Heiman G. Basic Statistics for the Behavioral Sciences
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CONTENTS
PREFACE xxii
1 Introduction to Statistics 1
GETTING STARTED 1
WHY IS IT IMPORTANT TO LEARN STATISTICS
(AND HOW DO YOU DO THAT?) 1
What Are Statistics? 1 Why Must I Learn Statistics? 1 What Do
Researchers Do with Statistics? 1 But I’m Not Interested in Research;
I Just Want to Help People! 2 But I Don’t Know Anything about Research! 2
What If I’m Not Very Good at Math? 2 What If I’m Not Very Good at
Statistics? 2 I Looked through the Book: Statistics Aren’t Written in English! 3
But, Those Formulas! 3 So All I Have to Do Is Learn How to Compute
the Answers? 3 What about Using a Computer to Do Statistics? 3
All Right, So How Do I Learn Statistics? 3
REVIEW OF MATHEMATICS USED IN STATISTICS 4
Basic Statistical Notation 4
Identifying Mathematical Operations 5 Determining the Order of
Mathematical Operations 5 Working with Formulas 5
Rounding Numbers 6
Transforming Scores 6
Proportions 7 Percents 7
Creating Graphs 7
PUTTING IT ALL TOGETHER 9
CHAPTER SUMMARY 9
KEY TERMS 10
REVIEW QUESTIONS 10
APPLICATION QUESTIONS 10
2 Statistics and the Research Process 12
GETTING STARTED 12
THE LOGIC OF RESEARCH 12
Samples and Populations 13
Obtaining Data by Measuring Variables 14
Understanding Relationships 15
The Strength of a Relationship 16 When No Relationship Is Present 17
CONTENTS
viii
A QUICK REVIEW 18
Graphing Relationships 18
APPLYING DESCRIPTIVE AND INFERENTIAL STATISTICS 20
Descriptive Statistics 20
Inferential Statistics 21
Statistics versus Parameters 21
UNDERSTANDING EXPERIMENTS AND CORRELATIONAL STUDIES 22
Experiments 22
The Independent Variable 22 Conditions of the Independent Variable 23
The Dependent Variable 23 Drawing Conclusions from Experiments 24
Correlational Studies 25
A QUICK REVIEW 26
A Word about Causality 26
THE CHARACTERISTICS OF SCORES 27
The Four Types of Measurement Scales 27
Continuous versus Discrete Scales 28
A QUICK REVIEW 29
STATISTICS IN PUBLISHED RESEARCH: USING STATISTICAL TERMS 30
PUTTING IT ALL TOGETHER 30
CHAPTER SUMMARY 31
KEY TERMS 32
REVIEW QUESTIONS 32
APPLICATION QUESTIONS 33
3 Frequency Distributions
and Percentiles 36
GETTING STARTED 36
NEW STATISTICAL NOTATION 36
WHY IS IT IMPORTANT TO KNOW ABOUT FREQUENCY DISTRIBUTIONS 37
SIMPLE FREQUENCY DISTRIBUTIONS 37
Presenting Simple Frequency in a Table 37
A QUICK REVIEW 38
Graphing a Simple Frequency Distribution 38
Bar Graphs 39 Histograms 40 Polygons 40
A QUICK REVIEW 42
TYPES OF SIMPLE FREQUENCY DISTRIBUTIONS 42
The Normal Distribution 42
Other Common Frequency Polygons 43
Labeling Frequency Distributions 45
A QUICK REVIEW 46
RELATIVE FREQUENCY AND THE NORMAL CURVE 47
Presenting Relative Frequency in a Table or Graph 48
Finding Relative Frequency Using the Normal Curve 49
Contents ix
A QUICK REVIEW 51
COMPUTING CUMULATIVE FREQUENCY AND PERCENTILE 51
Computing Cumulative Frequency 52
Computing Percentiles 52
Finding Percentile Using the Area Under the Normal Curve 53
A QUICK REVIEW 54
STATISTICS IN PUBLISHED RESEARCH: APA PUBLICATION RULES 54
A WORD ABOUT GROUPED FREQUENCY DISTRIBUTIONS 54
PUTTING IT ALL TOGETHER 55
Using the SPSS Appendix 55
CHAPTER SUMMARY 55
KEY TERMS 56
REVIEW QUESTIONS 56
APPLICATION QUESTIONS 57
INTEGRATION QUESTIONS 58
SUMMARY OF FORMULAS 59
4 Measures of Central Tendency:
The Mean, Median, and Mode 60
GETTING STARTED 60
NEW STATISTICAL NOTATION 60
WHY IS IT IMPORTANT TO KNOW ABOUT CENTRAL TENDENCY? 61
