Heiman G. Basic Statistics for the Behavioral Sciences

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SIXTH EDITION

Basic Statistics

for the Behavioral

Sciences

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Okay, so you’re taking a course in statistics. You probably are curious—and maybe a little

anxious—about what it involves. After all, statistics are math! Well, relax. Statistics do not

require that you be a math wizard. Students in the behavioral sciences throughout the

world take a course like the one you are about to take, and they get through it. In fact, sta-

tistics can be fun! They are challenging, there is an elegance to their logic, and you can do

nifty things with them. So, keep an open mind, be prepared to do a little work, and you’ll

be amazed by what happens. You’ll find that statistics are interesting and educational, they

help you to think logically, and they make behavioral research much easier to understand.

In this chapter we first deal with some common misconceptions that students have

about statistics. Then we’ll review the basic math that you’ll be using.

WHY IS IT IMPORTANT TO LEARN STATISTICS

(AND HOW DO YOU DO THAT?)

Here are some frequently asked questions that will teach you something about statistics

and your statistics course.

What Are Statistics? The term statistics is often used as a shorthand for statistical

procedures. These are formulas and calculations developed by statisticians that psy-

chologists and other behavioral researchers employ when “analyzing” the results of

their research. Also, some of the answers that we compute are called statistics.

Why Must I Learn Statistics? Statistics are an integral part of psychology and

other behavioral sciences, so statistics and statistical concepts are used every day.

Therefore, to understand your chosen field of study, you must understand statistics.

You’ve already experienced this if you’ve ever read a published research article—you

probably skipped the section titled “Results.” Now you won’t have to.

What Do Researchers Do with Statistics? Statistics are tools used in research.

They are needed because the behavioral sciences are based on empirical research.

The word empirical means that knowledge is obtained through observation and measure-

ment, and behavioral research measures behaviors. Such measurement results in num-

bers, or scores. These scores obtained in research are the data. (By the way, the word

Introduction to Statistics

GETTING STARTED

Your goals in this chapter are to learn

■

Why researchers learn statistics.

■

The general purpose of statistical procedures.

■

The basic math that’s needed.

2 CHAPTER 1 / Introduction to Statistics

data is plural, so say “the data are . . .”.) For example, to study intelligence, researchers

measure the IQ scores of different individuals; to study memory, we examine data that re-

flect the number of things that people remember or forget; to study social interactions,

we measure the distance people stand apart or their anxiety when meeting someone. And

so on. Thus, any study typically produces a very large batch of scores that must be made

manageable and meaningful. At this point, statistics are applied because they help us to

make sense out of the data. The procedures we will discuss do this in four ways. First,

some procedures organize the scores so that we can more clearly see any patterns in the

data. Often this simply involves creating a table or graph. Second, other statistics summa-

rize the scores. We don’t need to examine each of the hundreds of scores that may be

obtained in a study. Instead, a summary—such as the average score—allows us to quickly

and easily understand the general characteristics of the data. Third, statistics communi-

cate the results of a study. Researchers have created techniques and rules for this and,

because everyone uses the same rules, it is much easier for us to communicate with each

other, especially in published research reports. Finally, statistics are used to conclude

what the data indicate. All behavioral research is designed to answer a question about a

behavior and, ultimately, we must decide what the data tell us about that behavior.

But I’m Not Interested in Research; I Just Want to Help People! Even if you

are not interested in becoming a researcher, statistics are necessary for comprehending

other people’s research. Let’s say that you become a therapist or counselor. You hear of

a new therapy that says the way to “cure” people of some psychological problem is to

scare the living daylights out of them. This sounds crazy but what is important is the

research that does or does not support this therapy. As a responsible professional, you

would evaluate the research supporting this therapy before you would use it. You could

not do so without understanding statistics.

But I Don’t Know Anything about Research! This book is written for students

who have not yet studied how to conduct research. When we discuss each statistic, we also

discuss simple studies that employ the procedure, and this will be enough. Later, when you

study research methods, you will know the appropriate statistical procedures to use.

