Heiman G. Basic Statistics for the Behavioral Sciences

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FIGURE 1.2

Plot of height and

weight scores using a

different scale on the

axisY

Height Weight

63 130

64 140

65 155

66 160

67 165

68 170

8 CHAPTER 1 / Introduction to Statistics

066

Height (in inches)

Weight (in pounds)

200

100

67 68656463

63 65 66 67 68

Height (in inches)

Weight (in pounds)

170

160

150

140

130 Jane

640

In the body of the graph we plot the scores from the table on the left. Jane is 63 inches

tall and weighs 130 pounds, so we place a dot above the height of 63 and opposite the

weight of 130. And so on. Each dot on the graph is called a data point. Notice that you

read the graph by using the scores on one axis and the data points. For example, to find

the weight of the person who has a height of 67, travel vertically from 67 to the data

point and then horizontally to the axis: 165 is the corresponding weight.

In later chapters you will learn when to connect the data points with lines and when

to create other types of figures. Regardless of the final form of a graph, always label

the and axes to indicate what the scores measure (not just and ), and always give

your graph a title indicating what it describes.

When creating a graph, make the spacing between the labels for the scores on an axis

reflect the spacing between the actual scores. In Figure 1.1 the labels 64, 65, and 66 are

equally spaced on the graph because the difference between 64 and 65 is the same as

the difference between 65 and 66. However, in other situations, the labels may not be

equally spaced. For example, the labels 10, 20, and 40 would not be equally spaced

because the distance between these scores is not equal.

Sometimes there are so many different scores that we cannot include a label for each

one. Be careful here, because the units used in labeling each axis then determine the

impression the graph gives. Say that for the previous weight scores, instead of labeling

the axis in units of 10 pounds, we labeled it in units of 100 pounds, as shown in

Figure 1.2. This graph shows the same data as Figure 1.1, but changing the scale on the

Y axis creates a much flatter pattern of data points. This gives the misleading impression

YXYX

FIGURE 1.1

Plot of height and weight scores

Person Height Weight

Jane 63 130

Bob 64 140

Mary 65 155

Tony 66 160

Sue 67 165

Mike 68 170

Chapter Summary 9

that regardless of their height, the people all have about the same weight. However, look-

ing at the actual scores, you see that this is not the case. Thus, always label the axes in a

way that honestly presents the data, without exaggerating or minimizing the pattern

formed by the data points.

That’s all the basic math you’ll need to get started. You are now ready to begin learning

to use statistics. In fact, you already use statistics. If you compute your grade average

or if you ask your instructor to “curve” your grades, you are using statistics. When you

understand from the nightly news that Senator Fluster is projected to win the election

or when you learn from a television commercial that Brand X “significantly” reduces

tooth decay, you are using statistics. You simply do not yet know the formal names for

these statistics or the logic behind them. But you will.

PUTTING IT ALL

TOGETHER

CHAPTER SUMMARY

1. All empirical research is based on observation and measurement, resulting in

numbers, or scores. These scores are the data.

2. Statistical procedures are used to make sense out of data: They are used to

organize, summarize, and communicate data and to draw conclusions about

what the data indicate.

3. The goal in learning statistics is to know when to perform a particular procedure

and how to interpret the answer.

4 Statistical notation refers to the standardized code for symbolizing the

mathematical operations performed in the formulas and the answers obtained.

5. Unless otherwise indicated, the order of mathematical operations is to compute

inside parentheses first, then square or find square roots, then multiply or divide,

and then add or subtract.

6. Round off the final answer in a calculation to two more decimal places than are in

the original scores. If the digit in the next decimal place is equal to or greater than

5, round up; if the digit is less than 5, round down.

7.A transformation is a procedure for systematically converting one set of scores

into a different set of scores. Transformations make scores easier to work with

and make different kinds of scores comparable.