WHAT IS CENTRAL TENDENCY? 61
THE MODE 62
Uses of the Mode 63
THE MEDIAN 63
Uses of the Median 65
A QUICK REVIEW 65
THE MEAN 65
Uses of the Mean 66
A QUICK REVIEW 67
Comparing the Mean, Median, and Mode 67
TRANSFORMATIONS AND THE MEAN 69
DEVIATIONS AROUND THE MEAN 69
Using the Mean to Predict Scores 71
A QUICK REVIEW 72
DESCRIBING THE POPULATION MEAN 72
SUMMARIZING RESEARCH 73
Summarizing an Experiment 73
Graphing the Results of an Experiment 75
Line Graphs 75 Bar Graphs 76
A QUICK REVIEW 77
x Contents
Inferring the Relationship in the Population 78
STATISTICS IN PUBLISHED RESEARCH: USING THE MEAN 79
PUTTING IT ALL TOGETHER 79
Using the SPSS Appendix 79
CHAPTER SUMMARY 79
KEY TERMS 80
REVIEW QUESTIONS 80
APPLICATION QUESTIONS 81
INTEGRATION QUESTIONS 82
SUMMARY OF FORMULAS 83
5 Measures of Variability: Range,
Variance, and Standard Deviation 84
GETTING STARTED 84
NEW STATISTICAL NOTATION 84
WHY IS IT IMPORTANT TO KNOW ABOUT MEASURES OF VARIABILITY? 85
A QUICK REVIEW 86
THE RANGE 87
UNDERSTANDING THE VARIANCE AND STANDARD DEVIATION 87
The Sample Variance 88
The Sample Standard Deviation 89
A QUICK REVIEW 92
COMPUTING THE SAMPLE VARIANCE AND SAMPLE
STANDARD DEVIATION 93
Computing the Sample Variance 93
Computing the Sample Standard Deviation 94
A QUICK REVIEW 94
Mathematical Constants and the Standard Deviation 95
THE POPULATION VARIANCE AND THE POPULATION
STANDARD DEVIATION 95
Estimating the Population Variance and Population Standard Deviation 96
Computing the Estimated Population Variance and Standard Deviation 98
Interpreting the Estimated Population Variance and Standard Deviation 99
A QUICK REVIEW 99
A SUMMARY OF THE VARIANCE AND STANDARD DEVIATION 100
APPLYING THE VARIANCE AND STANDARD DEVIATION TO RESEARCH 101
Variability and Strength of a Relationship 101
Variability and Errors in Prediction 102
Accounting for Variance 103
STATISTICS IN PUBLISHED RESEARCH: REPORTING VARIABILITY 103
PUTTING IT ALL TOGETHER 104
Using the SPSS Appendix 104
Contents xi
CHAPTER SUMMARY 104
KEY TERMS 105
REVIEW QUESTIONS 105
APPLICATION QUESTIONS 106
INTEGRATION QUESTIONS 107
SUMMARY OF FORMULAS 108
6 z-Scores and the Normal
Curve Model 109
GETTING STARTED 109
NEW STATISTICAL NOTATION 109
WHY IS IT IMPORTANT TO KNOW ABOUT z-SCORES? 110
UNDERSTANDING z-SCORES 110
Computing z-Scores 112
Computing a Raw Score When z Is Known 113
A QUICK REVIEW 114
INTERPRETING z-SCORES USING THE z-DISTRIBUTION 115
USING z-SCORES TO COMPARE DIFFERENT VARIABLES 116
USING z-SCORES TO DETERMINE THE RELATIVE FREQUENCY
OF RAW SCORES 117
The Standard Normal Curve 118
Using the z-Table 120
A QUICK REVIEW 123
STATISTICS IN PUBLISHED RESEARCH: USING z-SCORES 123
USING z-SCORES TO DESCRIBE SAMPLE MEANS 124
The Sampling Distribution of Means 124
The Standard Error of the Mean 126
Computing a z-Score for a Sample Mean 127
Another Example Combining All of the Above 128
Describing the Relative Frequency of Sample Means 128
Summary of Describing a Sample Mean with a z-Score 129
A QUICK REVIEW 130
PUTTING IT ALL TOGETHER 130
Using the SPSS Appendix 130
CHAPTER SUMMARY 130
KEY TERMS 131
REVIEW QUESTIONS 131
APPLICATION QUESTIONS 132
INTEGRATION QUESTIONS 133
SUMMARY OF FORMULAS 134
xii Contents
7 The Correlation Coefficient 135
GETTING STARTED 135
NEW STATISTICAL NOTATION 135
WHY IS IT IMPORTANT TO KNOW ABOUT CORRELATION
COEFFICIENTS? 136
UNDERSTANDING CORRELATIONAL RESEARCH 136
Drawing Conclusions from Correlational Research 137
Distinguishing Characteristics of Correlational Analysis 138