What if I’m Not Very Good at Math? This is not a math course. We will discuss

some research tools that happen to involve mathematical operations. But it is simple

math: adding, subtracting, multiplying, dividing, finding square roots, and drawing

simple graphs. Also, we will continuously review the math operations as they are

needed. Best of all, statisticians have already developed the statistics we’ll discuss, so

we won’t be deriving formulas, performing proofs, or doing other “mystery” math. We

will simply learn when to use the procedure that statisticians say is appropriate for a

given situation, then compute the answer and then determine what it tells us about the

data. (Eventually you’ll understand the tables on the inside of the front cover that sum-

marize which procedures are used in which type of study.)

What if I’m Not Very Good at Statistics? This course is not a test of whether

you should change your college major! First, there are not all that many procedures to

learn, and these fancy sounding “procedures” include such simple things as computing

an average or drawing a graph. Second, researchers usually do not memorize the

formulas. (For quick reference, at the end of each chapter in this book is a list of the

formulas discussed.) Finally, statistics are simply a tool used in research, just like a

wrench is a tool used to repair automobile engines. A mechanic does not need to be an

expert wrencher who loves to wrench, and you do not need be an expert statistician

Why is It Important to Learn Statistics (and How Do You Do That?) 3

who loves statistics. Rather, in the same way that a mechanic must understand how to

correctly use a wrench, your goal is to be able to correctly use statistics.

I Looked through the Book: Statistics Aren’t Written in English! Statistics

do involve many strange symbols and unfamiliar terms. But these are simply the shorthand

“code” for communicating statistical results and for simplifying statistical formulas. A ma-

jor part of learning statistics is merely learning the code. Think of it this way: To understand

research you must speak the language, and you are about to learn the language of statistics.

Once you speak this language, much of the mystery surrounding statistics evaporates.

But, Those Formulas! What makes some formulas appear difficult is that they are

written in a code that communicates a sequence of operations: You first might square the

scores, then add them together, then divide by some other number, and so on. However,

most chapters begin with a section called “New Statistical Notation,” which explains the

symbols that you’ll encounter, and then each formula is presented with example data and

step-by-step instructions on how to work through it. (Each example involves an unrealis-

tically small number of scores, although real research involves large numbers of scores.)

There are also in-chapter “Quick Reviews” where you can practice what you have read

and at the end of each chapter are additional practice problems. With practice the formu-

las become easy, and then the rest of the mystery surrounding statistics will evaporate.

So All I Have to Do Is Learn How to Compute the Answers? No! Statistical

procedures are a tool that you must learn to apply. Ultimately you want to make sense

of data, and to do that, you must compute the appropriate statistic and then correctly

interpret it. More than anything else, you need to learn when and why to use each pro-

cedure and how to interpret its answer. Be sure to put as much effort into this as you

put into the calculations.

What about Using a Computer to Do Statistics? At first glance, you might

think that this book was written before the invention of computers. However, we focus

on using formulas to compute answers “by hand” because that is the only way for you

to understand statistics. (Having a computer magically produce an answer might sound

attractive, but with no idea how the answer was produced, you’d be overwhelmed.)

However, researchers usually do use computers to compute statistics. Therefore, Ap-

pendix B explains the basics of how to use “SPSS,” which is one of the leading statisti-

cal computer programs. Recently, the new version 17 was released and, for some

reason, the name was changed to PASW. However, we’ll still refer to it as SPSS. The

instructions in Appendix B are appropriate for version 17, and where needed, different

instructions are provided so they are appropriate for earlier versions also. Even if your

instructor cannot include this program in your statistics course, eventually you will

want to learn it or one like it. Then you can compute the statistics discussed in this book

quickly and accurately. (You’ll love it—it is soooo easy!)

