Usenet post (1996b) and in the paper for students (1997a).

APPENDIX E: THE DISTINCTION BETWEEN PROPERTIES AND VARIABLES

The discussion in Section 4.3 raises the following important question:

Are the concepts of 'property of an entity' and 'variable' merely interchangeable synonymous concepts?

I believe the two concepts are best viewed as being not synonymous for the following reasons:

• It is useful to speak of the difference between the value of a variable and the true value of the associated property of an entity. This difference is called the "error" in the value of the variable (Pearson 1902, Cochran 1968). If we merge the concepts of 'property' and 'variable', it makes less sense to speak of this difference.
• Children learn the concept of 'property' and the attendant concept of 'value of a property' before they enter kindergarten. (They unconsciously learn to unconsciously use the concepts.) Their knowledge of the concepts is reflected in their understanding and use of adjectives and adverbs, which (as discussed in Section 4.2) usually serve in language to express the values of properties. In contrast, children do not generally learn the concept of 'variable' until they first study algebra which (as discussed in Section 5.10) is usually in grade 6 or later. This supports the view that the two concepts are usefully viewed as separate concepts, with (given the developmental sequence) properties being more fundamental than variables.
• Variables can be used in algebraic expressions, where they are represented by symbols, such as x and y. In contrast, properties are not generally viewed as being usable in such expressions.
• The concept of 'variable' implicitly contains the concept of the value of the variable. (When a variable is used in an algebraic expression it usually stands for its value, as opposed to standing for the variable in some other sense.) In contrast, the concept of 'property' seems less inclined to (directly) contain the concept of its value.
• The concept of 'variable' in statistics comes from mathematics. The concept of 'variable' in mathematics originally came from the concept of 'property'. However, the concept of 'property' is generally not explicit in mathematical discussions of variables. This is because (as discussed in Appendix A.4) the abstract approach of ignoring the entities and properties enables the mathematical material to be substantially more general. Thus the concept of 'variable' has a strong abstract sense while the concept of 'property' is more concrete.

In light of these points, I recommend viewing the concept of 'property of an entity' as being fundamental and viewing the concept of 'variable' as being defined in terms of it -- a (statistical) variable is a formal representation of a property of entities.

I further discuss the distinction between properties and variables in a paper (1998a, app. E).

APPENDIX F: THE TERM "RELATIONSHIP"

Section 4.5 introduces the phrase "relationship between variables" and notes that several terms are available that we can use instead of the term "relationship" in this phrase.

For example, any of the following terms can replace the term "relationship" in the phrase: association, attachment, attunement, bond, consanguinity, commensurability, compatibility, complementarity, concord, concordance, concurrence, conformance, conformation, conformism, conformity, congruity, connection, connectiveness, consilience, correlation, correspondence, coupling, covariation, dependence, entanglement, equation, equivalence, function, harmony, homogeny, homology, interchange, intercommunication, interconnection, intercourse, interdependence, interlacing, interlinking, intermeshing, interpenetration, interplay, interrelation, interrelationship, intertwining, interweaving, interworking, kinship, liaison, link, linkage, linking, marriage, mimicking, mimicry, mutual dependence, mutualism, mutuality, nexus, parallel, parallelism, parity, partnership, propinquity, proportionality, rapprochement, reciprocality, reciprocalness, reciprocation, reciprocity , relatedness, relation, relativeness, relativism, relativity, simulacrum, symbiosis, symmetry, sympathy, tie, unity, yoke.

Which of these terms is preferred in general discussion? As before, a reasonable approach to determining the preferred term is to begin by considering the frequency of use of the terms in the literature typically encountered by high school and university students. Appendix B above introduces the Breland and Jenkins word frequency data (1997). Table F.1, which is extracted from those data, shows the frequency of use of some more commonly used synonyms for the term "relationship".

Table F.1
Estimated Frequencies of Selected Terms
Per Million Printed Terms Encountered by
High School and University Students

 Term Estimated Frequency marriage 81.4 relation 50.7 association 43.1 function 41.1 relationship 38.6 sympathy 37.2 connection 32.4 bond 23.7 parallel 21.9 unity 20.1 tie 17.8 harmony 17.6 intercourse 16.0 alliance 15.6 correspondence 13.4 link 12.6 concord 11.5 involvement 10.9 dependence 9.2 attachment 9.1 equation 6.0 accordance 5.9 yoke 5.8 affinity 5.6 symmetry 5.5
SOURCE: English Word Frequency Statistics: Analysis of a Selected Corpus of 14 Million Tokens by Hunter M. Breland & Laura M. Jenkins. College Entrance Examination Board, 1997. www.collegeboard.com Used with permission.

