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Applied Statistics I

Applied Statistics I
Basic Bivariate Techniques

Third Edition

January 2020 | 648 pages | SAGE Publications, Inc
Rebecca M. Warner’s bestselling Applied Statistics: From Bivariate Through Multivariate Techniques has been split into two volumes for ease of use over a two-course sequence. Applied Statistics I: Basic Bivariate Techniques, Third Edition is an introductory statistics text based on chapters from the first half of the original book. 

The author's contemporary approach reflects current thinking in the field, with its coverage of the "new statistics" and reproducibility in research. Her in-depth presentation of introductory statistics follows a consistent chapter format, includes some simple hand-calculations along with detailed instructions for SPSS, and helps students understand statistics in the context of real-world research through interesting examples. Datasets are provided on an accompanying website.

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Applied Statistics I + Applied Statistics II: Basic Bivariate Techniques, Third Edition 
Bundle Volume I and II ISBN: 978-1-0718-1337-9
An R Companion for Applied Statistics I: Basic Bivariate Techniques + Applied Statistics I

Bundle ISBN: 978-1-0718-1325-6

About the Author
1. Evaluating Numerical Information

Guidelines for Numeracy

Source Credibility

Message Content

Evaluating Generalizability

Making Causal Claims

Quality Control Mechanisms in Science

Biases of Information Consumers

Ethical Issues in Data Collection and Analysis

Lying with Graphs and Statistics

Degrees of Belief


2. Basic Research Concepts

Types of Variables

Independent and Dependent Variables

Typical Research Questions

Conditions for Causal Inference

Experimental Research Design

Nonexperimental Research Design

Quasi-Experimental Research Designs

Other Issues in Design and Analysis

Choice of Statistical Analysis (Preview)

Populations and Samples: Ideal Versus Actual Situations

Common Problems in Interpretation of Results

Appendix 2A: More About Levels of Measurement

Appendix 2B: Justification for the Use of Likert and Other Rating Scales as Quantitative Variables (in Some Situations)

3. Frequency Distribution Tables

Use of Frequency Tables for Data Screening

Frequency Tables for Categorical Variables

Elements of Frequency Tables

Using SPSS to Obtain a Frequency Table

Mode, Impossible Score Values, and Missing Values

Reporting Data Screening for Categorical Variables

Frequency Tables for Quantitative Variables

Frequency Tables for Categorical Versus Quantitative Variables

Reporting Data Screening for Quantitative Variables

What We Hope to See in Frequency Tables for Categorical Variables

What We Hope to See in Frequency Tables for Quantitative Variables


Appendix 3A: Getting Started in IBM SPSS® Version 25

Appendix 3B: Missing Values in Frequency Tables

Appendix 3C: Dividing Scores Into Groups or Bins

4. Descriptive Statistics

Questions about Quantitative Variables


Sample Median

Sample Mean (M)

An Important Characteristic of M: The Sum of Deviations From M = 0

Disadvantage of M: It is Not Robust Against Influence of Extreme Scores

Behavior of Mean, Median, and Mode in Common Real-World Situations

Choosing Among Mean, Median, and Mode

Using SPSS to Obtain Descriptive Statistics for a Quantitative Variable

Minimum, Maximum, and Range: Variation among Scores

The Sample Variance s2

Sample Standard Deviation (s or SD)

How a Standard Deviation Describes Variation Among Scores in a Frequency Table

Why Is There Variance?

Reports of Descriptive Statistics in Journal Articles

Additional Issues in Reporting Descriptive Statistics


Appendix 4A: Order of Arithmetic Operations

Appendix 4B: Rounding

5. Graphs: Bar Charts, Histograms, and Boxplots

Pie Charts for Categorical Variables

Bar Charts for Frequencies of Categorical Variables

Good Practice for Construction of Bar Charts

Deceptive Bar Graphs

Histograms for Quantitative Variables

Obtaining a Histogram Using SPSS

Describing and Sketching Bell-Shaped Distributions

Good Practices in Setting up Histograms

Boxplot (Box and Whiskers Plot)