Recognize that, although SPSS is a wizardlike program, it cannot select the appro-

priate statistical procedure for you nor interpret the answer. You really must learn when

to use each statistic and what the answer means.

All Right, So How Do I Learn Statistics? Avoid these pitfalls when studying

statistics:

■

Don’t skim the material. You do not speak the language yet, so you must translate

the terminology and symbols into things that you understand, and that takes time and

effort.

4 CHAPTER 1 / Introduction to Statistics

■

Don’t try to “cram” statistics. You won’t learn anything (and your brain will melt).

Instead, work on statistics a little every day, digesting the material in bite-sized

pieces. This is the most effective—and least painful—way to learn statistics.

■

Don’t skip something if it seems difficult. Concepts build upon one another, and

material in one chapter sets up a concept in a later chapter so that things are bite

sized for you. Go back and review when necessary. (The beginning of each chapter

lists what you should understand from previous chapters.)

An effective strategy in this course includes

■

Learn (memorize), understand, and use the terminology. You are learning a new

language, and you can do this only by practicing it. In this book, terminology printed

in boldface type is important, as is anything labeled REMEMBER. Use the glossary.

Making and studying flashcards is a good idea. At the end of a chapter, use the “Chapter

Summary” and “Key Terms” to improve your knowledge of the terminology.

■

Learn when and why to use each procedure. You’ll find this information at the begin-

ning of chapters in the section “Why Is It Important to Know About . . . ?”

■

Practice using the formulas. Master the formulas at each step because they often

reappear later as part of more complicated formulas. Test yourself with the in-chapter

“Quick Reviews.” If you cannot understand or perform a review, do not proceed! Go

back—you missed something.

■

Complete all of the problems at the end of each chapter, because the only way to

learn statistics is to do statistics. The “Review Questions” are for practicing the

terminology, definitions and concepts presented in the chapter. The “Application

Questions” give you practice computing and interpreting the answers from each

procedure. However, you must not forget about procedures from past chapters.

Therefore, starting in Chapter 3, you’ll see “Integration Questions,” which help you

to combine the information from different chapters. Remember, your task is to build

a complete mental list of all of the different procedures in this book. Eventually

you’ll be in faced with picking the one procedure that’s appropriate for a study from

all those we’ve discussed. (Often this occurs on your final exam.)

■

For all end-of-chapter problems, the answers to the odd-numbered problems are in

Appendix D and your instructor has the answers to the even-numbered problems.

Make a serious attempt at solving the problem first and only then look at the answer.

(This is the practice test before the real test.)

■

Pay extra attention to the sections titled “Interpreting the . . .” and “Statistics in

Published Research” so that you learn how to interpret statistical results, draw

conclusions, and understand what researchers do—and do not—say. These really

are the most important aspects of learning statistics.

REVIEW OF MATHEMATICS USED IN STATISTICS

The remainder of this chapter reviews the math used in performing statistical proce-

dures. As you’ll see, there are accepted systems for statistical notation, for rounding an

answer, for transforming scores, and for creating graphs.

Basic Statistical Notation

Statistical notation refers to the standardized code for symbolizing the mathematical

operations performed in the formulas and for symbolizing the answers we obtain.

Review of Mathematics Used in Statistics 5

Identifying Mathematical Operations

We write formulas in statistical notation

so that we can apply them to any data. We usually use the symbol or to stand for

each individual score obtained in a study. When a formula says to do something to , it

means to do it to all the scores you are calling . When a formula says to do something

to , it means to do it to all the scores called .

The mathematical operations we’ll perform are simple ones. Addition is indicated by

the plus sign, and subtraction is indicated by the minus sign. We read from left to right,

so is read as “ minus .” (I said this was simple!) This order is important be-

cause , for example, is , but is . With subtraction, pay attention to

what is subtracted from what and whether the answer is positive or negative.