Some of the synonyms in Table F.1 have multiple or special senses (which the Breland and Jenkins data do not distinguish). For example, the noun "marriage" is almost always used in a special (matrimonial) sense different from the sense of 'relationship' used in this paper. Similarly, the noun "association" can denote a "relationship" in the sense used in this paper, but it can also denote an organization of people who have a common interest. Similarly, the nouns "relation" and "function" have multiple senses.

On the other hand, study of the dictionary definition suggests that the term "relationship" has only one general sense -- denoting some link, connection, or involvement that exists between two or more entities. (In the present discussion the link is between two entities that are properties or variables.)

The frequency-of-use statistics together with the multiple and special meanings of some of the terms suggest that "relationship" is the preferred term.

Some readers may object to using the term "relationship" because they feel this term should be reserved for relationships that are "perfect" -- i.e., relationships between variables in which the error term in the model equation is zero. But such perfect relationships are never found in practice. Perfect relationships are not found due to measurement error, which is invariably present because (in general) no measuring instrument is perfect. Since we cannot completely eliminate measurement error, it is technically impossible to definitively conclude that the only error present in a model equation is true measurement error. Thus it is technically impossible to infer that a relationship between variables is "perfect". Thus instances of "perfect" relationships between variables cannot be shown to exist in practice. Thus it is a waste of the term "relationship" to reserve it for such non-existent-in-practice instances.

(Some readers may believe that some perfect relationships between variables are studied in the physical sciences. Certainly some physical relationships appear to be perfect when studied within the limits of current measurement technology. However, when more powerful measurement methods become available, we may learn that an apparently perfect relationship is not actually perfect, and other predictor variables are involved [albeit perhaps only very weakly] in determining the values of the response variable.)

Some readers may recall the important phrase "association does not prove causation" and lean toward the term "association" because that term appears in this phrase. However, the phrase seems to work at least as well if it is stated as "relationship does not prove causation".

I contrast the term "relationship" with the term "relation" in a Usenet post (1999, app. A).

The preceding points lead me to recommend using the noun "relationship" in the phrase "~ between variables".

Appendix I.2 discusses whether it should be "relationship between variables" or "relationship among variables".

APPENDIX G: THE TERMS "RESPONSE VARIABLE" AND "PREDICTOR VARIABLE"

This appendix evaluates some of the commonly used terms to name the response variable and the predictor variable(s) in the study of a relationship between variables. Let us first consider terms that are used to name the response variable.

This paper uses the term "response" to name the response variable. This term is effective because it conveys the important idea that something is being responded to. Thus the student wonders "response to what?", which links directly to the important concepts of 'predictor variable' and 'relationship between variables'.

The response variable in a relationship between variables is also sometimes called the "predicted" variable. The term "predicted" has the advantage over the term "response" that it is less likely to connote direct causation -- a danger with the term "response". It is important that students understand that many relationships between variables, although clearly mediated by causal connections, are not directly causal. However, we can easily protect against students mistakenly thinking that the term "response" implies that a relationship is directly causal by carefully discussing this point. I recommend that teachers discuss the point in the discussion of the difference between experiments and observational research projects.

The term "predicted variable" also has the advantage that it implies the concept of 'prediction' which, as discussed in Section 4.4, is an important concept in empirical research. However, the concept of 'prediction' is also implied by the term "predictor". Thus if we use the term "predictor" to name the predictor variable(s), we can imply the concept of 'prediction' even if we choose not to use the term "predicted" to name the response variable.

The term "predicted variable" has the disadvantage that it is less effective than the term "response variable" at suggesting that the variable is responding to a predictor variable. ("Responding" is sometimes meant only in the sense of "following" or "varying somewhat in step with".) As noted, the concept of 'responding' emphasizes the simple connections between the main concepts, which makes the material easier for students to understand.

The term "predicted variable" also has the disadvantage that it does not work well when the relationship is causal. In this case, we can usually do more than merely predict the values of the response variable -- we can control them. Thus in this case, the term "response variable", with its sense of "responding" to the predictor variable, is clearly more effective.

The above discussion suggests the possibility of using two terms -- "response variable" for experiments and "predicted variable" for observational research projects. However, most readers will agree that many symmetries exist between the study of causal and non-causal relationships between variables. (The key difference is that at least one of the predictor variables is "manipulated" by the experimenter in an experiment while all of the predictor variables are merely "observed" in an observational research project.) The existence of the many symmetries between experiments and observational research projects suggests that we can maximize student understanding by using only a single term to name the concept of 'response variable', which seems to be quite reasonably viewed as only a single concept.