Telling Stories About Distributions

Uses of Graphs in Actual Research

Data Screening: Separate Bar Charts or Histograms for Groups

Use of Bar Charts to Represent Group Means

Other Examples


6. The Normal Distribution and z Scores

Locations of Individual Scores in Normal Distributions

Standardized or z Scores

Converting z Scores Back Into X Units

Understanding Values of z

Qualitative Description of Normal Distribution Shape

More Precise Description of Normal Distribution Shape

Areas Under the Normal Distribution Curve Can Be Interpreted as Probabilities

Reading Tables of Areas for the Standard Normal Distribution

Dividing the Normal Distribution Into Three Regions: Lower Tail, Middle, Upper Tail

Outliers Relative to a Normal Distribution

Summary of First Part of Chapter

Why We Assess Distribution Shape

Departure from Normality: Skewness

Another Departure from Normality: Kurtosis

Overall Normality

Practical Recommendations for Preliminary Data Screening and Descriptions of Scores for Quantitative Variables

Reporting Information About Distribution Shape, Missing Values, Outliers, and Descriptive Statistics for Quantitative Variables


Appendix 6A: The Mathematics of the Normal Distribution

Appendix 6B: How to Select and Remove Outliers in SPSS

Appendix 6C: Quantitative Assessments of Departure From Normality

Appendix 6D: Why Are Some Real-World Variables Approximately

7. Sampling Error and Confidence Intervals
Descriptive Versus Inferential Uses of Statistics

Notation for Samples Versus Populations

Sampling Error and the Sampling Distribution for Values of M

Prediction Error

Sample Versus Population (Revisited)

The Central Limit Theorem: Characteristics of the Sampling Distribution of M

Factors That Influence Population Standard Error (sM)

Effect of N on Value of the Population Standard Error

Describing the Location of a Single Outcome for M Relative to Population Sampling Distribution (Setting Up a z Ratio)

What We Do When s Is Unknown

The Family of t Distributions

Tables for t Distributions

Using Sampling Error to Set Up a Confidence Interval

How to Interpret a Confidence Interval

Empirical Example: Confidence Interval for Body Temperature

Other Applications for Confidence Intervals

Error Bars in Graphs of Group Means


8. The One-Sample t test: Introduction to Statistical Significance Tests

Significance Tests as Yes/No Questions About Proposed Values of Population Means

Stating a Null Hypothesis

Selecting an Alternative Hypothesis

The One-Sample t Test

Choosing an Alpha (a) Level

Specifying Reject Regions on the Basis of a, Halt, and df

Questions for the One-Sample t Test

Assumptions for the Use of the One-Sample t Test

Rules for the Use of NHST

First Analysis of Mean Driving Speed Data (Using a Nondirectional Test)

SPSS Analysis: One-Sample t Test for Mean Driving Speed (Using a Nondirectional or Two-Tailed Test)

“Exact” p Values

Reporting Results for a Two-tailed One-Sample t Test

Second Analysis of Driving Speed Data Using a One-Tailed or Directional Test

Reporting Results for a One-tailed One-Sample t Test

Advantages and Disadvantages of One-Tailed Tests

Traditional NHST Versus New Statistics Recommendations

Things You Should Not Say About p Values


9. Issues in Significance Tests: Effect Size, Statistical Power, and Decision Errors
Beyond p Values

Cohen’s d: An Effect Size Index

Factors that Affect the Size of t Ratios

Statistical Significance Versus Practical Importance

Statistical Power

Type I and Type II Decision Errors

Meanings of “Error”

Use of NHST in Exploratory Versus Confirmatory Research

Inflated Risk for Type I Decision Error for Multiple Tests

Interpretation of Null Outcomes

Interpretation of Statistically Significant Outcomes

Understanding Past Research

Planning Future Research

Guidelines for Reporting Results

What You Cannot Say


Appendix 9A: Further Explanation of Statistical Power

10. Bivariate Pearson Correlation
Research Situations Where Pearson’s r Is Used

Correlation and Causal Inference

How Sign and Magnitude of r Describe an X, Y Relationship

Setting Up Scatterplots

Most Associations Are Not Perfect

Different Situations in Which r = .00

Assumptions for Use of Pearson’s r

Preliminary Data Screening for Pearson’s r

Effect of Extreme Bivariate Outliers

Research Example

Data Screening for Research Example

Computation of Pearson’s r

How Computation of Correlation Is Related to Pattern of Data Points in the Scatterplot