8.A proportion is a decimal between 0 and 1 that indicates a fraction of the total. To

transform a score to a proportion, divide the score by the total. To determine the

score that produces a particular proportion, multiply the proportion times the total.

9. To transform a proportion to a percent, multiply the proportion times 100. To

transform an original score to a percent, find the proportion by dividing the score

by the total and then multiplying by 100.

10. To transform a percent to a proportion, divide the percent by 100. To find the

original score that corresponds to a particular percent, transform the percent to

a proportion and then multiply the proportion times the total.

11. A data point is a dot plotted on a graph to represent a pair of and scores.YX

10 CHAPTER 1 / Introduction to Statistics

KEY TERMS

data 1

data point 8

percent 7

proportion 7

statistical notation 4

transformation 6

REVIEW QUESTIONS

(Answers for odd-numbered problems are in Appendix D.)

1. Why do researchers need to learn statistics?

2. What does the term statistical notation refer to?

3. (a) To how many places should you round a final answer? (b) If you are rounding

to two decimal places, what are the rules for rounding up or down?

4. (a) What is a transformation? (b) Why do we transform data?

5. If given no other information, what is the order in which to perform mathematical

operations?

6. What is a percentage?

7. What is a data point?

8. A researcher measures the IQ scores of a group of college students. What four

things will the researcher use statistics for?

9. What is a proportion and how is it computed?

10. How do you transform a percentage to a proportion?

APPLICATION QUESTIONS

11. (a) What proportion is 5 out of 15? (b) What proportion of 50 is 10? (c) One in a

thousand equals what proportion?

12. For each of the following, to how many places will you round off your final

answer? (a) When measuring the number of questions students answered cor-

rectly on a test. (b) When measuring what proportion of the total possible points

students have earned in a course. (c) When counting the number of people having

various blood types. (d) When measuring the number of dollar bills possessed by

each person in a group.

13. Transform each answer in question 11 to a percent.

14. The intermediate answers from some calculations based on whole-number scores

are and . We now want to find After

rounding, what values of and do we use?

15. Round off the following numbers to two decimal places: (a) 13.7462, (b) 10.043,

16. For find the value of Q when and .

17. Using the formula in question 16, find when and .

18. For and , find :

19. Using the formula in question 18, find for and .

20. Of the 40 students in a gym class, 13 played volleyball, 12 ran track (4 of whom

did a push-up), and the remainder were absent. (a) What proportion of the class

Y 524X 5 9D

D 5 a

X 2 Y

b12X2

DY 5 4.8X 5 14

Y 522X 5 8Q

Y 5 5X 5 3Q 5 1X 1 Y21X

1 Y

.Y 5 3.3333X 5 4.3467892

Application Questions 11

ran track? (b) What percentage played volleyball? (c) What percentage of the

runners did a push-up? (d) What proportion of the class was absent?

21. In your statistics course, there are three exams: I is worth 40 points, II is worth

35 points, and III is worth 60 points. Your professor defines a passing grade as

earning 60% of the points. (a) What is the smallest score you must obtain on each

exam to pass it? (b) In total you can earn 135 points in the course. How many

points must you earn from the three exams combined to pass the course? (c) You

actually earn a total of 115 points during the course. What percent of the total did

you earn?

22. There are 80 students enrolled in statistics. (a) You and 11 others earned the same

score. What percent of the class received your score? (b) Forty percent of the class

received a grade of C. How many students received a C? (c) Only 7.5% of the

class received a D. How many students is this? (d) A student claims that .5% of

the class failed. Why is this impossible?

23. (a) How do you space the labels on the or axis of a graph? (b) Why must you

be careful when selecting the amounts used as these labels?

24. Create a graph showing the data points for the following scores.

X Score Y Score

20 10

25 30

35 20

45 60

25 55

40 70

45 30

Because statistics are part of the process of conducting research, to understand statis-

tics you need to first understand a little about research. Therefore, this chapter dis-

cusses the basics of behavioral research, the general ways that statistics are used in

research, and the major aspects of how we conduct a study that influence which statis-

tics are used.