The above points lead me to prefer a single term to name the response variable and to prefer the term "response variable" over the term "predicted variable".

Some discussions of empirical research refer to the response variable as the "dependent" variable, which reflects the terminology of mathematical functions. The dependent variable of a mathematical function is (except in the trivial case of a constant function) always dependent on the argument(s) or "independent" variables of the function. Thus in discussions of mathematical functions, the term "dependent" is fully appropriate.

On the other hand, in statistical discussions of empirical research the term "dependent" implicitly assumes what we are often directly interested in determining, which is whether (and how) the response variable depends on the predictor variable(s). Thus the term "dependent" begs (i.e., assumes the truth of) the very question we are often trying to answer. Since the term "dependent" often begs the question, it confuses students, and is therefore less effective than other terms.

The point of the preceding paragraph also applies to the term "response", but to a much lesser extent. That is, it seems confusing and counterintuitive to think that a "dependent" variable may or may not actually "depend" on a predictor variable. But it is reasonable and consistent with the everyday sense of the concept of "respond" to think that a "response" variable may or may not actually "respond" to a predictor variable -- no response is an allowable "response".

The response variable in a relationship between variables is also sometimes call the "output" variable, especially in the study of physical processes. The term "output" suggests the notion of a relationship because a reference to an "output" variable suggests that an "input" variable ought also to be present. However, the term "response" seems to more directly convey the important notion that the response variable is "responding" (or is hoped to respond) to the predictor variable(s), thereby more strongly implying the sense of 'relationship'. Also, the term "response" seems to work effectively in many (all?) situations in which the term "output" is used.

The response variable in a relationship between variables is also sometimes called the "criterion" variable in recognition of the fact that it is important -- it is the variable we wish to learn how to better predict or control. However, the term "criterion", apart from implying the importance of the variable, is somewhat empty. In particular, it lacks the important idea of responding to the predictor variable(s) that is inherent in the term "response".

The above points lead me to recommend that teachers use the term "response" to name the response variable in a relationship between variables.

Let us now consider some of the terms used to name the predictor variable(s) in a relationship. This paper uses the term "predictor" to name the predictor variable(s). This term has the advantage that it directly suggests the important concept of 'prediction' which, as discussed in Section 4.4, is a key goal of empirical research. Furthermore, the "or" suffix on the term "predictor" enables the term to differentiate itself from what is being predicted, even for beginners. Thus the term leads beginners to wonder about the identity of the "predictee" variable, thereby leading to the ideas of 'response variable' and 'relationship'.

A predictor variable is also sometimes called an "explanatory" variable. However, the term "explanatory" emphasizes the concept of 'explanation' which (as noted in Section 4.4) is generally subordinate to the concept of 'prediction'.

A predictor variable is also sometimes called an "input" variable. As with the term "output", the term "input" does suggest the concept of 'relationship between variables'. However, the term "input" does not suggest the important concept of 'prediction' as well as the term "predictor".

A predictor variable is also sometimes called an "independent" variable. As with the term "dependent", the term "independent" reflects the terminology of mathematical functions. The term "independent" has the disadvantage that it suggests an idea that that we almost always hope is false. That is, we almost always hope that a predictor variable is not statistically independent of the response variable, so calling such a variable "independent" is confusing.

The above points lead me to recommend that teachers use the term "predictor" to name the predictor variable(s) in a relationship between variables.

The paper for students lists other terms that are sometimes used to name the response and predictor variables in the study of a relationship between variables (1997a, sec. 7.6). Similar arguments to the above can be given why these terms are also less effective than the terms "response" and "predictor".

APPENDIX H: DO RESEARCH PROJECTS STUDY RELATIONSHIPS BETWEEN VARIABLES?

Section 4.5 introduces the concept of 'relationship between variables' and states that most empirical research projects can be easily and usefully viewed as studying relationships between variables (as a means to predict and control the values of variables). Some readers may be unaware that the concept of 'relationship between variables' can be used as widely as this paper claims. (Probably more than one-half of all empirical research projects are not viewed by their authors as studying relationships between variables.) I encourage these readers to seek examples of research projects that apparently cannot be viewed as studies of relationships between variables and then to consider the examples in terms of the nine questions in Section 4.5. One finds that many such examples can be viewed as studying one or more relationships between variables, and this fresh point of view substantially increases understanding.

The remainder of this appendix discusses six aspects of empirical research projects that may at first appear not to be studying relationships between variables, but which can nevertheless be usefully viewed as either studying relationships or assisting in the study of relationships.

