Introduction to Statistics in Criminal Justice and Criminology

Arthur J. Lurigio, PhD, is Professor of Psychology and Criminal Justice and Criminology at Loyola University Chicago. A distinguished scholar with more than 500 publications, he has received numerous awards recognizing his contributions to criminal justice research, mental health, and applied scholarship. Michael Perry, PhD, is Senior Lecturer of Mathematics and Statistics at Loyola University. He teaches a range of statistics courses and has published research across polymerization, public health, and quantitative modeling, bringing applied mathematical expertise to criminal justice education. Nathan M. Lutz, PhD, is a pediatric psychologist and Clinical Assistant Professor at Nationwide Children’s Hospital in Columbus, Ohio. His research focuses on improving outcomes for youth involved in child welfare and enhancing measurement-based behavioral health care. George K. Thiruvathukal, PhD, is Professor and Chairperson of Computer Science at Loyola University Chicago and Visiting Computer Scientist at Argonne National Laboratory. Author or co-author of more than 200 publications and 6 books, his research spans high-performance computing, distributed systems, and artificial intelligence.

• About the Authors xiiiAcknowledgements xv1 Introduction 11.1 Statistics 11.2 Types of Statistics: Descriptive and Inferential 11.3 Basic Statistical Concepts and Terminology 21.3.1 Population, Element, Census, and Sample 21.3.2 Data and Variables 31.3.3 Variable Types 41.3.4 Levels of Variables and Scales 41.3.5 Nominal Scales 41.3.6 Ordinal Scales 51.3.7 Interval Scales 61.3.8 Ratio Scales 61.3.9 Likert Scales 61.3.10 The Hierarchy of Measurement Scales 71.4 Dependent and Independent Variables 101.5 Practical Application of Statistics 111.6 Introduction to SAS 111.7 Summary 121.8 Exercises 121.9 Answers to Exercises 14References 162 Organizing and Describing Data 172.1 The Jail Report 172.2 Organizing and Describing Data 172.3 Frequency Distributions 182.3.1 Absolute Frequency Distributions 182.3.2 Relative Frequency Distributions 192.3.3 Relative Frequency Distributions 202.3.4 Cumulative Relative Frequency Distributions 222.3.5 Interval Frequency Distributions 232.3.6 Frequency Distributions in SAS 262.4 Visual Techniques 272.4.1 Bar Graphs 282.4.2 Bar Graphs in SAS 302.4.3 Pie Charts 312.4.4 Pie Charts in SAS 332.4.5 Histograms 332.4.6 Histograms in SAS 332.4.7 Polygons 352.4.8 Polygons in SAS 362.5 Properties of Distributions 372.5.1 Central Tendency 372.5.2 Variability 372.5.3 Skewness 372.5.4 Kurtosis 392.6 Summary 412.7 Exercises 422.8 Answers to Exercises 43References 473 Comparative Statistics 483.1 Re-electing the County Sheriff 483.1.1 Comparative Statistics 483.1.2 Crime Rates 493.1.3 Crime-Specific Rates 513.1.4 Percent Change 523.2 Trend Analyses 543.3 Summary 613.4 Importing Data from a Database in SAS 623.5 Exercises 623.6 Answers to Exercises 65References 684 Descriptive Statistics: Measures of Central Tendency and Variability 694.1 Analysis of Domestic Violence Cases 694.2 Measures of Central Tendency 694.3 Central Tendency 704.4 The Mean 704.4.1 Calculating the Mean 704.4.2 Computing the Mean from Frequency Distributions 724.4.3 Estimating the Mean of an Interval Frequency Distribution 734.5 The Median 754.5.1 Calculating the Median 754.5.2 Computing the Median from Frequency Distributions 764.5.3 Estimating the Median of an Interval Frequency Distribution 764.6 The Mode 764.6.1 Calculating the Mode 764.6.2 Calculating the Mode from a Frequency Distribution 774.6.3 Comparing the Mean, Median, and Mode: Skewness 774.7 When to Use Measures of Central Tendency: A Final Word 834.7.1 Comparing Mean, Median, and Mode 844.8 Characteristics of the Mean and Median 844.9 Variability 854.10 The Range 864.11 Measures of Deviation 874.12 Variance 874.13 Standard Deviation 884.13.1 Standard Deviation from Samples 894.13.1.1 When to Use the Sample or Population Standard Deviation 894.13.1.2 What Type of Data Should You Use When You Calculate a Standard Deviation? 