Testing the Hypothesis That p0 = 0

Reporting Many Correlations and Inflated Risk for Type I Error

Obtaining Confidence Intervals for Correlations

Pearson’s r and r2 as Effect Sizes and Partition of Variance

Statistical Power and Sample Size for Correlation Studies

Interpretation of Outcomes for Pearson’s r

SPSS Example: Relationship Survey

Results Sections for One and Several Pearson’s r Values

Reasons to Be Skeptical of Correlations


Appendix 10A: Nonparametric Alternatives to Pearson’s r

Appendix 10B: Setting Up a 95% CI for Pearson’s r by Hand

Appendix 10C: Testing Significance of Differences Between Correlations

Appendix 10D: Some Factors That Artifactually Influence Magnitude of r

Appendix 10E: Analysis of Nonlinear Relationships

Appendix 10F: Alternative Formula to Compute Pearson’s r

11. Bivariate Regression
Research Situations Where Bivariate Regression Is Used

New Information Provided by Regression

Regression Equations and Lines

Two Versions of Regression Equations

Steps in Regression Analysis

Preliminary Data Screening

Formulas for Bivariate Regression Coefficients

Statistical Significance Tests for Bivariate Regression

Confidence Intervals for Regression Coefficients

Effect Size and Statistical Power

Empirical Example Using SPSS: Salary Data

SPSS Output: Salary Data

Results Section: Hypothetical Salary Data

Plotting the Regression Line: Salary Data

Using a Regression Equation to Predict Score for Individual (Joe’s Heart Rate Data)

Partition of Sums of Squares in Bivariate Regression

Why Is There Variance (Revisited)?

Issues in Planning a Bivariate Regression Study

Plotting Residuals

Standard Error of the Estimate


Appendix 11A: Review: How to Graph a Line From Two Points Obtained From an Equation

Appendix 11B: OLS Derivation of Equation for Regression Coefficients

Appendix 11C: Alternative Formula for Computation of Slope

Appendix 11D: Fully Worked Example: Deviations and SS

12. The Independent-Samples t Test
Research Situations Where the Independent-Samples t Test Is Used

Hypothetical Research Example

Assumptions for Use of Independent-Samples t Test

Preliminary Data Screening: Evaluating Violations of Assumptions and Getting to Know Your Data

Computation of Independent-Samples t Test

Statistical Significance of Independent-Samples t Test

Confidence Interval Around M1 – M2

SPSS Commands for Independent-Samples t Test

SPSS Output for Independent-Samples t Test

Effect Size Indexes for t

Factors that Influence the Size of t

Results Section

Graphing Results: Means and CIs

Decisions About Sample Size for the Independent-Samples t Test

Issues in Designing a Study


Appendix 12A: A Nonparametric Alternative to the Independent-Samples t Test

13. One-Way Between-Subjects Analysis of Variance
Research Situations Where One-Way ANOVA Is Used

Questions in One-Way Between-S ANOVA

Hypothetical Research Example

Assumptions and Data Screening for One-Way ANOVA

Computations for One-Way Between-S ANOVA

Patterns of Scores and Magnitudes of SSbetween and SSwithin

Confidence Intervals for Group Means

Effect Sizes for One-Way Between-S ANOVA

Statistical Power Analysis for One-Way Between-S ANOVA

Planned Contrasts

Post Hoc or “Protected” Tests

One-Way Between-S ANOVA in SPSS

Output From SPSS for One-Way Between-S ANOVA

Reporting Results From One-Way Between-S ANOVA

Issues in Planning a Study


Appendix 13A: ANOVA Model and Division of Scores Into Components

Appendix 13B: Expected Value of F When H0 Is True

Appendix 13C: Comparison of ANOVA and t Test

Appendix 13D: Nonparametric Alternative to One-Way Between-S ANOVA: Independent-Samples Kruskal-Wallis Test