H.1 Physics or Chemistry Experiments

Suppose a physics or chemistry experiment has discovered that if ingredients A and B are mixed together under conditions C, then ingredient D appears. Does this experiment study a relationship between variables? Yes. The predictor variables are "amount of ingredient A", "amount of ingredient B", and the variables that reflect the conditions C. The response variable is "amount of ingredient D that is produced". Viewing the experiment in terms of a relationship between variables has the benefit of reminding us that in addition to being interested in the fact that ingredient D is produced, we also wish to know how much of ingredient D is produced for given values of A, B, and C. This knowledge gives us better prediction or control capability of D.

(The population of entities in the example is all the instances [trials, runs, cases] ever in which the ingredients A and B are brought together under conditions C, and the sample is the set of those instances that occur in the experiment.)

H.2 Parameter Estimation

Some readers may wonder whether a research project that performs "parameter estimation" can be viewed as studying a relationship between variables. One type of parameter-estimation research project is intended to determine (i.e., estimate) the values of one or more "population parameters", where a population parameter is simply some property of a population, such as the average of the values of some property of the entities in the population. (In more general terms, the procedure is studying the univariate distribution of the parameter.) Any standard procedure for estimating the value of a population parameter can be usefully viewed as the study of a degenerate case of a relationship between variables in which the response variable is present, but the set of predictor variables is empty. This point of view is consistent at a deep level with the underlying mathematics. That is, the mathematics of any standard population-parameter-estimation procedure (or any procedure for studying univariate distributions) is (rigorously) the limiting case of the mathematics of some particular standard method of studying relationships between variables, when the number of predictor variables is reduced to zero.

Note that if we adopt the foregoing point of view of parameter estimation, we are naturally led to ask in particular cases why the set of predictor variables is empty. Asking that question may prompt us to find some predictor variables. If predictor variables are found that are related to the response variable, knowledge of the relationship will give us a more complete understanding of the response variable than we could obtain by studying equivalent instances of the univariate distribution of that variable in isolation. In particular, the knowledge will give us the ability to make more accurate predictions or the ability to exercise more accurate control of the values of the response variable.

A second type of parameter-estimation research project can be performed to determine (i.e., estimate) the values of one or more parameters of a model (equation) of a relationship between variables. Because model equations are directly related to the study of relationships between variables, this second type of parameter estimation is simply a particular aspect of the study of relationships between variables.

H.3 Interval Estimation

Some readers may wonder whether a research project that performs "interval estimation" can be viewed as studying a relationship between variables. Here, given the links established in Appendix H.2 between parameter estimation and relationships between variables, it is easy to see similar links between interval estimation and relationships between variables. The links occur because the intervals in interval estimation are simply intervals in which we determine (with a stated level of "confidence") that the associated parameters probably lie.

H.4 Statistical Tests

Section 7.10 states that most practical instances of statistical tests of hypotheses can be usefully viewed as testing for evidence of the existence of a relationship between variables (or testing for evidence of the existence of an extension to an already known relationship between variables). The following discussion expands this point.

Consider a research project that uses an independent-samples t-test to determine if there is evidence of a difference between female and male medical patients in their response to a particular medical treatment. Can this research project be viewed as studying a relationship between variables?

Yes. The response variable is the measured value of the "response" to the treatment for each patient who participated in the research project. The predictor variable is the variable "gender", which reflects an important property of the patients. Thus we can view this research project (and the associated t-test) as an attempt to see if a relationship exists in patients between the variables "gender" and "response". In other words, the research project is an attempt to determine whether a patient's response "depends" on their gender.

Similarly, in a research project in which the entities are cross-classified in two or more different ways, each classification (i.e., each subscript or margin) represents a different predictor variable. These predictor variables may reflect properties of the entities that are under study, or they may reflect properties of the entities' environment.

In the case of a particular treatment that is applied to the entities, the associated predictor variable reflects a property of the entities after they have received the treatment. That is, the predictor variable reflects the amount of the treatment -- possibly zero -- that an entity received.

The preceding points together with consideration of the standard statistical tests suggest that we can view many instances of the standard tests of hypotheses (e.g., the t-test, many tests of analysis of variance, regression analysis, categorical analysis, time series analysis, etc.) as techniques for detecting evidence of a relationship (or an extension to a relationship) (in the population of entities under study) between the response variable and one or more of the predictor variables studied in a research project.

But even if we agree that we can view many instances of statistical tests as techniques for detecting evidence of relationships between variables, a more fundamental question remains: Is it empirically useful to view these statistical tests this way?