We indicate division by forming a fraction, such as . The number above the di-

viding line is called the numerator, and the number below the line is called the denomi-

nator. Always express fractions as decimals, dividing the denominator into the

numerator. (After all, 1/2 equals .5, not 2!)

Multiplication is indicated in one of two ways. We may place two components next

to each other: means “multiply times .” Or we may indicate multiplication using

parentheses: 4(2) and (4)(2) both mean “multiply 4 times 2.”

The symbol means square the score, so if is 4, is 16. Conversely, means

“find the square root of ,” so is 2. (The symbol also means “use your calculator.”)

Determining the Order of Mathematical Operations Statistical formulas of-

ten call for a series of mathematical steps. Sometimes the steps are set apart by paren-

theses. Parentheses mean “the quantity,” so always find the quantity inside the

parentheses first and then perform the operations outside of the parentheses on that

quantity. For example, (2)(4  3) indicates to multiply 2 times “the quantity 4 plus 3.”

So first add, which gives (2)(7), and then multiply to get 14.

A square root sign also operates on “the quantity,” so always compute the quantity

inside the square root sign first. Thus, means find the square root of the quan-

tity 2  7; so becomes , which is 3.

Most formulas are giant fractions. Pay attention to how far the dividing line is drawn

because the length of a dividing line determines the quantity that is in the numerator

and the denominator. For example, you might see a formula that looks like this:

The longest dividing line means you should divide the square root of 64 into the quan-

tity in the numerator. The dividing line in the fraction in the numerator is under only

the 6, so first divide 6 by 3, which is 2. Then add 14, for a total of 16. In the denomina-

tor, the square root of 64 is 8. After dividing, the final answer is 2.

If you become confused in reading a formula, remember that there is an order of prece-

dence of mathematical operations. Often this is summarized with PEMDAS meaning that,

unless otherwise indicated, first compute inside any Parentheses, then compute Exponents

(squaring and square roots), then Multiply or Divide, and finally, Add or Subtract. Thus,

for (2)(4)  5, multiply 2 times 4 first and then add 5. For 2

 3

, square first, which

gives 4  9, which is then 13. On the other hand, (2  3)

is 5

, which is 25.

Working with Formulas We use a formula to find an answer, and we have sym-

bols that stand for that answer. For example, in the formula , the stands

for the answer we will obtain. The symbol for the unknown answer is always isolated

BB 5 AX 1 K

1 14

264

2 1 14

264

5 2

2922 1 7

22 1 7

24X

2XX

YXXY

X>Y

264 2 101610 2 4

YXX 2 Y

6 CHAPTER 1 / Introduction to Statistics

on one side of the equal sign, but we will know the numbers to substitute for the

symbols on the other side of the equal sign. For example, to find , say that ,

, and . In working any formula, the first step is to copy the formula and

then rewrite it, replacing the symbols with their known values. Thus, start with

Filling in the numbers gives

Rewrite the formula after performing each mathematical operation. Above, multiplication

takes precedence over addition, so multiply and then rewrite the formula as

After adding,

For simple procedures, you may have an urge to skip rewriting the formula after each

step. Don’t! That’s a good way to introduce errors.

Rounding Numbers

Close counts in statistics, so you must carry out calculations to the appropriate num-

ber of decimal places. Usually, you must “round off” your answer. The rule is this:

Always carry out calculations so that your final answer after rounding has two more

decimal places than the original scores. Usually, we have whole-number scores

(e.g., 2 and 11) so the final answer contains two decimal places. But say the original

scores contain one decimal place (e.g., 1.4 and 12.3). Here the final answer should

contain three decimal places.

However, do not round off at each intermediate step in a formula; round off only at

the end! Thus, if the final answer is to contain two decimal places, round off your

intermediate answers to at least three decimal places. Then after you’ve completed all

calculations, round off the final answer to two decimal places.

To round off a calculation use the following rules:

If the number in the next decimal place is 5 or greater, round up. For example,

to round to two decimal places, an answer of 2.366 is rounded to 2.370, which

becomes 2.37.