THE LOGIC OF RESEARCH

Behavioral scientists study the “laws of nature” regarding the behavior of living organ-

isms. That is, researchers assume there are specific influences that govern every behavior

of all members of a particular group. Although any single study is a very small step in this

process, the goal is to understand every factor that influences behavior. Thus, when re-

searchers study such things as the mating behavior of sea lions, social interactions between

humans, or neural firing in a rat’s brain, they are ultimately studying the laws of nature.

The reason a study is a small step is because nature is very complex. Therefore,

research involves a series of translations in which we simplify things so that we can

examine a specific influence on a specific behavior in a specific situation. Then, using

our findings, we generalize back to the broader behaviors and laws we began with. For

example, here’s an idea for a simple study. Say that we think a law of nature is that

people must study information in order to learn it. We translate this into the more

Statistics and the

Research Process

GETTING STARTED

To understand this chapter, recall the following:

■

From Chapter 1, (1) that we use statistics to make sense out of data and

(2) how to create and interpret graphs.

Your goals in this chapter are to learn

■

The logic of samples and populations in behavioral research.

■

How to recognize a relationship between scores and what is meant by the

strength of a relationship.

■

What descriptive statistics are used for.

■

What inferential statistics are used for.

■

What the difference is between an experiment and a correlational study and

how to recognize the independent variable, the conditions, and the dependent

variable in an experiment.

■

What the four scales of measurement are.

The Logic of Research 13

specific hypothesis that “the more you study statistics, the better you’ll learn them.”

Next we will translate the hypothesis into a situation where we can observe and meas-

ure specific people who study specific material in different amounts, to see if they do

learn differently. Based on what we observe, we will have evidence for working back

to the general law regarding studying and learning.

Part of this translation process involves samples and populations.

Samples and Populations

When researchers want to talk of a behavior occurring in some group in nature, they

say it occurs in the population. A population is the entire group of individuals to which

a law of nature applies. The population might be broadly defined (such as all animals,

all mammals, or all humans), but it can be more narrowly defined (such as all women,

all four-year-old English-speaking children in Canada, or all presidents of the United

States). For our studying research, the population might be all college students taking

statistics. Notice that, although ultimately researchers discuss the population of indi-

viduals, we sometimes talk of the population of scores, as if we have already measured

the behavior of everyone in the population in a particular situation.

The population contains all past, present, and future members of the group, so we

usually consider it to be infinitely large. However, to measure an infinitely large popu-

lation would take roughly forever! Instead, we measure a sample from the population.

A sample is a relatively small subset of a population that is intended to represent, or

stand in for, the population. Thus, we might study the students in your statistics class

as a sample representing the population of all college students enrolled in statistics. The

individuals measured in a sample are called the participants (or sometimes, the sub-

jects) and it is the scores from the sample(s) that constitute our data. As with a popula-

tion, sometimes we discuss a sample of scores as if we have already measured the

participants in a particular situation.

Notice that the definitions of a sample and a population depend on your perspective.

Say that we study the students in your statistics class. If these are the only individuals

we are interested in, then we have measured the population of scores. Or if we are in-

terested in the population of all college students studying statistics, then we have a sam-

ple of scores that represent that population. But if we are interested in both the

populations of college men and college women who are studying statistics, then the

men in the class are one sample and the women in the class are another sample, and

each represents its respective population. Finally, scores from one student can be a

sample representing the population of all scores that the student might produce. Thus, a

population is any complete group of scores that would be found in a particular situa-

tion, and a sample is a subset of those scores that we actually measure in that situation.