To answer that question, note that it is precisely instances of the concept of a relationship between variables in entities that empirical researchers are usually interested in detecting when they use statistical tests in research projects. That is, (using the definition of a relationship between variables in the paper for students [1997a, sec. 7.10]) researchers are usually precisely interested in determining whether the expected value in entities of some variable y depends on the values of certain other variables x1,...,xp. Researchers are interested in making this determination because if they find such a relationship, one can confidently use the knowledge of the relationship to make predictions or exercise control.

(Another valid way of viewing some statistical tests is to say that they are techniques for detecting differences between subpopulations, or differences between groups, or significant differences between sample means [e.g., Abelson 1995, p. 27, Jones and Tukey 2000]. Thus in the t-test example above we can say that the test determines whether there is a difference between the subpopulation of females and the subpopulation of males in their response to the treatment. However, saying that certain statistical tests are techniques for detecting differences between subpopulations is not as general as saying that the tests are techniques for detecting relationships between variables. Specifically, all statistical tests that can be viewed as detecting differences between subpopulations can also easily be viewed as detecting relationships between variables by [1] viewing the relevant property that is measured in each entity in the sample as the response variable, and [2] by viewing whatever distinguishes the different subpopulation groups in the research project from each other as constituting the [set of] predictor variable[s]. On the other hand, not all statistical tests that can be viewed as detecting relationships between variables can be viewed as detecting differences between subpopulations. Specifically, in some statistical tests that detect relationships between variables [especially in observational research projects as opposed to experiments, and especially when continuous variables are used] no subpopulation groups exist, and thus the statistical tests cannot be easily viewed as detecting differences between [distinct] subpopulations, not even subpopulations of one.)

H.5 The Human Genome Project

The Human Genome Project has recently identified or "mapped" almost all the relevant portions of the sequence of three billion "base pairs" in the human genome. This sequence is generally viewed as the genetic blueprint for the human species.

From an easily-adopted high-level point of view, the Human Genome Project does not directly study a relationship between variables (although considerations about relationships between variables permeate the project). Instead, this research project studies various different types of entities and it studies the value of an important property of one of these types.

Some of the types of entities relevant to the Human Genome Project are (in rough decreasing order of conceptual or physical containment) species, humans, chromosomes, DNA, genomes (i.e., complete DNA sequences), genes, proteins, and base pairs. Base pairs can be four different types (which are labeled A, T, C, and G), and the linear sequence of these four types in the human genome is believed to encode (in one important sense) all the information needed to produce a human being.

A reasonable way to view the human genome is to view the sequence of base pairs in the genome as a property of the genome. Under this point of view the Human Genome Project has almost completely determined the relevant aspects of the value of a (complicated) property of the human genome -- the linear sequence of its three billion base pairs.

A next step after determining the sequence of base pairs is to identify more genes, which are linear subsequences of base pairs in the human genome that are associated with inherited traits. Other steps are to identify the proteins that are activated by the genes and to identify drugs or other approaches that can retard or increase the production of these proteins. Control of production of these proteins will help doctors to achieve a main goal of this research, which is to control (i.e., improve) the values of variables that reflect measures of human health.

As noted, it is reasonable to view the Human Genome Project as studying a property of entities, and not as studying a relationship between properties or a relationship between variables. But, as also noted, a main societal goal of this project is to control the values of variables that reflect measures of human health. This control will be achieved through discovery of new relationships between variables -- discovery that will be aided by knowledge of the human genome. Thus although the Human Genome Project does not directly study relationships between variables, it is an intermediate step toward the goal of study of such relationships as a means to predict and control human health.

H.6 Other Research Projects that Do Not Study Relationships Between Variables

Appendix I.2 discusses some other empirical research methods that do not study relationships between variables, but instead study relationships among variables. I discuss other empirical research projects that study entities, relationships between entities, or simple properties in two Usenet posts (1997d, app. A; 1998e, app. D).

APPENDIX I: DOES THE EPR APPROACH UNIFY STATISTICAL METHODS?

Section 5.4 lists twenty-one statistical methods and then makes two claims about these methods. This appendix provides support for the claims and discusses some related issues.

I.1 Is the List of Twenty-one Methods Comprehensive?

One claim in Section 5.4 is that the list of twenty-one methods contains almost all of the currently popular statistical methods. This is supported by the fact that if one surveys empirical research projects that use statistical methods, one finds that a large majority of these research projects use as their main statistical method(s) one or more of the methods in the list.

I.2 Response-Variable and No-Response-Variable Statistical Methods

I call the statistical methods in the list in Section 5.4 "response-variable" methods. My criterion for calling a method a response-variable method is

A statistical method is a response-variable method if it focuses on a single response variable and zero or more predictor variables.

All the methods in the list can be easily viewed as satisfying this criterion.

