The New Statistical Analysis of Data

The Nature of the Book This book is a text for a first course in statistical concepts and methods. It introduces the analysis of data and statistical inference and explains various methods in enough detail so that the student can apply them. Little mathematical background is required; only high school algebra is used. No mathematical proof is given in the body of the text, although algebraic demonstrations are given in appendices at the ends of some chapters. The exposition is based on logic, verbal explanations, figures, and numerical examples. The verbal and conceptual levels are higher than the mathematical leveL The concepts and methods of statistical analysis are illustrated by more than 100 interesting real-life data sets. Some examples are taken from daily life; many deal with the behavioral sciences, some with business, the health sciences, the physical sciences, and engineering. The exercises are of varying degrees of difficulty. This book is suitable for undergraduates in many majors and for grad­ uate students in the health sciences, behavioral sciences, and education. It has grown out of our experience over many years of teaching such courses. An earlier text by T. W. Anderson and S. L. Sclove, The Statistical Analysis of Data, had similar objectives and was of a similar nature.

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.