Chapter 1 Introduction 1

1.1 Why Statistics? 1

1.2 Descriptive Statistics 2

1.3 Inferential Statistics 3

1.4 The Role of Statistics in Educational Research 4

1.5 Variables and Their Measurement 5

1.6 Some Tips on Studying Statistics 8

PART 1 DESCRIPTIVE STATISTICS 13

Chapter 2 Frequency Distributions 14

2.1 Why Organize Data? 14

2.2 Frequency Distributions for Quantitative Variables 14

2.3 Grouped Scores 15

2.4 Some Guidelines for Forming Class Intervals 17

2.5 Constructing a Grouped-Data Frequency Distribution 18

2.6 The Relative Frequency Distribution 19

2.7 Exact Limits 21

2.8 The Cumulative Percentage Frequency Distribution 22

2.9 Percentile Ranks 23

2.10 Frequency Distributions for Qualitative Variables 25

2.11 Summary 26

Chapter 3 Graphic Representation 34

3.1 Why Graph Data? 34

3.2 Graphing Qualitative Data: The Bar Chart 34

3.3 Graphing Quantitative Data: The Histogram 35

3.4 Relative Frequency and Proportional Area 39

3.5 Characteristics of Frequency Distributions 41

3.6 The Box Plot 44

3.7 Summary 45

Chapter 4 Central Tendency 52

4.1 The Concept of Central Tendency 52

4.2 The Mode 52

4.3 The Median 53

4.4 The Arithmetic Mean 54

4.5 Central Tendency and Distribution Symmetry 57

4.6 Which Measure of Central Tendency to Use? 59

4.7 Summary 59

Chapter 5 Variability 66

5.1 Central Tendency Is Not Enough: The Importance of Variability 66

5.2 The Range 67

5.3 Variability and Deviations From the Mean 68

5.4 The Variance 69

5.5 The Standard Deviation 70

5.6 The Predominance of the Variance and Standard Deviation 71

5.7 The Standard Deviation and the Normal Distribution 72

5.8 Comparing Means of Two Distributions: The Relevance of Variability 73

5.9 In the Denominator: n Versus n −1 75

5.10 Summary 76

Chapter 6 Normal Distributions and Standard Scores 81

6.1 A Little History: Sir Francis Galton and the Normal Curve 81

6.2 Properties of the Normal Curve 82

6.3 More on the Standard Deviation and the Normal Distribution 82

6.4 z Scores 84

6.5 The Normal Curve Table 87

6.6 Finding Area When the Score Is Known 88

6.7 Reversing the Process: Finding Scores When the Area Is Known 91

6.8 Comparing Scores From Different Distributions 93

6.9 Interpreting Effect Size 94

6.10 Percentile Ranks and the Normal Distribution 96

6.11 Other Standard Scores 97

6.12 Standard Scores Do Not “Normalize” a Distribution 98

6.13 The Normal Curve and Probability 98

6.14 Summary 99

Chapter 7 Correlation 106

7.1 The Concept of Association 106

7.2 Bivariate Distributions and Scatterplots 106

7.3 The Covariance 111

7.4 The Pearson r 117

7.5 Computation of r: The Calculating Formula 118

7.6 Correlation and Causation 120

7.7 Factors Influencing Pearson r 122

7.8 Judging the Strength of Association: r² 125

7.9 Other Correlation Coefficients 127

7.10 Summary 127

Chapter 8 Regression and Prediction 134

8.1 Correlation Versus Prediction 134

8.2 Determining the Line of Best Fit 135

8.3 The Regression Equation in Terms of Raw Scores 138

8.4 Interpreting the Raw-Score Slope 141

8.5 The Regression Equation in Terms of z Scores 141

8.6 Some Insights Regarding Correlation and Prediction 142

8.7 Regression and Sums of Squares 145

8.8 Residuals and Unexplained Variation 147

8.9 Measuring the Margin of Prediction Error: The Standard Error of Estimate 148

8.10 Correlation and Causality (Revisited) 152

8.11 Summary 153

PART 2 INFERENTIAL STATISTICS 163

Chapter 9 Probability and Probability Distributions 164

9.1 Statistical Inference: Accounting for Chance in Sample Results 164

