# Chapter 1 Practice Problems

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Displaying the Order in a Group of Numbers Using Tables and Graphs

Chapter Outline

✪ The Two Branches of Statistical Methods 2

✪ Some Basic Concepts 3

✪ Frequency Tables 7

✪ Histograms 10

✪ Shapes of Frequency Distributions 15

✪ Controversy: Misleading Graphs 19

✪ Frequency Tables and Histograms in Research Articles 21

Welcome to Statistics for Psychology. We imagine you to be like other stu-dents we have known who have taken this course. You have chosen tomajor in psychology or a related field because you are fascinated by people—by the visible behaviors of the people around you, perhaps too by their inner lives as well as by your own. Some of you are highly scientific sorts; others are more intuitive. Some of you are fond of math; others are less so, or even afraid of it. What- ever your style, we welcome you. We want to assure you that if you give this book some special attention (perhaps a little more than most textbooks require), you will learn statistics. The approach used in this book has successfully taught all sorts of stu- dents before you, including those who had taken statistics previously and done poorly. With this book and your instructor’s help, you will learn statistics and learn it well.

More importantly, we want to assure you that whatever your reason for studying psychology or a related field, this course is not a waste of time. Learning about statistics

✪ Summary 23

✪ Key Terms 24

✪ Example Worked-Out Problems 24

✪ Practice Problems 25

✪ Using SPSS 29

✪ Chapter Note 32

CHAPTER 1

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helps you to read the work of other psychologists, to do your own research if you so choose, and to hone both your reasoning and intuition. Formally, statistics is a branch of mathematics that focuses on the organization, analysis, and interpretation of a group of numbers. But really what is statistics? Think of statistics as a tool that has evolved from a basic thinking process employed by every human: you observe a thing; you wonder what it means or what caused it; you have an insight or make an intuitive guess; you ob- serve again, but now in detail, or you try making little changes in the process to test your intuition. Then you face the eternal problem: was your hunch confirmed or not? What are the chances that what you observed this second time will happen again and again, so that you can announce your insight to the world as something probably true?

Statistics is a method of pursuing truth. As a minimum, statistics can tell you the likelihood that your hunch is true in this time and place and with these sorts of people. This pursuit of truth, or at least its future likelihood, is the essence of psychol- ogy, of science, and of human evolution. Think of the first research questions: what will the mammoths do next spring? What will happen if I eat this root? It is easy to see how the early accurate “researchers” survived. You are here today because your ancestors exercised brains as well as brawn. Do those who come after you the same favor: think carefully about outcomes. Statistics is one good way to do that.

Psychologists use statistical methods to help them make sense of the numbers they collect when conducting research. The issue of how to design good research is a topic in itself, summarized in a Web Chapter (Overview of the Logic and Language of Psychology Research) available on the Web site for this book http://www. pearsonhighered.com/. But in this text we confine ourselves to the statistical meth- ods for making sense of the data collected through research.

Psychologists usually use a computer and statistical software to carry out statis- tical procedures, such as the ones you will learn in this book. However, the best way to develop a solid understanding of statistics is to learn how to do the procedures by hand (with the help of a calculator). To minimize the amount of figuring you have to do, we use relatively small groups of numbers in each chapter’s examples and prac- tice problems. We hope that this will also allow you to focus more on the underlying principles and logic of the statistical procedure, rather than on the mathematics of each practice problem (such as subtracting 3 from 7 and then dividing the result by 2 to give an answer of 2). (See the Introduction to the Student on pp. xvi–xviii for more information on the goals of this book.) Having said that, we also recognize the importance of learning how to do statistical procedures on a computer, as you most likely would when conducting your own research. So, at the end of relevant chap- ters, there is a section called Using SPSS (see also the Study Guide and Computer Workbook that accompanies this text and that includes a guide to getting started with SPSS). SPSS statistical software is commonly used by psychologists and other behavioral and social scientists to carry out statistical analyses. Check with your instructor to see if you have access to SPSS at your institution.

The Two Branches of Statistical Methods There are two main branches of statistical methods.

1. Descriptive statistics: Psychologists use descriptive statistics to summarize and describe a group of numbers from a research study.

2. Inferential statistics: Psychologists use inferential statistics to draw conclu- sions and to make inferences that are based on the numbers from a research study but that go beyond the numbers. For example, inferential statistics allow researchers to make inferences about a large group of individuals based on a re- search study in which a much smaller number of individuals took part.

descriptive statistics procedures for summarizing a group of scores or other- wise making them more comprehensible.

inferential statistics procedures for drawing conclusions based on the scores collected in a research study but going beyond them.

statistics branch of mathematics that focuses on the organization, analysis, and interpretation of a group of numbers.