(Section 7.11 and Appendix H.2 discuss the study of univariate distributions -- i.e., the group of response-variable methods in which a researcher uses a single response variable and zero predictor variables Graphical methods, Bayesian methods, and meta-analysis are usually used in conjunction with response-variable methods, but these methods themselves are more general than response-variable methods.)

The criterion above states that response-variable methods focus on a single response variable. Some multivariate methods (e.g., multivariate analysis of variance, multivariate multiple regression) still have easily identifiable response and predictor variables, but instead of having only a single response variable they have multiple response variables.

In the following discussion of multivariate methods I exclude the important and frequently used method of "repeated measurements" or "repeated measures". Although research projects that use repeated measurements may be reasonably analyzed using multivariate methods, they are perhaps best viewed as having only a single response variable whose value is measured repeatedly (more than once) in each research entity. (Researchers use repeated measurements because this method is an inexpensive way to substantially increase the power of statistical tests.)

Multivariate methods (excluding repeated measurements) are used only rarely in real empirical research. However, if these methods are used, the set of response variables can usually be reasonably viewed as a unity -- as a single response variable that is also a vector. This vector usually represents a reasonable single "property" of the entities under study. Thus it is reasonable to view these methods as having only a single response variable and to include them in the response-variable methods.

I further discuss research projects with multiple response variables in a Usenet post (2002).

In considering the twenty-one response-variable methods the question arises whether we should speak of relationships "between" the variables or relationships "among" the variables. I recommend using the preposition "between" when referring to relationships studied by the response-variable methods. This is because in any individual use of one of these methods only a single response variable (which may on rare occasion be a vector) is under study, and the relationship under study is between that variable and the predictor variable(s).

I call the statistical methods that do not focus on a single response variable "no-response-variable" methods. Decision theory (discussed in Section 7.4) is a no-response-variable method because it focuses on making decisions rather than focusing on relationships between variables.

Cluster analysis, factor analysis, principal components analysis (including dual scaling and correspondence analysis), and a few other less frequently used methods are also no-response-variable methods because they do not focus (either explicitly or implicitly) on a specific single response variable. These methods still study variables (properties of entities) and in a loose sense they also study relationships between or among variables, even though they do not focus on a specific single response variable.

I estimate that the no-response-variable methods are used in total in less than three percent of reported empirical research projects that use statistical methods. Thus although the no-response-variable methods are important in a few research projects, I believe they are not important topics for discussion in an introductory statistics course.

On the other hand, the correlation coefficient and the study of contingency tables are (implicitly) included in the list of twenty-one response-variable methods even though when we use these two methods it is generally mathematically unnecessary to specify which variable is the response variable and which variable(s) is (are) the predictor variable(s). In the case of these "mathematically symmetric" statistical procedures the mathematics is oblivious to any distinction between response and predictor variables. However, in the underlying empirical research (assuming it is real research) one of the variables is almost always identifiable as the response variable and the other variable(s) is (are) identifiable as predictor variable(s). Thus although it is an interesting side issue that the mathematics of some statistical procedures is symmetric, the underlying empirical research (assuming it is real or realistic) usually still has response and predictor variables, which it is helpful to identify as such.

In exploratory data analysis one is "looking around" in data, often without a particular response variable in mind. Thus the question arises whether we should view exploratory data analysis as a response-variable method or as a no-response-variable method. One answer is that if an exploratory data analysis is to be put to any practical use, a response variable and zero or more predictor variables will usually be (implicitly or explicitly) determined. Thus when exploratory data analysis is put to a practical use, it can usually be viewed as a response-variable method.

I.3 Are the Four Groups of Techniques The Only Techniques Response-Variable Methods Can Perform?

Section 5.4 lists the following four groups of techniques that statistical methods can perform:

1. techniques for detecting relationships between variables
2. techniques for illustrating relationships between variables
3. techniques for predicting and controlling the values of variables on the basis of relationships between variables, and
4. miscellaneous techniques for the study of variables and relationships between or among variables.

Section 5.4 then lists twenty-one response-variable methods and claims that the only techniques that the methods can perform are the four groups of techniques. When considering this claim, note that (to emphasize the practical functions of the statistical techniques) the EPR approach views parameter-estimation techniques as a subset of prediction and control techniques. The paper for students gives further discussion of the four groups of techniques (1997a, sec. 8 - 13).

I cannot directly prove my claim that the twenty-one methods do no more than what is described by the four groups of techniques. I cannot prove the claim because it is a statement that something that is logically possible (i.e., another important statistical technique) does not exist. In general, such statements cannot be empirically or theoretically supported. However, if my claim is incorrect, one can easily disprove it by stating a viable counterexample. I am unaware of any viable counterexamples.

