Fundamentals of Statistical Reasoning in Education

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: r 2 125     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    When    Is Known: The One-Sample z Test 199     11.1 Testing a Hypothesis About   : 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 (  ) 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    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    When    Is Not Known: The One-Sample t Test 235     13.1 Reality:    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    When    Is Not Known 246     13.10 Summary 247     Chapter 14 Comparing the Means of Two Populations: Independent Samples 253     14.1 From One Mu (  ) to Two 253     14.2 Statistical Hypotheses 254     14.3 The Sampling Distribution of Differences Between Means 255     14.4 Estimating   X1 X2 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   1       2 262     14.8 Appraising the Magnitude of a Difference: Measures of Effect Size for X1   X2 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   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 Multiple t 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   i       j 308     16.10 One-Way ANOVA: Summarizing the Steps 309     16.11 Estimating the Strength of the Treatment Effect: Effect Size (     2) 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    to    322     17.2 The Sampling Distribution of r When    0 322     17.3 Testing the Statistical Hypothesis That    0 324     17.4 An Example 324     17.5 In Brief: Student   s t Distribution and 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    0 329     17.10 Interval Estimation of    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   2: A Measure of Discrepancy Between Expected and Observed Frequencies 340     18.4 The Sampling Distribution of   2 341     18.5 Completion of the Voter Survey Problem: The   2 Goodness-of-Fit Test 343     18.6 The   2 Test of a Single Proportion 344     18.7 Interval Estimate of a Single Proportion 345     18.8 When There Are Two Variables: The   2 Test of Independence 347     18.9 Finding Expected Frequencies in the Two-Variable Case 348     18.10 Calculating the Two-Variable   2 350     18.11 The   2 Test of Independence: Summarizing the Steps 351     18.12 The 2 x 2 Contingency Table 352     18.13 Testing a Difference Between Two Proportions 353     18.14 The Independence of Observations 353     18.15   2 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