14. Paired-Samples t Test
Independent- Versus Paired-Samples Designs

Between-S and Within-S or Paired-Groups Designs

Types of Paired Samples

Hypothetical Study: Effects of Stress on Heart Rate

Review: Data Organization for Independent Samples

New: Data Organization for Paired Samples

A First Look at Repeated-Measures Data

Calculation of Difference (d) Scores

Null Hypothesis for Paired-Samples t Test

Assumptions for Paired-Samples t Test

Formulas for Paired-Samples t Test

SPSS Paired-Samples t Test Procedure

Comparison Between Results for Independent-Samples and Paired-Samples t Tests

Effect Size and Power

Some Design Problems in Repeated-Measures Analyses

Results for Paired-Samples t Test: Stress and Heart Rate

Further Evaluation of Assumptions


Appendix 14A: Nonparametric Alternative to Paired-Samples t: Wilcoxon Signed Rank Test

15. One-Way Repeated-Measures Analysis of Variance

Null Hypothesis for Repeated-Measures ANOVA

Preliminary Assessment of Repeated-Measures Data

Computations for One-Way Repeated-Measures ANOVA

Use of SPSS Reliability Procedure for One-Way Repeated-Measures ANOVA

Partition of SS in Between-S Versus Within-S ANOVA

Assumptions for Repeated-Measures ANOVA

Choices of Contrasts in GLM Repeated Measures

SPSS GLM Procedure for Repeated-Measures ANOVA

Output of GLM Repeated-Measures ANOVA

Paired-Samples t Tests as Follow-Up


Effect Size

Statistical Power

Counterbalancing in Repeated-Measures Studies

More Complex Designs


Appendix 15A: Test for Person-by-Treatment Interaction

Appendix 15B: Nonparametric Analysis for Repeated Measures (Friedman Test)

16. Factorial Analysis of Variance
Research Situations Where Factorial Design Is Used

Questions in Factorial ANOVA

Null Hypotheses in Factorial ANOVA

Screening for Violations of Assumptions

Hypothetical Research Situation

Computations for Between-S Factorial ANOVA

Computation of SS and df in Two-Way Factorial ANOVA

Effect Size Estimates for Factorial ANOVA

Statistical Power

Follow-Up Tests

Factorial ANOVA Using the SPSS GLM Procedure

SPSS Output


Design Decisions and Magnitudes of SS Terms


Appendix 16A: Fixed Versus Random Factors

Appendix 16B: Weighted Versus Unweighted Means

Appendix 16C: Unequal Cell n’s in Factorial ANOVA: Computing Adjusted Sums of Squares

Appendix 16D: Model for Factorial ANOVA

Appendix 16E: Computation of Sums of Squares by Hand

17. Chi-Square Analysis of Contingency Tables
Evaluating Association Between Two Categorical Variables

First Example: Contingency Tables for Titanic Data

What Is Contingency?

Conditional and Unconditional Probabilities

Null Hypothesis for Contingency Table Analysis

Second Empirical Example: Dog Ownership Data

Preliminary Examination of Dog Ownership Data

Expected Cell Frequencies If H0 Is True

Computation of Chi Squared Significance Test

Evaluation of Statistical Significance of x2

Effect Sizes for Chi Squared

Chi Squared Example Using SPSS

Output From Crosstabs Procedure

Reporting Results

Assumptions and Data Screening for Contingency Tables

Other Measures of Association for Contingency Tables


Appendix 17A: Margin of Error for Percentages in Surveys

Appendix 17B: Contingency Tables With Repeated Measures: McNemar Test

Appendix 17C: Fisher Exact Test

Appendix 17D: How Marginal Distributions for X and Y Constrain Maximum Value of f

Appendix 17E: Other Uses of x2

18. Selection of Bivariate Analyses and Review of Key Concepts
Selecting Appropriate Bivariate Analyses

Types of Independent and Dependent Variables (Categorical Versus Quantitative

Parametric Versus Nonparametric Analyses

Comparisons of Means or Medians Across Groups (Categorical IV and Quantitative DV)