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Displaying the Order in a Group of Numbers 3

variable characteristic that can have different values.

values possible number or category that a score can have.

score particular person’s value on a variable.

In this chapter and the next, we focus on descriptive statistics. This topic is im- portant in its own right, but it also prepares you to understand inferential statistics. Inferential statistics are the focus of the remainder of the book.

In this chapter we introduce you to some basic concepts, and then you will learn to use tables and graphs to describe a group of numbers. The purpose of descriptive statistics is to make a group of numbers easy to understand. As you will see, tables and graphs help a great deal.

Some Basic Concepts Variables, Values, and Scores As part of a larger study (Aron, Paris, & Aron, 1995), researchers gave a question- naire to students in an introductory statistics class during the first week of the course. One question asked was, “How stressed have you been in the last 21⁄2 weeks, on a scale of 0 to 10, with 0 being not at all stressed and 10 being as stressed as possible?” (How would you answer?) In this study, the researchers used a survey to examine students’ level of stress. Other methods that researchers use to study stress include measuring stress-related hormones in human blood or conducting controlled laboratory studies with animals.

In this example, level of stress is a variable, which can have values from 0 to 10, and the value of any particular person’s answer is the person’s score. If you answered 6, your score is 6; your score has a value of 6 on the variable called “level of stress.”

More formally, a variable is a condition or characteristic that can have different values. In short, it can vary. In our example, the variable was level of stress, which can have the values of 0 through 10. Height is a variable, social class is a variable, score on a creativity test is a variable, type of psychotherapy received by patients is a variable, speed on a reaction time test is a variable, number of people absent from work on a given day is a variable, and so forth.

A value is just a number, such as 4, –81, or 367.12. A value can also be a category, such as male or female, or a psychiatric diagnosis—major depression, post-traumatic stress disorder—and so forth.

Finally, on any variable, each person studied has a particular number or score that is his or her value on the variable. As we’ve said, your score on the stress vari- able might have a value of 6. Another student’s score might have a value of 8.

Psychology research is about variables, values, and scores (see Table 1–1). The formal definitions are a bit abstract, but in practice, the meaning usually is clear.

Levels of Measurement (Kinds of Variables) Most of the variables psychologists use are like those in the stress ratings example: the scores are numbers that tell you how much there is of what is being measured. In the stress ratings example, the higher the number is, the more stress there is. This is

Table 1–1 Some Basic Terminology

Term Definition Examples

Variable Condition or characteristic that can have different values Stress level, age, gender, religion

Value Number or category 0, 1, 2, 3, 4, 25, 85, female, Catholic

Score A particular person’s value on a variable 0, 1, 2, 3, 4, 25, 85, female, Catholic

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equal-interval variable variable in which the numbers stand for approximately equal amounts of what is being measured.

ratio scale an equal-interval variable is measured on a ratio scale if it has an absolute zero point, meaning that the value of zero on the variable indicates a complete absence of the variable.

discrete variable variable that has specific values and that cannot have values between these specific values.

continuous variable variable for which, in theory, there are an infinite number of values between any two values.

an example of a numeric variable. Numeric variables are also called quantitative variables.

There are several kinds of numeric variables. In psychology research the most important distinction is between two types: equal-interval variables and rank-order variables. An equal-interval variable is a variable in which the numbers stand for approximately equal amounts of what is being measured. For example, grade point average (GPA) is a roughly equal-interval variable, since the difference between a GPA of 2.5 and 2.8 means about as much as the difference between a GPA of 3.0 and 3.3 (each is a difference of 0.3 of a GPA). Most psychologists also consider scales like the 0-to-10 stress ratings as roughly equal interval. So, for example, a difference between stress ratings of 4 and 6 means about as much as the difference between 7 and 9.

Some equal-interval variables are measured on what is called a ratio scale. An equal-interval variable is measured on a ratio scale if it has an absolute zero point. An absolute zero point means that the value of zero on the variable indicates a com- plete absence of the variable. Most counts or accumulations of things use a ratio scale. For example, the number of siblings a person has is measured on a ratio scale, because a zero value means having no siblings. With variables that are measured on a ratio scale, you can make statements about the difference in magnitude between values. So, we can say that a person with four siblings has twice as many siblings as a person with two siblings. However, most of the variables in psychology are not on a ratio scale.