The four groups of techniques listed in the preceding subsection are broader than the twenty-one response-variable statistical methods listed in Section 5.4. In particular, we can view the no-response-variable methods as falling in the category of miscellaneous techniques for the study of variables and relationships between or among variables. This is because all these methods clearly study variables and relationships among variables.

Since the four groups of techniques encompass both the response-variable and no-response-variable statistical methods, they encompass a large proportion of the field of statistics.

APPENDIX J: FUTURE SYSTEMS FOR
STUDYING RELATIONSHIPS
BETWEEN VARIABLES

Nowadays a researcher wishing to use statistical methods to study relationships between variables must be skilled in the use of those methods (even though statistical software can do most of the arithmetic). And when unskilled researchers try to use the methods they often, through misunderstanding, make serious blunders. To prevent these blunders, and to help improve the quality of empirical research, it seems likely that statisticians will develop expert software that will interactively guide an unskilled researcher through the steps of properly designing an empirical research project, performing it, and interpreting the results. I call this software "research guidance software".

To help researchers design a research project, research guidance software will help them to select the response and predictor variables and help them to determine the important details of the design. Some important goals are to design research projects

• that are unequivocal
• that have minimum cost
• whose statistical tests have maximum power (Appendix B in the paper for students [1997a])
• whose subsequent predictions or control will have maximum accuracy (Appendix E in the paper for students [1997a]).

To help researchers perform a research project, the software will be capable of controlling any instruments used in the research project and capable of obtaining values of variables directly from the instruments.

To help researchers interpret the results of a research project, the software will automatically guide them through the phases of checking data for anomalies, detecting relationships between variables, displaying relationships, and prediction or control on the basis of relationships.

The output of a research guidance software system will likely appear in a web browser, since web browsing software has advanced tools for efficiently displaying and linking many forms of information. The output will include custom text and graphics that reflect specific details of the research project under study. (The text will be generated from pre-written templates.) To bring the user interface up to the visual resolution of a printed textbook, the software and video system will be capable of simultaneously displaying the equivalent of at least two full pages of highly legible text and graphics from a standard textbook. (Computer monitors are beginning to approach this capability.) In addition, the system will be capable of displaying standard television-quality video segments in which a narrator discusses and illustrates a concept.

(As a prolific writer of notes in the margins of the books I buy and the computer output I generate, I recommend that the software provide a convenient way for users to "write notes in the margin" of its on-screen output [including output of the help system]. I also recommend that the software be able to automatically preserve these notes and details of their linkages when it is upgraded.)

In the past, most researchers and teaching facilities lacked access to computer hardware that could run the type of system described in the preceding paragraphs. Thus the market for research guidance software was too small to justify the development cost of a comprehensive system. However, nowadays the necessary computer hardware is within most researchers' and many teaching facilities' budgets.

Furthermore, it is now easier to develop a research guidance software system because such a system can use an existing statistical software product as its data analysis engine. (At least one leading statistical software vendor has developed an "output delivery system" that can provide all output from its statistical procedures in formats that are easily used as input by other programs [SAS Institute 2000].) This enables research guidance software developers to concentrate on other important tasks, such as creating the research-project-design and high-level-analysis modules, and writing the many necessary text templates for the system.

The default path through any output from a research guidance software system will be on the highest conceptual level, which will focus the user on the important points. However, customized full information about the underlying details (including tutorials in the associated statistical concepts) will be only a keystroke or two away.

Research guidance software will enable researchers to get much closer to their data because in interpreting the results of a research project the software will automatically examine many different graphical views of the data and will present the most interesting views (as defined by research in human perception) to the researcher. (Most researchers would be incapable of generating and examining all the necessary views manually.)

Research guidance software will be designed by teams of statisticians, programmers, and editors. (In the past, the statisticians and editors would have developed statistics textbooks.) The quality of the writing, graphics, and layout will be equal to that of a superior textbook.

For a research guidance software system to be complete, its developers must codify a complete general research design and data analysis strategy, a difficult but not impossible task. (The codification will involve extensive consultation with experts in research design and data analysis and extensive testing with inexperienced users.) The approach must be general enough and well-enough presented to enable a well-motivated neophyte to properly design, perform, and analyze efficient simple empirical research projects.

Rudimentary research guidance software systems have begun to appear. Silvers, Herrmann, Godfrey, Roberts, and Cerys (1994) discuss one such system and give references to important earlier systems. Gale and Pregibon (1984) were the pioneers.