The logic behind samples and populations is this: We use the scores in a sample to

infer—to estimate—the scores we would expect to find in the population, if we could

measure them. Then, by translating the scores back into the behaviors they reflect, we

can infer the behavior of the population. Thus, when the television news uses a survey

to predict who will win the presidential election, they are using the scores from a sam-

ple (usually containing about 1200 voters) to infer the voting behavior of the population

of over 100 million voters. Likewise, if we observe that greater studying leads to better

learning for a sample of statistics students, we will infer that similar scores and behav-

iors would be found in the population of all statistics students. Then, because the popu-

lation is the entire group to which the law of nature applies, we are describing how

nature works. Thus, whenever we say a finding applies to the population, we are really

describing how a law of nature applies to everyone out there in the world.

14 CHAPTER 2 / Statistics and the Research Process

REMEMBER The population is the entire group of individuals—and scores—

to which our conclusions apply, based on our observation of a sample, which

is a subset of the population.

Recognize that the above logic assumes that our sample is representative of the pop-

ulation. We will discuss this issue in detail in Chapter 9, but put simply, a representa-

tive sample accurately reflects the individuals, behaviors, and scores found in the

population. Essentially, a representative sample is a good example—a miniversion—of

the larger population. With such a sample, our inferences about the scores and behav-

iors found in the population will also be accurate, and so we can believe what our data

seem to be telling us about nature. Thus, if your class is representative of all statistics

students, then the scores in the class are a good example of the scores that the popula-

tion would produce, and we can believe that everyone would behave as the class does.

Researchers try to create a representative sample by freely allowing the types of

individuals found in the population to occur in the sample. To accomplish this, we cre-

ate a random sample: the individuals in our sample are randomly selected from the

population. This means that who gets chosen depends simply on the luck of the draw

(like drawing names from a hat). Because we don’t influence which participants are

selected, the different types of individuals are free to occur in our sample as they do in

the population, so the sample’s characteristics “should” match the population.

However, random sampling is not foolproof because it may not produce a representa-

tive sample: Just by the luck of the draw, we may select participants whose characteris-

tics do not match those of the population. Then the sample will be unrepresentative,

inaccurately reflecting the behavior of the population. For example, maybe unknown to

us, a large number of individuals happen to be in your statistics class who do not behave

at all like typical students in the population—they are too bright, too lazy, or whatever.

If so, we should not believe what such a sample indicates about our law of nature

because the evidence it provides will be misleading and our conclusions will be wrong!

Therefore, as you’ll see, researchers always deal with the possibility that their conclu-

sions about the population might be incorrect because their sample is unrepresentative.

Nonetheless, after identifying the population and sample, the next step is to define

the specific situation and behaviors to observe and measure. We do this by selecting our

variables.

Obtaining Data by Measuring Variables

In our example research, we asked: Does studying statistics improve your learning of

them? Now we must decide what we mean by “studying” and how to measure it, and

what we mean by “learning” and how to measure it. In research the factors we measure

that influence behaviors—as well as the behaviors themselves—are called variables. A

variable is anything that, when measured, can produce two or more different scores.

A few of the variables found in behavioral research include your age, race, gender, and

intelligence; your personality type or political affiliation; how anxious, angry, or ag-

gressive you are; how attractive you find someone; how hard you will work at a task; or

how accurately you recall a situation.

Variables fall into two general categories. If a score indicates the amount of a variable

that is present, the variable is a quantitative variable. A person’s height, for example, is a

quantitative variable. Some variables, however, cannot be measured in amounts, but in-

stead a score classifies an individual on the basis of some characteristic. Such variables are

called qualitative, or classification, variables. A person’s gender, for example, is a qualita-

tive variable, because the “score” of male or female indicates a quality, or category.

The Logic of Research 15

For our study, we might measure “studying” using such variables as how much

effort is put into studying or the number of times a chapter is read, but say we select

the variable of the number of hours spent studying for a particular statistics test. We

might measure “learning” by measuring how well statistical results can be interpreted

or how quickly a specific procedure can be performed, but say we select the variable of

grades on the statistics test.

As in any research, we then study the law of nature by studying the relationship

between our variables.

