The New Statistical Analysis of Data

I: Introduction.- 1—The Nature of Statistics.- 1.1 Some Examples of the Use of Statistics.- Political Polls.- The Polio Vaccine Trial.- Smoking and Health.- Project “Head Start”.- “A Minority of One Versus a Unanimous Majority”.- Deciding Authorship.- 1.2 Basic Concepts of Statistics.- Experimental and Nonexperimental Research.- Populations and Samples.- Descriptive and Inferential Statistics.- Planning Statistical Investigations.- Exercises.- II: Descriptive Statistics.- 2 — Organization of Data.- 2.1 Kinds of Variables: Scales.- Categorical Variables.- Numerical Variables.- Scales*.- 2.2 Organization of Categorical Data.- Frequencies.- Frequency Graphs.- 2.3 Organization of Numerical Data.- Discrete Data.- Continuous Data.- Frequency Distributions for Continuous Data.- Exercises.- 3 — Measures of Location.- 3.1 The Mode.- Definition and Interpretation of the Mode.- Mode of Grouped Data.- 3.2 The Median and Other Percentiles.- The Median.- Quartiles.- Deciles Percentiles and Other Quantiles.- 3.3 The Mean.- Definition and Interpretation of the Mean.- Use of Notation.- Calculating the Mean from a Frequency Distribution.- The Proportion as a Mean.- Other Properties of the Mean.- Effects of Change of Scale.- 3.4 Choosing Among Measures of Location Shape of Distributions.- Appendices.- Appendix 3A Computing the Median and Other Quantiles of Grouped Continuous Data.- Appendix 3B Rules for Summation.- Appendix 3C Change of Scale.- Appendix 3D Significant Digits.- Exercises.- 4 — Measures of Variability.- 4.1 Ranges.- The Range.- The Interquartile Range.- 4.2 The Mean Deviation*.- 4.3 The Standard Deviation.- Definitions.- Reasons for Dividing by One Less than the Sample Size.- Interpreting the Standard Deviation.- 4.4 Formulas for the Standard Deviation.- Computing Formula.- Calculating the Standard Deviation from a Frequency Distribution.- Effects of Change of Scale.- 4.5 Some Uses of Location and Dispersion Measures Together.- Standard Scores.- Box-and-Whisker Plots.- Appendices.- Appendix 4A Proofs of Some Algebraic Principles.- Appendix 4B Adjusting Data to Maintain Computational Accuracy.- Exercises.- 5 — Summarizing Multivariate Data: Association Between Numerical Scales.- 5.1 Association of Two Numerical Variables.- Scatter Plots.- Other Information Revealed by Scatter Plots.- The Correlation Coefficient.- Rank Correlation.- 5.2 More than Two Variables.- Profiles.- Correlation Matrix.- Appendices.- Appendix 5A Computational Form for the Covariance.- Appendix 5B Change of Scale.- Exercises.- 6 — Summarizing Multivariate Data: Association Between Categorical Variables.- 6.1 Two-by-Two Frequency Tables.- Organization of Data into Two-by-Two Tables.- Calculation of Percentages.- Interpretation of Frequencies.- 6.2 Larger Two-Way Frequency Tables.- Organization of Data for Two Categorical Variables.- Interpretation of Frequencies.- 6.3 Three Categorical Variables.- Organization of Data for Three Yes-No Variables.- Larger Three-Way Frequency Tables.- 6.4 Effects of a Third Variable.- Association and Interpretation.- Independence in Subtables.- Similar Association in Subtables.- Reversal of Association in Subtables.- Hidden Relationships.- Exercises.- III: Probability.- 7 — Basic Ideas of Probability.- 7.1 Intuitive Examples of Probability.- Physical Devices Which Approximate Randomness.- The Draft Lottery.- Probability and Everyday Life.- 7.2 Probability and Statistics.- 7.3 Probability in Terms of Equally Likely Cases.- 7.4 Events and Probabilities in General Terms.- Outcomes Events and Probabilities.- Addition of Probabilities of Mutually Exclusive Events.- Addition of Probabilities.- 7.5 Interpretation of Probability: Relation to Real Life.- 7.6 Conditional Probability.- 7.7 Independence.- 7.8 Random Sampling; Random Numbers.- Random Devices.- Random Numbers.- Sampling with Replacement.- Sampling without Replacement.- 7.9 Bayes’ Theorem*.- Bayes’ Theorem in a Simplified Case.- Examples 260 Use of Subjective Prior Probabilities in Bayes’ Theorem.- Exercises.- 8 — Probability Distributions.- 8.1 Random Variables.- 8.2 Cumulative Probability.- 8.3 The Mean and Variance of a Probability Distribution.- The Mean of a Discrete Random Variable.- The Variance of a Discrete Random Variable.- The Mean and Variance of a Continuous Random Variable.- 8.4 Uniform Distributions.- The Discrete Uniform Distribution.- The Continuous Uniform Distribution.- 8.5 The Family of Normal Distributions.- The Normal Distributions.- Different Normal Distributions.- The Standard Normal Distribution.- Other Normal Distributions.- Exercises.- 9 — Sampling Distributions.- 9.1 Sampling from a Population.- Random Samples.- Sampling Distributions.- Independence of Random Variables.- Sampling from a Probability Distribution.- 9.2 Sampling Distributions of a Sum and of a Mean.- Sampling Distribution of a Sum.- Sampling Distribution of the Sample Mean.- 9.3 The Binomial Distribution.- Sampling Distribution of the Number of Heads.