Equal-interval variables can also be distinguished as being either discrete vari- ables or continuous variables. A discrete variable is one that has specific values and cannot have values between the specific values. The number of times you went to the dentist in the last 12 months is a discrete variable. You may have gone 0, 1, 2, 3, or more times, but you can’t have gone 1.72 times or 2.34 times. With a continuous variable, there are in theory an infinite number of values between any two values. So, even though we usually answer the question “How old are you?” with a specific age, such as 19 or 20, you could also answer it by saying that you are 19.26 years old. Height, weight, and time are examples of other continuous variables.

The other main type of numeric variable, a rank-order variable, is a variable in which the numbers stand only for relative ranking. (Rank-order variables are also called ordinal variables.) A student’s standing in his or her graduating class is an ex- ample. The amount of difference in underlying GPA between being second and third in class standing could be very unlike the amount of difference between being eighth and ninth.

A rank-order variable provides less information than an equal-interval variable. That is, the difference from one rank to the next doesn’t tell you the exact difference in amount of what is being measured. However, psychologists often use rank-order vari- ables because they are the only information available. Also, when people are being asked to rate something, it is sometimes easier and less arbitrary for them to make rank-order ratings. For example, when rating how much you like each of your friends, it may be easier to rank them by how much you like them than to rate your liking for them on a scale. Yet another reason researchers often use rank-order variables is that asking people to do rankings forces them to make distinctions. For example, if asked to rate how much you like each of your friends on a 1-to-10 scale, you might rate sev- eral of them at exactly the same level, but ranking would avoid such ties.

Another major type of variable used in psychology research, which is not a nu- meric variable at all, is a nominal variable in which the values are names or cate- gories. The term nominal comes from the idea that its values are names. (Nominal

rank-order variable numeric variable in which the values are ranks, such as class standing or place finished in a race. Also called ordinal variable.

numeric variable variable whose values are numbers (as opposed to a nominal variable). Also called quantita- tive variable.

nominal variable variable with values that are categories (that is, they are names rather than numbers). Also called categorical variable.

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variables are also called categorical variables because their values are categories.) For example, for the nominal variable gender, the values are female and male. A per- son’s “score” on the variable gender is one of these two values. Another example is psychiatric diagnosis, which has values such as major depression, post-traumatic stress disorder, schizophrenia, and obsessive-compulsive disorder.

These different kinds of variables are based on different levels of measurement (see Table 1–2). Researchers sometimes have to decide how they will measure a par- ticular variable. For example, they might use an equal-interval scale, a rank-order scale, or a nominal scale. The level of measurement selected affects the type of sta- tistics that can be used with a variable. Suppose a researcher is studying the effects of a particular type of brain injury on being able to recognize objects. One approach the researcher might take would be to measure the number of different objects an injured person can observe at once. This is an example of an equal-interval level of measurement. Alternately, the researcher might rate people as able to observe no objects (rated 0), only one object at a time (rated 1), one object with a vague sense of other objects (rated 2), or ordinary vision (rated 3). This would be a rank-order approach. Finally, the researcher might divide people into those who are completely blind (rated B), those who can identify the location of an object but not what the ob- ject is (rated L), those who can identify what the object is but not locate it in space (rated I), those who can both locate and identify an object but have other abnormali- ties of object perception (rated O), and those with normal visual perception (rated N). This is a nominal level of measurement.

In this book, as in most psychology research, we focus mainly on numeric, equal-interval variables (or variables that roughly approximate equal-interval variables). We discuss statistical methods for working with nominal variables in Chapter 13 and methods for working with rank-order variables in Chapter 14.

levels of measurement types of underlying numerical information provided by a measure, such as equal- interval, rank-order, and nominal (categorical).

Table 1–2 Levels of Measurement

Level Definition Example

Equal-interval Numeric variable in which differences between values correspond to differences in the underlying thing being measured

Stress level, age

Rank-order Numeric variable in which values correspond to the relative position of things measured

Class standing, position finished in a race

Nominal Variable in which the values are categories Gender, religion

How are you doing?

1. A father rates his daughter as a 2 on a 7-point scale (from 1 to 7) of cranki- ness. In this example, (a) what is the variable, (b) what is the score, and (c) what is the range of values?

2. What is the difference between a numeric and a nominal variable? 3. What is the difference between a discrete and a continuous variable? 4. Give the level of measurement of each of the following variables: (a) a person’s

nationality (Mexican, Spanish, Ethiopian, Australian, etc.), (b) a person’s score on a standard IQ test, (c) a person’s place on a waiting list (first in line, second in line, etc.).