Understanding Relationships

If nature relates those mental activities that we call studying to those mental activities

that we call learning, then different amounts of learning should occur with different

amounts of studying. In other words, there should be a relationship between studying

and learning. A relationship is a pattern in which, as the scores on one

variable change, the corresponding scores on the other variable change

in a consistent manner. In our example, we predict the relationship in

which the longer you study, the higher your test grade will be.

Say that we asked some students how long they studied for a test and

their subsequent grades on the test. We might obtain the data in Table 2.1.

To see the relationship, first look at those people who studied for 1 hour

and see their grades. Then look at those whose score is 2 hours and see

their grades. And so on. These data form a relationship because, as study-

time scores change (increase), test grades also change in a consistent fash-

ion (also increase). Further, when study-time scores do not change (for

example, Gary and Bob both studied for 1 hour), grades do not change

either (they both received Fs). In statistics, we use the term association

when talking about relationships. Here, low study times are associated with

low test grades and high study times are associated with high test grades.

REMEMBER In a relationship, as the scores on one variable change, the

scores on the other variable change in a consistent manner.

Because this relationship occurs in the sample data, we have evidence that the

amount that people study does make a difference in their test grades. Therefore, assum-

ing that the sample is representative, we can generalize this finding to the broader pop-

ulation so that we can talk about how people learn in general. In the same way, most

research investigates relationships because a relationship is the telltale sign of a law of

nature at work: When nature ties behaviors or events together, we see a relationship

between the variables that measure those behaviors and events.

Thus, an important step in any research is to determine if there is a relationship in

the sample data that matches the relationship that we predict. A major use of statistical

procedures is to help us understand the relationship, examining the scores and the pat-

tern they form. The simplest relationships fit one of two patterns. Sometimes the pat-

tern fits “The more you , the more you ,” with higher scores paired with higher

scores. Thus, the old saying “The bigger they are, the harder they fall” describes such

a relationship, as does “The more often you speed, the more traffic tickets you accumu-

late.” At other times, the pattern fits “The more you , the less you , with higher YX

XYX

Study Time

Student in Hours Test Grades

Gary 1 F

Bob 1 F

Sue 2 D

Jane 2 D

Tony 3 C

Sidney 3 C

Ann 4 B

Rose 4 B

Lou 5 A

TABLE 2.1

Scores Showing a

Relationship between

the Variables of Study

Time and Test Grades

The data presented in this book are a work of fiction. Any resemblance to real data is purely

a coincidence.

16 CHAPTER 2 / Statistics and the Research Process

scores paired with lower scores.” Thus, that other old saying “The more you prac-

tice statistics, the less difficult they are” describes a relationship, as does “The more al-

cohol you consume, the less coordinated you are.”

Relationships may also form more complicated patterns where, for example, more

at first leads to more , but beyond a certain point more leads to less . For example,

the more you exercise, the better you feel, but beyond a certain point more exercise

leads to feeling less well, as pain and exhaustion set in.

Although the above examples involve quantitative variables, relationships can also in-

volve qualitative variables. For example, men typically are taller than women. If you think

of male and female as “scores” on the variable of gender, then this is a relationship, be-

cause as gender scores change (going from male to female), height scores tend to decrease.

We can study any combination of qualitative and quantitative variables in a relationship.

The Strength of a Relationship The data back in Table 2.1 show a perfectly con-

sistent association between study time and test grades: In a perfectly consistent rela-

tionship, each score on one variable is paired with only one score on the other variable.

In Table 2.1, all those who studied the same amount received the same grade. In the real

world, however, not everyone who studies the same amount will receive the same

grade. (Life is not fair.) However, a relationship can be present even if there is only

some degree of consistency so that, as the scores on one variable change, the scores on

the other variable tend to change in a consistent fashion. The degree of consistency in a

relationship is called its strength, and a less consistent relationship is called a weaker

relationship. For example, Table 2.2 shows two relationships between the number of

hours spent studying and the number of errors made on a test.