894.14 Calculating Central Tendencies and Variability in SAS 904.15 Summary 934.16 Exercises 934.17 Answers to Exercises 955 Normal Distributions 985.1 Theoretical Distributions 985.1.1 Characterizing Shapes 985.2 Normal Distribution PDF 995.3 Binomial Distribution PMF 1005.4 The Standard Normal Distribution 1045.4.1 Z-Scores 1055.5 Properties of Z-Scores 1065.5.1 A Unitless Metric 1065.5.2 Indication of Relative Standing 1075.5.3 Standardization 1095.5.4 Z-Scores and Probability 1095.5.5 Standard Normal in SAS 1145.6 Summary 1165.7 Exercises 1165.8 Answers to Exercises 1176 Hypothesis Testing: z- andt-Tests 1196.1 Introduction 1196.2 Hypothesis Testing, One Sample z 1206.2.1 Comparing Means in Hypothesis Testing 1206.2.2 Sampling Distributions 1206.2.3 Confidence Intervals (CI) 1246.2.4 The Central Limit Theorem (CLT) 1266.2.5 The Sampling Distribution of the Mean and z-Scores 1266.2.6 The Single Sample z-Test 1276.2.7 Hypothesis Testing 1286.2.8 Level of Significance (Alpha) 1296.2.9 Type I and Type II Errors 1306.2.10 Type II Error 1316.2.11 Steps in Testing for Statistical Significance 1316.3 Hypothesis Testing, One Sample t 1316.3.1 Single Sample t-Test 1316.3.2 Hypothesis Testing 1326.3.3 Degrees of Freedom 1326.3.4 The Sampling Distribution of t 1336.3.5 One Sample t-Test and Confidence Intervals in SAS 1346.3.6 Independent Samples (or Unpaired) t-Test 1356.3.6.1 Assumptions 1376.3.6.2 The Steps in Testing the Hypothesis That Two Teaching Methods Do Not Differ in Their Effectiveness 1386.3.7 Independent Sample t-Test in SAS 1396.3.8 t-Test for Related (Paired) Samples 1396.3.8.1 Paired t-Test in SAS 1426.4 Summary 1436.5 Exercises 1446.6 Answers to Exercises 1457 Analyzing Categorical Data: Chi-Square Test 1527.1 Comparing Proportions 1527.1.1 Comparing Proportions–One Sample 1527.1.2 Confidence Interval for One-Sample Proportions 1527.1.3 Hypothesis Testing for One-Sample Proportions 1537.2 Comparing Proportions – Two Samples 1557.2.1 Confidence Interval for Two-Sample Proportions 1557.2.2 Hypothesis Test for Two-Sample Proportions 1567.3 The Chi-Square Test 1567.3.1 Chi-Square Goodness of Fit Test 1577.3.2 Chi-Square Goodness of Fit Test in SAS 1597.3.3 Chi-Square Test of Independence 1597.3.3.1 Hypotheses for Chi-Square Test of Independence 1627.3.4 Chi-Square Test of Independence in SAS 1647.4 Summary 1667.5 Exercises 1667.6 Answers to Exercises 167Reference 1758 Correlation Coefficient 1768.1 Relationships Between Variables 1768.1.1 Pearson Correlation Coefficient 1778.1.2 Relationships Between Variables 1778.1.3 Scatter Diagrams 1778.1.4 Meaning of Covariation 1798.1.5 Interpretation of Covariance 1798.1.6 The Calculation of Covariance 1808.1.7 Finding the Correlation Coefficient Covariance Formula 1818.1.8 Computational Formula 1828.1.9 Pearson r as a General Measure of Association 1838.1.10 The Range and Direction of Pearson’s r 1848.1.11 The Strength of Pearson’s r 1848.1.12 A Picture of Perfection 1858.2 Interpreting the Meaning and Significance of r 1858.2.1 The Effect of Outlying Scores on Pearson’s r 1868.2.2 Testing the Statistical Significance of r 1868.2.3 Using SAS to Calculate Correlations 1878.2.4 Nonlinear Relationships 1888.2.5 A Scatterplot Depicting the Nonlinear Relationship Between Anxiety and the Percentage of Target Center Hits During Weapons Qualifications 1888.2.5.1 Important Messages About Correlation Coefficient 1888.3 Summary 1898.4 Exercises 1938.5 Answers to Exercises 1939 Linear Regression and Prediction 1969.1 The Concept of Prediction 1969.2 Basic Assumptions and Terminology 1979.2.1 A Linear Regression Problem 1979.2.2 The Formula for a Straight Line 2009.2.3 Finding the Regression Line 2019.3 Linear Regression Using SAS 2039.3.1 The REG Procedure 2049.3.2 The Accuracy of Prediction 2059.3.3 Linear Regression with Z-Scores 2079.3.4 Taking Another Look at Prediction Error 2099.4 Multiple Linear Regression 2119.4.1 Multiple Regression in SAS 2169.4.1.1 Regression to the Mean 2169.5 Summary 2189.6 Exercises 2189.7 Answers to Exercises 219Reference 22210 Analysis of Variance 22310.1 Introduction 22310.1.1 Inmate Fights on Tier 3 22310.2 A Return to Variance 22410.2.1 The Context For ANOVA 22410.2.2 ANOVA: Basic Concepts and Terminology 22510.2.3 ANOVA Terminology and Computational Terms 22610.2.4 A Simple ANOVA Problem 22810.3 Steps in Hypothesis Testing 22910.3.1 Computing ANOVA for the Tier 3 Study 23010.3.2 Hypotheses 23010.3.3 Steps in ANOVA Testing 23110.3.4 The Research Question 23110.3.5 ANOVA Calculations 23110.3.6 Conclusion 23210.3.7 Scores by Treatment Group 23210.3.8 Assumptions for Conducting ANOVA 23410.3.9 After ANOVA: Comparing Group Means 23510.3.10 Computing the Amount of Explained Variance 23610.4 ANOVA in SAS 23710.5 Summary 23910.6 Exercises 23910.7 Answers to Exercises 24111 Survival Analysis 24911.1 Introduction 24911.1.1 Recidivism Analyses 24911.1.2 Survival Analysis 24911.1.3 Censored Data 25011.2 Survival Function 25011.2.1 The Kaplan–Meier Method 25211.2.2 Using SAS to Estimate the Survival Function 25511.2.3 Log-Rank Test 25711.2.4 Log-Rank Test in SAS 25911.2.5 Cox Proportional Model 25911.2.6 Using SAS to Calculate the Cox Proportional Hazard Model 26311.3 Summary 26311.4 Exercises 26311.5 Answers to Exercises 265Appendices 268Index 295