The inevitability of useful research guidance software arises from the fact that today's best research project designers and data analysts work by simply proceeding through a complicated (often subconscious, sometimes vague) decision tree. Thus three steps are necessary:

1. Decision trees must be elicited from master designers and data analysts.
2. Decision trees must have any vagueness removed (or perhaps parameterized under user control, with consensually chosen default values for the parameters).
3. Software must be developed that will (in the manner of a wise and friendly consultant, and using different levels of detail depending on the user's knowledge) guide a researcher through the decision tree.

Once these steps are completed, the ability of research guidance software to help researchers perform research projects will approach (and likely someday surpass) that of an expert.

Research guidance software amounts to copying both statistical thought and research thought from experts' minds and from textbooks into the computer. Once (properly) captured in the computer, the thought can be automatically customized by the computer for the situation at hand, and the researcher or student can actively interact with the thought. Carefully designed customized interaction guarantees better understanding.

The teaching component will be an important part of any research guidance software system. In the better systems the teaching component will be easy to understand, comprehensive, and compliant with norms of statistical practice.

Meeker presents a similar vision for statistical technology in an article by Moore, Cobb, Garfield, and Meeker (1995, sec 2.2).

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(version of January 30, 2002)

CONTENTS

1. INTRODUCTION *

2. A DEFINITION OF "EMPIRICAL RESEARCH" *

3. COURSE GOALS *

3.1 The Value of Emphasizing Goals *

3.2 Topic-Based Goals Have a Significant Drawback *

3.3 Recommended Goals (A Lasting Appreciation of the Role of Statistics) *

4. SIX CONCEPTS *

4.1 Entities *

4.2 Properties of Entities *

4.3 Variables *

4.4 A Goal of Empirical Research: To Predict and Control the Values of Variables *

4.5 Relationships Between Variables as a Key to Prediction and Control *

4.6 Statistical Techniques for Studying Relationships Between Variables as a Means to Accurate Prediction and Control *

4.7 General Comments *

5. EVALUATING THE EPR APPROACH *

5.1 Main Differences Between the EPR Approach and Other Approaches *

5.2 The Concepts of the Approach Are Easy to Understand *

5.3 The Approach Provides a Deep and Broad Foundation for Statistical Concepts *

5.4 The Approach Unifies Statistical Methods *

5.5 The Approach Links Well with General Concepts of Science *

5.6 The Approach Unifies Empirical Research *

5.7 The Approach Links Well with General Concepts of Commerce *

5.8 The Approach Links Well with Language *

5.9 The Concepts of the Approach Are Fundamental *

5.10 Easy-to-Understand Fundamental Concepts Should Be Taught First *

5.11 The Approach Gives Students a Lasting Appreciation of Statistics *

5.12 The Approach Links Well with the Concept of 'Data Analysis' *

5.13 The Concepts Are Old But the Approach Is New *

5.14 The Approach Links Well with Other Approaches to the Introductory Course *

5.15 Responses to Criticisms of the EPR Approach *

6. TESTING THE EPR APPROACH *

6.1 Methods of Testing *

6.2 Testing of the EPR Approach *

7. IMPLEMENTING THE EPR APPROACH *

7.1 Motivating Students on the First Day of Class *

7.2 What Topics Should Follow the Six Concepts? *

7.3 A Syllabus *

7.4 "Basis for Action" Versus "Decision Procedure" *

7.5 Practical Examples *

7.6 Generalization and Instantiation *

7.7 Feedback Systems *

7.8 Realistic Data Versus Real Data *

7.9 The Discussion of Mathematics in the Introductory Statistics Course *

7.10 Hypothesis Testing *

7.11 Univariate Distributions *

7.12 Implementation With Software Support *

8. SUMMARY *

APPENDIX A: THE PRIORITY OF THE CONCEPT OF 'ENTITY' *

APPENDIX B: THE TERM "PROPERTY" *

APPENDIX C: THE EVOLUTION OF ENTITIES AND PROPERTIES IN HUMAN THOUGHT *

APPENDIX D: DEFINING THE TERM "VARIABLE" *

APPENDIX E: THE DISTINCTION BETWEEN PROPERTIES AND VARIABLES *

APPENDIX F: THE TERM "RELATIONSHIP" *

APPENDIX G: THE TERMS "RESPONSE VARIABLE" AND "PREDICTOR VARIABLE" *

APPENDIX H: DO RESEARCH PROJECTS STUDY RELATIONSHIPS BETWEEN VARIABLES? *

APPENDIX I: DOES THE EPR APPROACH UNIFY STATISTICAL METHODS? *

APPENDIX J: FUTURE SYSTEMS FOR STUDYING RELATIONSHIPS BETWEEN VARIABLES *

REFERENCES *