Problems With Selective Reporting of Evidence and Analyses

Limitations of Statistical Significance Tests and p Values

Statistical Versus Practical Significance

Generalizability Issues

Causal Inference

Results Sections

Beyond Bivariate Analyses: Adding Variables

Some Multivariable or Multivariate Analyses

Degree of Belief

Appendix A: Proportions of Area Under a Standard Normal Curve

Appendix B: Critical Values for t Distribution

Appendix C: Critical Values of F

Appendix D: Critical Values of Chi-Square

Appendix E: Critical Values of the Pearson Correlation Coefficient

Appendix F: Critical Values of the Studentized Range Statistic

Appendix G: Transformation of r (Pearson Correlation) to Fisher’s Z



Instructor Teaching Site
Password-protected Instructor Resources include the following:
  • Editable, chapter-specific Microsoft® PowerPoint® slides offer you complete flexibility in easily creating a multimedia presentation for your course. 
  • Test banks in Word and LMS-ready formats provide a diverse range of pre-written options as well as the opportunity to edit any question and/or insert your own personalized questions to effectively assess students’ progress and understanding.
  • Tables and figures from the printed book are available in an easily-downloadable format for use in papers, hand-outs, and presentations.
  • Answers to comprehension questions from the text.

Open-access Student Resources include flashcards, web resources, and data sets provided by the author for student download for completing in-chapter exercises.


“Combined, these texts provide both simplistic explanations of analyses, and also in-depth exploration of them with examples. Thus, they prove to be a useful resource to beginning statistics students all the way through the dissertation level, and even for faculty conducting research.”

Karla Hamlen Mansour
Cleveland State University

“This book presents statistical complexity in a friendly and uncomplicated way with friendly text and plenty of helpful diagrams and tables.”

Beverley Hale
University of Chichester, U.K.

“Well-written, comprehensive statistics book. A very valuable resource for advanced undergraduate and graduate students.”

Dan Ispas
Illinois State University

“Warner's textbook is ideal for graduate or advanced undergraduate students providing extensive, yet highly accessible, coverage of important issues in fundamental research design and statistical analysis and newer recommendations in how to conduct statistical analysis and report results ethically. She writes extremely well and my students find her book very readable and useful.”

Paul F. Tremblay
University of Western Ontario

“The book is well-written and focuses on practical applications of the concepts rather than typical ‘textbook’ applications. The focus on meaning rather than the mechanics of computation is also a strength.”

Linda M. Bajdo
Wayne State University
Key features
  • New content includes more detailed coverage of frequency distributions, graphs, sampling error, and confidence intervals. 
  • Standard deviations are marked on frequency tables and histograms to make it clear how they describe dispersion of scores. 
  • Students are encouraged to find their own z scores and think about locations of their own scores. 
  • Clear guidelines are provided for decisions about problems with data (such as outliers). 
  • Common misconceptions about p values and confidence intervals are discussed, along with lists of things you can and should never say. 
  • Coverage of bivariate techniques (e.g., correlation, ANOVA) has been simplified from the previous edition. 
  • Computation of CI and effect sizes not given by SPSS are included in response to New Statistics advocates’ call for more information on the topic.
  • An expanded introduction to IBM SPSS version 25 ensures students will not need a separate SPSS book.


  • The book’s applied approach to introduction to statistics shows that real data have problems (such as missing values, outliers, or violations of assumptions). Discussion of ways in which actual practice differs from ideal situations helps students understand statistics in the context of real-world research.
  • Each chapter follows the same format: discussion of the types of questions this analysis can answer; worked examples with by-hand computation for small data sets; screenshots for SPSS menu selections and output; and results sections. 
  • A focus on the New Statistics guidelines places more emphasis on confidence intervals, effect sizes, and the need to document decisions made during analysis. 
  • Additional technical and supplemental information, including nonparametric alternatives, is provided in appendices at the ends of most chapters so that it doesn’t get in the way of the primary discussion.

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