- Proportion of Heads in Bernoulli Trials.- The Mean and Variance of the Binomial Probability Distribution.- 9.4 The Law of Averages (Law of Large Numbers)*.- 9.5 The Normal Distribution of Sample Means.- The Central Limit Theorem.- Normal Approximation to the Binomial Distribution.- Appendix 9A The Correction for Continuity.- Exercises.- IV: Statistical Inference.- 10 — Using a Sample to Estimate Characteristics of One Population.- 10.1 Estimation of a Mean by a Single Number.- 10.2 Estimation of Variance and Standard Deviation.- 10.3 An Interval of Plausible Values for a Mean.- Confidence Intervals when the Standard Deviation is Known.- Confidence Intervals when the Standard Deviation is Estimated.- 10.4 Estimation of a Proportion.- Point Estimation of a Proportion.- Interval Estimation of a Proportion.- 10.5 Estimation of a Median.- Point Estimation of a Median.- Interval Estimation of a Median*.- 10.6 Paired Measurements.- Mean of a Population of Differences.- Matched Samples.- 10.7 Importance of Size of Population Relative to Sample Size*.- Appendix 10A The Continuity Adjustment.- Exercises.- 11 — Answering Questions about Population Characteristics.- 11.1 Testing a Hypothesis About a Mean.- An Example and Terminology.- Hypothesis Testing Procedures.- Deciding Whether a Population Mean Differs from a Given Value.- Relation of Two-Tailed Tests to Confidence Intervals.- Validity Conditions.- 11.2 Errors and Power.- Types of Error.- Probability of a Type I Error.- Probability of a Type II Error and Power.- 11.3 Testing Hypotheses About a Mean when the Standard Deviation Is Unknown.- 11.4 P Values: Another Way to Report Tests of Significance.- P Values for Two-Tailed Tests.- P Values when the Population Standard Deviation Is Unknown.- 11.5 Testing Hypotheses About a Proportion.- Testing Hypotheses About the Probability of a Success.- Example of a Two-Tailed Test.- 11.6 Testing Hypotheses About a Median: The Sign Test*.- 11.7 Paired Measurements.- Testing Hypotheses About the Mean of a Population of Differences.- Testing the Hypothesis of Equality of Proportions.- Appendix 11A The Continuity Correction.- Exercises.- 12 — Differences Between Populations.- 12.1 Comparison of Two Independent Sample Means When the Population Standard Deviations Are Known.- One-Tailed Tests.- Two-Tailed Tests.- Confidence Intervals.- Validity Conditions.- 12.2 Comparison of Two Independent Sample Means When the Population Standard Deviations are Unknown but Treated as Equal.- Confidence Intervals.- 12.3 Comparison of Two Independent Sample Means When the Population Standard Deviations Are Unknown and Not Treated as Equal*.- 12.4 Comparison of Two Independent Sample Proportions.- Hypothesis Tests.- Confidence Intervals.- 12.5 The Sign Test for a Difference in Locations*.- Appendix 12A The Continuity Adjustment.- Exercises.- 13 — Variability in One Population and in Two Populations.- 13.1 Variability in One Population.- The Sampling Distribution of the Sum of Squared Deviations.- Testing the Hypothesis that the Variance Equals a Given Number.- Confidence Intervals for the Variance*.- 13.2 Variability in Two Populations.- The Sampling Distribution of the Ratio of Two Sample Variances.- Testing the Hypothesis of Equality of Two Variances 512 Confidence Intervals for the Ratio of Two Variances*.- Exercises.- V: Statistical Methods for Other Problems.- 14 — Inference on Categorical Data.- 14.1 Tests of Goodness of Fit.- Two Categories—Dichotomous Data.- Any Number of Categories.- Combining Categories.- 14.2 Chi-Square Tests of Independence.- Two-by-Two Tables.- Two-Way Tables in General.- Combining Categories.- 14.3 Measures of Association.- The Phi Coefficient.- A Coefficient Based on Prediction.- A Coefficient Based on Ordering*.- Appendix 14A The Continuity Adjustment.- Exercises.- 15 — Simple Regression Analysis.- 15.1 Functional Relationship.- 15.2 Statistical Relationship.- 15.3 Least-Squares Estimates.- 15.4 Statistical Inference for ?.- 15.5 The Correlation Coefficient: A Measure of Linear Relationship.- The Bivariate Normal Distribution and Test of Significance for a Correlation.- Exercises.- 16 — Comparison of Several Populations.- 16.1 One-Way Analysis of Variance.- A Complete Example.- The Algebra of ANOVA.- An Example with Unequal Sample Sizes.- More About the Analysis of Variance.- 16.2 Which Groups Differ from Which and by How Much?.- Comparing Two Means.- Comparing More Than Two Means.- 16.3 Analysis of Variance of Ranks*.- Exercises.- 17 — Sampling from Populations: Sample Surveys.- 17.1 Simple Random Sampling.- 17.2 Stratified Random Sampling.- Two Strata.- More than Two Strata.- 17.3 Cluster Sampling.- 17.4 Systematic Sampling with a Random Start.- 17.5 Systematic Subsampling with Random Starts.- Exercises.- Answers to Selected Exercises.- References.- Appendices.- Appendix I Table of the Standard Normal Distribution.- Appendix II Table of Binomial Probabilities.- Appendix IV Percentage Points of Chi-Square Distributions.- Appendix VI Table of Random Digits.