9.2 Probability: The Study of Chance 165

9.3 Definition of Probability 166

9.4 Probability Distributions 168

9.5 The OR/addition Rule 169

9.6 The AND/Multiplication Rule 171

9.7 The Normal Curve as a Probability Distribution 172

9.8 “So What?”—Probability Distributions as the Basis for Statistical Inference 174

9.9 Summary 175

Chapter 10 Sampling Distributions 179

10.1 From Coins to Means 179

10.2 Samples and Populations 180

10.3 Statistics and Parameters 181

10.4 Random Sampling Model 181

10.5 Random Sampling in Practice 183

10.6 Sampling Distributions of Means 184

10.7 Characteristics of a Sampling Distribution of Means 185

10.8 Using a Sampling Distribution of Means to Determine Probabilities 188

10.9 The Importance of Sample Size (n) 191

10.10 Generality of the Concept of a Sampling Distribution 193

10.11 Summary 193

Chapter 11 Testing Statistical Hypotheses About μ When σ Is Known: The One-Sample z Test 199

11.1 Testing a Hypothesis About μ: Does “Homeschooling” Make a Difference? 199

11.2 Dr. Meyer’s Problem in a Nutshell 200

11.3 The Statistical Hypotheses: H₀ and H₁ 201

11.4 The Test Statistic z 202

11.5 The Probability of the Test Statistic: The p Value 203

11.6 The Decision Criterion: Level of Significance (α) 204

11.7 The Level of Significance and Decision Error 207

11.8 The Nature and Role of H₀ and H₁ 209

11.9 Rejection Versus Retention of H₀ 209

11.10 Statistical Significance Versus Importance 210

11.11 Directional and Nondirectional Alternative Hypotheses 212

11.12 The Substantive Versus the Statistical 214

11.13 Summary 215

Chapter 12 Estimation 222

12.1 Hypothesis Testing Versus Estimation 222

12.2 Point Estimation Versus Interval Estimation 223

12.3 Constructing an Interval Estimate of μ 224

12.4 Interval Width and Level of Confidence 226

12.5 Interval Width and Sample Size 227

12.6 Interval Estimation and Hypothesis Testing 228

12.7 Advantages of Interval Estimation 230

12.8 Summary 230

Chapter 13 Testing Statistical Hypotheses About μ When σ Is Not Known: The One-Sample t Test 235

13.1 Reality: σ Often Is Unknown 235

13.2 Estimating the Standard Error of the Mean 236

13.3 The Test Statistic t 237

13.4 Degrees of Freedom 238

13.5 The Sampling Distribution of Student’s t 239

13.6 An Application of Student’s t 242

13.7 Assumption of Population Normality 244

13.8 Levels of Significance Versus p Values 244

13.9 Constructing a Confidence Interval for μ When σ Is Not Known 246

13.10 Summary 247

Chapter 14 Comparing the Means of Two Populations: Independent Samples 253

14.1 From One Mu (μ) to Two 253

14.2 Statistical Hypotheses 254

14.3 The Sampling Distribution of Differences Between Means 255

14.4 Estimating σ_x̄1-x̄2 257

14.5 The t Test for Two Independent Samples 259

14.6 Testing Hypotheses About Two Independent Means: An Example 260

14.7 Interval Estimation of μ₁ − μ₂ 262

14.8 Appraising the Magnitude of a Difference: Measures of Effect Size for − 264

14.9 How Were Groups Formed? The Role of Randomization 268

14.10 Statistical Inferences and Nonstatistical Generalizations 269

14.11 Summary 270

Chapter 15 Comparing the Means of Dependent Samples 278

15.1 The Meaning of “Dependent” 278

15.2 Standard Error of the Difference Between Dependent Means 279

15.3 Degrees of Freedom 281

15.4 The t Test for Two Dependent Samples 281

15.5 Testing Hypotheses About Two Dependent Means: An Example 283

15.6 Interval Estimation of μ_D 286

15.7 Summary 287

Chapter 16 Comparing the Means of Three or More Independent Samples: One-Way Analysis of Variance 294