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BOX 1–1 Important Trivia for Poetic Statistics Students The word statistics comes from the Italian word statista, a person dealing with affairs of state (from stato, “state”). It was originally called “state arithmetic,” involving the tabulation of information about nations, especially for the purpose of taxation and planning the feasibility of wars.

Statistics were needed in ancient times to figure the odds of shipwrecks and piracy for marine insurance that would encourage voyages of commerce and exploration to far-flung places. The modern study of mortality rates and life insurance descended from the 17th-century plague pits—counting the bodies of persons cut down in the bloom of youth. The theory of errors (covered in Chapter 12) began in astronomy, that is, with stargazing; the theory of correlation (Chapter 11) has its roots in bi- ology, from the observation of parent and child differ- ences. Probability theory (Chapter 3) arose in the tense environs of the gambling table. The theory of analysis of experiments (Chapters 7 to 10) began in breweries and out among waving fields of wheat, where correct guesses determined not only the survival of a tasty beer but of thousands of marginal farmers. Theories of measurement and factor analysis (Chapter 15) derived from personality psychology, where the depths of human character were first explored with numbers. And chi-square (Chapter 13) came to us from sociology, where it was often a question of class.

In the early days of statistics, it was popular to use the new methods to prove the existence of God. For example, John Arbuthnot discovered that more male than female babies were born in London between 1629 and 1710. In

what is considered the first use of a statistical test, he proved that the male birthrate was higher than could be expected by chance (assuming that 50:50 was chance) and concluded that there was a plan operating, since males face more danger to obtain food for their families, and only God, he said, could do such planning.

In 1767, John Michell also used probability theory to prove the existence of God when he argued that the odds were 500,000 to 1 against six stars being placed as close together as those in the constellation Pleiades; so their placement had to have been a deliberate act of the Creator.

Statistics in the “state arithmetic” sense are legally en- dorsed by most governments today. For example, the first article of the U.S. Constitution requires a census. And statistics helped the United States win the Revolu- tionary War. John Adams obtained critical aid from Holland by pointing out certain vital statistics, carefully gathered by the clergy in local parishes, demonstrating that the colonies had doubled their population every 18 years, adding 20,000 fighting men per annum. “Is this the case of our enemy, Great Britain?” Adams wrote. “Which then can maintain the war the longest?”

Similar statistics were observed by U.S. President Thomas Jefferson in 1786. He wrote that his people “be- come uneasy” when there are more of them than 10 per square mile and that given the population growth of the new country, within 40 years these restless souls would fill up all of their country’s “vacant land.” Some 17 years later, Jefferson doubled the size of the United States’ “vacant” land through the Louisiana Purchase.

Answers

1.(a) crankiness, (b) 2, (c) 1 to 7. 2.A numeric variable has values that are numbers that tell you the degree or

extent of what the variable measures; a nominal variable has values that are different categories and have no particular numerical order.

3.A discrete variable has specific values and has no values between the spe- cific values. A continuous variable has, in theory, an infinite number of values between any two values.

4.(a) nominal, (b) equal-interval, (c) rank-order.

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Frequency Tables An Example Let’s return to the stress ratings example. Recall that in this study, students in an in- troductory statistics class during the first week of the course answered the question, “How stressed have you been in the last 21⁄2 weeks, on a scale of 0 to 10, with 0 being not at all stressed and 10 being as stressed as possible?” The actual study included scores from 151 students. To ease the learning for this example, we are going to use a representative subset of scores from 30 of the 151 students (this also saves you time if you want to try it for yourself). The 30 students’ scores (their ratings on the scale) are:

8, 7, 4, 10, 8, 6, 8, 9, 9, 7, 3, 7, 6, 5, 0, 9, 10, 7, 7, 3, 6, 7, 5, 2, 1, 6, 7, 10, 8, 8.

Looking through all these scores gives some sense of the overall tendencies, but this is hardly an accurate method. One solution is to make a table showing how many stu- dents used each of the 11 values that the ratings can have (0, 1, 2, and so on, through 10). We have done this in Table 1–3. We also figured the percentage each value’s fre- quency is of the total number of scores. Tables like this sometimes give only the raw- number frequencies, not the percentages, or only the percentages and not the raw-number frequencies. In addition, some frequency tables include, for each value, the total number of scores with that value and all values preceding it. These are called cumulative frequencies because they tell how many scores are accumulated up to this pointon the table. Ifpercentagesareused,cumulativepercentagesalsomaybe included (for an example, see Figure 1–18 in the Using SPSS section on page 30). Cumulative percentages give, for each value, the percentage of scores up to and including that value. The cumulative percentage for any given value (or for a score having that value) is also called a percentile. Cumulative frequencies and cumulative percentages allow you to see where a particular score falls in the overall group of scores.