First look at Part A on the left side of Table 2.2. Again note the error scores paired

with each study-time score. Two aspects of the data produce a less consistent relation-

ship: (1) Not everyone who studies the same amount receives the same error score

(1 hour of study produced 13, 12, or 11 errors), and (2) sometimes the same error score

is paired with different studying scores (11 errors occur with 1 and 2 hours). Nonethe-

less, a reasonably clear pattern is still here in which one batch of similar error scores

YXY

X Y

Study Time Errors

Student in Hours on Test

1113

2112

3111

4211

5210

6210

7310

839

949

10 4 8

11 5 7

12 5 6

X Y

Study Time Errors

Student in Hours on Test

1113

2111

319

4212

5210

629

739

837

949

10 4 7

11 5 8

12 5 6

TABLE 2.2

Data Showing a Stronger

(A) and Weaker (B)

Relationship between

Study Time and Number

of Errors on a Test

The Logic of Research 17

tends to occur at one study time, but a different, lower batch of similar error scores

tends to occur at the next study time. Therefore, this is a reasonably strong relationship.

The data in Part B of Table 2.2 show a weaker relationship: (1) Each study time is

paired with a wider range of error scores (here 1 hour of study produced anywhere

between 13 and 9 errors), and (2) the same error scores occur with a greater variety of

study times (here 9 errors occur with 1, 2, 3 or 4 hours). These aspects produce greater

overlap between the error scores at one study time and those at the next, so there is

closer to the same batch of error scores at each study time. This produces a pattern of

decreasing errors that is harder to see.

Thus, the strength of a relationship is the extent to which one or close to one value

of tends to be consistently associated with only one value of . Conversely, in a

weaker relationship, a greater variety of scores is associated with each score and/or

the same score is paired with different scores.

REMEMBER A stronger relationship occurs the more that one group of simi-

lar scores is associated with one score and a different group of similar

scores is associated with the next score.

Two factors produce a relationship that is not perfectly consistent. First, extraneous

influences are operating. For example, say that distracting noises occurred while some-

one studied for 1 hour, but not when someone else studied for 1 hour. Because of this,

their studying might not be equally effective, resulting in different error scores paired

with the same study-time score. Second, individual differences are operating. Individ-

ual differences refer to the fact that no two individuals are identical because of differ-

ences in genetics, experience, intelligence, personality, and many other variables. Thus,

test performance will be influenced by a person’s intelligence, aptitude, and motiva-

tion. Because students exhibit individual differences in these characteristics, they will

each be influenced differently by the same amount of studying and so will produce dif-

ferent error scores at the same study-time score.

Theoretically, a relationship can have any degree of strength. However, perfectly

consistent relationships do not occur in real research because individual differences and

extraneous variables are always operating. Despite this, the less consistent relationships

back in Table 2.2 still support our original hypothesis about how nature operates: They

show that, at least to some degree, nature does relate studying and test errors as we pre-

dicted. Therefore, our next step would be to measure the degree to which nature does

this. Likewise, in any research, it is never enough to say that you have observed a rela-

tionship; you must also determine the strength of the relationship. (Later chapters dis-

cuss statistical procedures for describing the strength of a relationship.)

REMEMBER Research is concerned not only with the existence of a relation-

ship but also with the strength of the relationship.

When No Relationship Is Present At the other extreme, when there is no consis-

tent pattern between two variables, there is no relationship. For example, there is not

(I think) a relationship between the number of chocolate bars people consume each day

and the number of times they blink each minute. If we measure individuals on these

two variables, we might have the data shown in Table 2.3. Mentally draw in horizontal

lines so that you look at the batch of eye-blink scores paired with one chocolate score

at a time. Here there is no consistent change in the blink scores as the scores on the

chocolate variable change. Instead, very similar—but not identical—groups of blinking

scores are paired with each chocolate score. Because there is no relationship in this

sample, we do not have evidence that these variables are linked in nature.