16.1 Comparing More Than Two Groups: Why Not Multiplet Tests? 294

16.2 The Statistical Hypotheses in One-Way ANOVA 295

16.3 The Logic of One-Way ANOVA: An Overview 296

16.4 Alison’s Reply to Gregory 299

16.5 Partitioning the Sums of Squares 300

16.6 Within-Groups and Between- Groups Variance Estimates 303

16.7 The F Test 304

16.8 Tukey’s “HSD” Test 306

16.9 Interval Estimation of μ_i − μ_j 308

16.10 One-Way ANOVA: Summarizing the Steps 309

16.11 Estimating the Strength of the Treatment Effect: Effect Size (ω²) 311

16.12 ANOVA Assumptions (and Other Considerations) 312

16.13 Summary 313

Chapter 17 Inferences about the Pearson Correlation Coefficient 322

17.1 From μ to ρ 322

17.2 The Sampling Distribution of r When ρ = 0 322

17.3 Testing the Statistical Hypothesis That ρ = 0 324

17.4 An Example 324

17.5 In Brief: Student’s t Distribution and the Regression Slope (b) 326

17.6 Table E 326

17.7 The Role of n in the Statistical Significance of r 328

17.8 Statistical Significance Versus Importance (Again) 329

17.9 Testing Hypotheses Other Than ρ = 0 329

17.10 Interval Estimation of ρ 330

17.11 Summary 332

Chapter 18 Making Inferences From Frequency Data 338

18.1 Frequency Data Versus Score Data 338

18.2 A Problem Involving Frequencies: The One-Variable Case 339

18.3 χ²: A Measure of Discrepancy Between Expected and Observed Frequencies 340

18.4 The Sampling Distribution of χ² 341

18.5 Completion of the Voter Survey Problem: The χ² Goodness-of-Fit Test 343

18.6 The χ² Test of a Single Proportion 344

18.7 Interval Estimate of a Single Proportion 345

18.8 When There Are Two Variables: The χ² Test of Independence 347

18.9 Finding Expected Frequencies in the Two-Variable Case 348

18.10 Calculating the Two-Variable χ² 350

18.11 The χ² Test of Independence: Summarizing the Steps 351

18.12 The 2 × 2 Contingency Table 352

18.13 Testing a Difference Between Two Proportions 353

18.14 The Independence of Observations 353

18.15 χ2 and Quantitative Variables 354

18.16 Other Considerations 355

18.17 Summary 355

Chapter 19 Statistical “Power” (and How to Increase It) 363

19.1 The Power of a Statistical Test 363

19.2 Power and Type II Error 364

19.3 Effect Size (Revisited) 365

19.4 Factors Affecting Power: The Effect Size 366

19.5 Factors Affecting Power: Sample Size 367

19.6 Additional Factors Affecting Power 368

19.7 Significance Versus Importance 369

19.8 Selecting an Appropriate Sample Size 370

19.9 Summary 373 Epilogue A Note on (Almost) Assumption-Free Tests 379

References 380

Appendix A Review of Basic Mathematics 382

A.1 Introduction 382

A.2 Symbols and Their Meaning 382

A.3 Arithmetic Operations Involving Positive and Negative Numbers 383

A.4 Squares and Square Roots 383

A.5 Fractions 384

A.6 Operations Involving Parentheses 385

A.7 Approximate Numbers, Computational Accuracy, and Rounding 386

Appendix B Answers to Selected End-of-Chapter Problems 387

Appendix C Statistical Tables 408

Glossary 421

Index 427

Useful Formulas 433

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