Plotting Correlational Data: The Scatterplot 138
TYPES OF RELATIONSHIPS 139
Linear Relationships 139
Nonlinear Relationships 141
How the Correlation Coefficient Describes the Type of Relationship 142
STRENGTH OF THE RELATIONSHIP 142
Perfect Association 143
Intermediate Association 144
Zero Association 146
A QUICK REVIEW 147
THE PEARSON CORRELATION COEFFICIENT 147
A QUICK REVIEW 150
THE SPEARMAN RANK-ORDER CORRELATION COEFFICIENT 151
A QUICK REVIEW 153
THE RESTRICTION OF RANGE PROBLEM 153
STATISTICS IN PUBLISHED RESEARCH: CORRELATION
COEFFICIENTS 154
PUTTING IT ALL TOGETHER 155
Using the SPSS Appendix 155
CHAPTER SUMMARY 155
KEY TERMS 156
REVIEW QUESTIONS 156
APPLICATION QUESTIONS 157
INTEGRATION QUESTIONS 159
SUMMARY OF FORMULAS 159
8 Linear Regression 160
GETTING STARTED 160
NEW STATISTICAL NOTATION 160
WHY IS IT IMPORTANT TO KNOW ABOUT LINEAR REGRESSION? 161
UNDERSTANDING LINEAR REGRESSION 161
THE LINEAR REGRESSION EQUATION 163
Contents xiii
Computing the Slope 164
Computing the Y Intercept 166
Describing the Linear Regression Equation 166
Plotting the Regression Line 166
Computing Predicted Y Scores 167
A QUICK REVIEW 168
DESCRIBING THE ERRORS IN PREDICTION 168
Computing the Variance of the Y Scores around 169
Computing the Standard Error of the Estimate 170
A QUICK REVIEW 171
Interpreting the Standard Error of the Estimate 171
The Strength of a Relationship and Prediction Error 173
COMPUTING THE PROPORTION OF VARIANCE ACCOUNTED FOR 174
Using r to Compute the Proportion of Variance Accounted For 176
Applying the Proportion of Variance Accounted For 177
A QUICK REVIEW 178
A WORD ABOUT MULTIPLE CORRELATION AND REGRESSION 179
STATISTICS IN PUBLISHED RESEARCH: LINEAR REGRESSION 179
PUTTING IT ALL TOGETHER 179
Using the SPSS Appendix 180
CHAPTER SUMMARY 180
KEY TERMS 181
REVIEW QUESTIONS 181
APPLICATION QUESTIONS 181
INTEGRATION QUESTIONS 183
SUMMARY OF FORMULAS 184
9 Using Probability to Make
Decisions about Data 185
GETTING STARTED 185
NEW STATISTICAL NOTATION 185
WHY IS IT IMPORTANT TO KNOW ABOUT PROBABILITY? 186
THE LOGIC OF PROBABILITY 186
COMPUTING PROBABILITY 188
Creating Probability Distributions 188
Factors Affecting the Probability of an Event 188
A QUICK REVIEW 189
OBTAINING PROBABILITY FROM THE STANDARD NORMAL CURVE 189
Determining the Probability of Individual Scores 190
Determining the Probability of Sample Means 191
A QUICK REVIEW 192
RANDOM SAMPLING AND SAMPLING ERROR 193
Y¿
xiv Contents
DECIDING WHETHER A SAMPLE REPRESENTS A POPULATION 194
A QUICK REVIEW 196
Setting Up the Sampling Distribution 197
Identifying the Critical Value 198
Deciding if the Sample Represents the Population 199
Other Ways to Set up the Sampling Distribution 200
A QUICK REVIEW 202
PUTTING IT ALL TOGETHER 202
CHAPTER SUMMARY 203
KEY TERMS 203
REVIEW QUESTIONS 204
APPLICATION QUESTIONS 204
INTEGRATION QUESTIONS 205
SUMMARY OF FORMULAS 206
10 Introduction to Hypothesis Testing 207
GETTING STARTED 207
NEW STATISTICAL NOTATION 207
WHY IS IT IMPORTANT TO KNOW ABOUT THE z-TEST? 208
THE ROLE OF INFERENTIAL STATISTICS IN RESEARCH 208
SETTING UP INFERENTIAL PROCEDURES 209
Creating the Experimental Hypotheses 209
Designing a One-Sample Experiment 210
Creating the Statistical Hypotheses 211
The Alternative Hypothesis 211 The Null Hypothesis 212
A QUICK REVIEW 213
The Logic of Statistical Hypothesis Testing 213
PERFORMING THE z-TEST 215
Setting Up the Sampling Distribution for a Two-Tailed Test 215
Computing z 216