Table 1–3 is called a frequency table because it shows how frequently (how many times) each score was used. A frequency table makes the pattern of numbers easy to see. In this example, you can see that most of the students rated their stress level around 7 or 8, with few rating it very low.

How to Make a Frequency Table There are the four steps in making a frequency table.

❶ Make a list down the page of each possible value, from lowest to highest. In the stress ratings results, the list goes from 0, the lowest possible rating, up to 10, the highest possible rating.1 Note that even if one of the ratings between 0 and 10 is not used, you still include that value in the listing, showing it as hav- ing a frequency of 0. For example, if no one gives a stress rating of 2, you still include 2 as one of the values on the frequency table.

❷ Go one by one through the scores, making a mark for each next to its value on your list. This is shown in Figure 1–1.

❸ Make a table showing how many times each value on your list is used. That is, add up the number of marks beside each value.

❹ Figure the percentage of scores for each value. To do this, take the frequency for that value, divide it by the total number of scores, and multiply by 100. You may need to round off the percentage. We recommend that you round percent- ages to one decimal place. Note that because of the rounding, your percentages do not usually add up to exactly 100% (but they should be close).

Table 1–3 Frequency Table of Number of Students Rating Each Value of the Stress Scale

Stress Rating Frequency Percent

0 1 3.3

1 1 3.3

2 1 3.3

3 2 6.7

4 1 3.3

5 2 6.7

6 4 13.3

7 7 23.3

8 5 16.7

9 3 10.0

10 3 10.0

Source: Data based on Aron et al. (1995).

frequency table listing of number of individuals having each of the different values for a particular variable.

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Frequency Tables for Nominal Variables The preceding steps assume you are using numeric variables, the most common situa- tion. However, you can also use a frequency table to show the number of scores in each value (or category) of a nominal variable. For example, researchers (Aron, Aron, & Smollan, 1992) asked 208 students to name the closest person in their life.As shown in Table 1–4, 33 students selected a family member, 76 a nonromantic friend, 92 a roman- tic partner, and 7 selected some other person.Also in Table 1–4, the values listed on the left hand side of the frequency table are the values (the categories) of the variable.

Another Example Tracy McLaughlin-Volpe and her colleagues (2001) had 94 introductory psychology students keep a diary of their social interactions for a week during the regular semester. Each time a participant had a social interaction lasting 10 minutes or longer, he or she would fill out a card. The card had questions about various aspects of the conversation and the conversation partner. Excluding family and work situations, the number of so- cial interactions 10 minutes or longer over a week for these students were as follows:

48, 15, 33, 3, 21, 19, 17, 16, 44, 25, 30, 3, 5, 9, 35, 32, 26, 13, 14, 14, 47, 47, 18, 11, 5, 19, 24, 17, 6, 25, 8, 18, 29, 1, 18, 22, 3, 22, 29, 2, 6, 10, 29, 10, 29, 21, 38, 41, 16, 17, 8, 40, 8, 10, 18, 7, 4, 4, 8, 11, 3, 23, 10, 19, 21, 13, 12, 10, 4, 17, 11, 21, 9, 8, 7, 5, 3, 22, 14, 25, 4, 11, 10, 18, 1, 28, 27, 19, 24, 35, 9, 30, 8, 26.

Now, let’s follow our four steps for making a frequency table.

❶ Make a list down the page of each possible value, from lowest to highest. The lowest possible number of interactions is 0. In this study, the highest num- ber of interactions could be any number. However, the highest actual number in this group is 48; so we can use 48 as the highest value. Thus, the first step is to list these values down a page. (It might be good to use several columns so that you can have all the scores on a single page.)

❷ Go one by one through the scores, making a mark for each next to its value on your list. Figure 1–2 shows the results of this step.

❸ Make a table showing how many times each value on your list is used. Table 1–5 is the result.

8, 7, 4, 10, 8, 6, 8, 9, 9, 7, 3, 7, 6, 5, 0, 9, 10, 7, 7, 3, 6, 7, 5, 2, 1, 6, 7, 10, 8, 8

STRESS RATING FREQUENCY

0 1 2 3 4 5 6 7 8 9

10

Figure 1–1 Making a frequency table for the stress ratings …