If the number in the next decimal place is less than 5, round down: an answer of

3.524 is rounded to 3.520, which becomes 3.52.

We add zeroes to the right of the decimal point to indicate the level of precision we

are using. For example, rounding 4.996 to two decimal places produces 5, but to show

we used the precision of two decimal places, we report it as 5.00.

REMEMBER Round off your final answer to two more decimal places than

are in the original scores.

Transforming Scores

Many statistical procedures are nothing more than elaborate transformations. A

transformation is a mathematical procedure for systematically converting a set of

B 5 47

B 5 44 1 3

B 5 411121 3

B 5 AX 1 K

K 5 3X 5 11

A 5 4B

Review of Mathematics Used in Statistics 7

scores into a different set of scores. Adding 5 to each score is a transformation, or

converting “number correct” into “percent correct” is a transformation.

We transform data for one of two reasons. First, transformations make scores easier

to work with. For example, if all of the scores contain a decimal, we might multiply

every score by 10 to eliminate the decimals. Second, transformations make different

kinds of scores comparable. For example, if you obtained 8 out of 10 on a statistics test

and 75 out of 100 on an English test, it would be difficult to compare the two scores.

However, if you transformed each grade to percent correct, you could then directly

compare performance on the two tests.

In statistics, we rely heavily on transformations to proportions and percents.

Proportions A proportion is a decimal number between 0 and 1 that indicates a

fraction of the total. To transform a number to a proportion, simply divide the number

by the total. If 4 out of 10 people pass an exam, then the proportion of people passing

the exam is 4/10, which equals .4. Or, if you score 6 correct on a test out of a possible

12, the proportion you have correct is 6/12, which is .5.

We can also work in the opposite direction from a known proportion to find the num-

ber out of the total it represents. Here, multiply the proportion times the total. Thus, to

find how many questions out of 12 you must answer correctly to get .5 correct, multi-

ply .5 times 12, and voilà, the answer is 6.

Percents We can also transform a proportion into a percent. A percent (or percent-

age) is a proportion multiplied by 100. Above, your proportion correct was .5, so you

had (.5)(100) or 50% correct. Altogether, to transform the original test score of 6 out of

12 to a percent, first divide the score by the total to find the proportion and then multi-

ply by 100. Thus, (6/12)(100) equals 50%.

To transform a percent back into a proportion, divide the percent by 100 (above,

50/100 equals .5). Altogether, to find the test score that corresponds to a certain per-

cent, transform the percent to a proportion and then multiply the proportion times the

total number possible. Thus, to find the score that corresponds to 50% of 12, transform

50% to the proportion, which is .5, and then multiply .5 times 12. So, 50% of 12 is

equal to (50/100)(12), which is 6.

Recognize that a percent is a whole unit: Think of 50% as 50 of those things called

percents. On the other hand, a decimal in a percent is a proportion of one percent. Thus,

.2% is .2, or two-tenths, of one percent, which is .002 of the total.

Creating Graphs

One type of statistical procedure is none other than plotting graphs. In case it’s been a

long time since you’ve drawn one, recall that the horizontal line across the bottom of a

graph is the axis, and the vertical line at the left-hand side is the axis. (Draw the

axis so that it is about 60 to 75% of the length of the axis.) Where the two axes in-

tersect is always labeled as a score of zero on and a score of zero on . On the axis,

scores become larger positive scores as you move to the right. On the axis, scores be-

come larger positive scores as you move upward.

Say that we measured the height and weight of several people. We decide to place

weight on the axis and height on the axis. (How to decide this is discussed later.)

We plot the scores as shown in Figure 1.1. Notice that because the lowest height score

is 63, the lowest label on the axis is also 63. The symbol // in the axis indicates that

we cut out the part between 0 and 63. We do this with either axis when there is a large

gap between 0 and the lowest score we are plotting.

XYX