Comparing the Obtained z to the Critical Value 217
INTERPRETING SIGNIFICANT RESULTS 218
INTERPRETING NONSIGNIFICANT RESULTS 219
SUMMARY OF THE z-TEST 220
A QUICK REVIEW 221
THE ONE-TAILED TEST 221
The One-Tailed Test for Increasing Scores 221
The One-Tailed Test for Decreasing Scores 223
A QUICK REVIEW 223
STATISTICS IN PUBLISHED RESEARCH: REPORTING
SIGNIFICANCE TESTS 224
ERRORS IN STATISTICAL DECISION MAKING 224
Contents xv
Type I Errors: Rejecting H
0
When H
0
Is True 224
Type II Errors: Retaining H
0
When H
0
Is False 226
Comparing Type I and Type II Errors 227
Power 228
A QUICK REVIEW 229
PUTTING IT ALL TOGETHER 229
CHAPTER SUMMARY 230
KEY TERMS 231
REVIEW QUESTIONS 231
APPLICATION QUESTIONS 231
INTEGRATION QUESTIONS 233
SUMMARY OF FORMULAS 233
11 Performing the One-Sample t-Test
and Testing Correlation Coefficients 234
GETTING STARTED 234
WHY IS IT IMPORTANT TO KNOW ABOUT t-TESTS? 234
PERFORMING THE ONE-SAMPLE t-TEST 235
Computing t
obt
236
A QUICK REVIEW 237
The t-Distribution and Degrees of Freedom 238
Using the t-Tables 240
Interpreting the t-Test 240
The One-Tailed t-Test 241
Some Help When Using the t-Tables 242
A QUICK REVIEW 242
ESTIMATING BY COMPUTING A CONFIDENCE INTERVAL 243
Computing the Confidence Interval 244
A QUICK REVIEW 246
Summary of the One-Sample t-Test 246
STATISTICS IN PUBLISHED RESEARCH: REPORTING THE t-TEST 246
SIGNIFICANCE TESTS FOR CORRELATION COEFFICIENTS 247
Testing the Pearson r 247
The Sampling Distribution of r 249 Interpreting r 250
One-Tailed Tests of r 251
Testing the Spearman r
S
252
Summary of Testing a Correlation Coefficient 253
A QUICK REVIEW 253
MAXIMIZING THE POWER OF STATISTICAL TESTS 253
PUTTING IT ALL TOGETHER 255
Using the SPSS Appendix 255
CHAPTER SUMMARY 255
KEY TERMS 256
xvi Contents
REVIEW QUESTIONS 256
APPLICATION QUESTIONS 257
INTEGRATION QUESTIONS 258
SUMMARY OF FORMULAS 259
12 The Two-Sample t-Test 260
GETTING STARTED 260
NEW STATISTICAL NOTATION 260
WHY IS IT IMPORTANT TO KNOW ABOUT THE TWO-SAMPLE t-TEST? 261
UNDERSTANDING THE TWO-SAMPLE EXPERIMENT 261
THE INDEPENDENT-SAMPLES t-TEST 262
Statistical Hypotheses for the Independent-Samples t-Test 262
The Sampling Distribution for the Independent-Samples t-Test 263
Computing the Independent-Samples t-Test 264
Estimating the Population Variance 264 Computing the Standard Error of
the Difference 265 Computing t
obt
266
A QUICK REVIEW 267
Interpreting the Independent-Samples t-Test 267
Confidence Interval for the Difference between Two s 268
Performing One-Tailed Tests with Independent Samples 269
SUMMARY OF THE INDEPENDENT-SAMPLES t-TEST 270
A QUICK REVIEW 270
THE RELATED-SAMPLES t-TEST 271
The Logic of Hypotheses Testing in the Related-Samples t-Test 272
STATISTICAL HYPOTHESES FOR THE RELATED-SAMPLES t-TEST 273
Computing the Related-Samples t-Test 274
Interpreting the Related-Samples t-Test 275
Computing the Confidence Interval for
D
277
Performing One-Tailed Tests with Related Samples 277
SUMMARY OF THE RELATED-SAMPLES t-TEST 278
A QUICK REVIEW 278
DESCRIBING THE RELATIONSHIP IN A TWO-SAMPLE EXPERIMENT 279
Graphing the Results of a Two-Sample Experiment 279
Measuring Effect Size in the Two-Sample Experiment 280
Effect Size Using Cohen’s d 280 Effect Size Using Proportion of Variance
Accounted For 282
STATISTICS IN PUBLISHED RESEARCH: THE TWO-SAMPLE EXPERIMENT 283
PUTTING IT ALL TOGETHER 284
Using the SPSS Appendix 284
CHAPTER SUMMARY 284
KEY TERMS 285
REVIEW QUESTIONS 285
APPLICATION QUESTIONS 286
Contents xvii