Measurement and Statistics for Teachers

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  • Measurement and Statistics for Teachers

    Written in a student-friendly style, this modestly priced text shows teachers how to usemeasurement and statistics wisely in their classes. Although there is some discussion oftheory, emphasis is given to the practical, everyday uses of measurement and statisticssuch as how to develop and use eective classroom tests, how to carry out informalassessments, performance assessments, portfolio assessments, and how to use and inter-pret standardized tests. Part II provides a more complete coverage of basic descriptivestatistics and their use in the classroom than any text now available.

    Malcolm Van Blerkom has been teaching various college courses in Psychology andEducational Psychology for the past 28 years, and is currently at the University ofPittsburgh at Johnstown. He has also published research on topics as varied as cognitivestyles, class attendance, and study strategies.

  • Measurement and Statisticsfor Teachers

    Malcolm L. Van BlerkomUniversity of Pittsburgh at Johnstown

  • First published 2009by Routledge270 Madison Ave, New York, NY 10016

    Simultaneously published in the UKby Routledge2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN

    Routledge is an imprint of the Taylor & Francis Group, an informa business

    2009 Taylor & Francis

    All rights reserved. No part of this book may be reprinted or reproduced or utilizedin any form or by any electronic, mechanical, or other means, now known orhereafter invented, including photocopying and recording, or in any informationstorage or retrieval system, without permission in writing from the publishers.

    Trademark Notice: Product or corporate names may be trademarks or registeredtrademarks, and are used only for identification and explanation without intentto infringe.

    Library of Congress Cataloging in Publication DataVan Blerkom, Malcolm L.

    Measurement and statistics for teachers / Malcolm L. Van Blerkom.1st. ed.v. cm.

    Includes bibliographical references and index.1. Educational tests and measurements. 2. Educational statistics. I. Title.LB3051. V25 2008371.26dc22


    ISBN10: 0415995655 (hbk)ISBN10: 0805864571 (pbk)ISBN10: 0203887867 (ebk)

    ISBN13: 9780415995658 (hbk)ISBN13: 9780805864571 (pbk)ISBN13: 9780203887868 (ebk)

    This edition published in the Taylor & Francis e-Library, 2008.

    To purchase your own copy of this or any of Taylor & Francis or Routledgescollection of thousands of eBooks please go to

    ISBN 0-203-88786-7 Master e-book ISBN

  • DedicationFor my wife, Diane,

    for all of her love, help,and encouragement



    Preface xvii

    Acknowledgements xxi


    Section I Basic Issues in Measurement 3

    Chapter 1 Introduction to Measurement 5Introduction 5The Role of Measurement 5Measurement, Assessment, and Evaluation 6

    Assessment 6Measurement 6Evaluation 7

    Formal vs. Informal Assessment 8Classroom Assessment 8

    Preliminary or Placement Assessment 8Diagnostic Assessment 9Formative Assessment 9Summative Assessment 10

    Maximum vs. Typical Performance Measures 11Uses of Measurement 11Summary 12Exercises 13Spotlight on the Classroom 13Study Tips: Setting Eective Academic Goals 14


  • Chapter 2 Frames of Reference: Interpreting Test Scores 16Introduction 16Four Frames of Reference 16

    Ability-Referenced Interpretations 16Growth-Referenced Interpretations 17Norm-Referenced Interpretations 19Criterion-Referenced Interpretations 20A Comparison of Norm-Referenced and

    Criterion-Referenced Interpretations 21Choosing a Frame of Reference for Assessment 22Characteristics of Norm-Referenced and

    Criterion-Referenced Tests 23Item Diculty 24Number of Items on the Test 24

    Summary 25Exercises 25Spotlight on the Classroom 26Study Tips: Time Management 26

    Chapter 3 Developing Objectives 28Introduction 28Standards 28Planning by Using Goals and Objectives 29

    Goals vs. Objectives 29Benjamin Blooms Taxonomy of Objectives 31

    Knowledge Level Objectives 32Comprehension Level Objectives 32Application Level Objectives 32Analysis Level Objectives 32Synthesis Level Objectives 32Evaluation Level Objectives 33

    Robert Magers Instructional Objectives 33Norman Gronlunds Instructional Objectives 34Robert Gagns Learning Outcomes 35

    Gagns Categories 35Using Objectives 36Summary 37Exercises 38Spotlight on the Classroom 39

    Chapter 4 Reliability 40Introduction 40

    viii Contents

  • What is Reliability? 40Theoretical Model of Reliability 41

    Computing Reliability 45Reliability and Validity 45Estimating Reliability 46

    TestRetest Reliability 47Alternate Form Reliability 48Internal Consistency Reliability 48

    Interpreting Reliabilities 51Improving Test Reliability 52

    Reducing Subject Eects 52Reducing Test Eects 53Reducing Environmental Eects 54

    Some Final Comments about Reliability 55Summary 55Exercises 55Spotlight on the Classroom 56

    Chapter 5 Validity 57Introduction 57Perspectives on Validity 57

    Content-Related Evidence of Validity 58Criterion-Related Evidence of Validity 61Construct-Related Evidence of Validity 62Which Perspective is the Most Important? 63

    Reliability and Validity 64Summary 65Exercises 65Spotlight on the Classroom 66

    Section II Classroom Testing 67

    Chapter 6 Completion and Short-Answer Items 69Introduction 69Short-Answer Items 69Advantages and Limitations of Short-Answer Items 71

    Advantages 71Limitations 71

    Attributes Desired in Short-Answer Items 72Evaluating Short-Answer Items 76Summary 77Exercises 77Spotlight on the Classroom 77

    Contents ix

  • Chapter 7 Essay Items 79Introduction 79Advantages and Limitations of the Essay Format 79

    Advantages 79Limitations 80

    Types of Essay Item 81Scoring Essay Items 82

    Holistic Scoring 82Analytic Scoring 83General Recommendations for Scoring Essay Answers 83

    Attributes Desired in Essay Items 84Evaluating Essay Items 86Summary 87Exercises 87Spotlight on the Classroom 88

    Chapter 8 Multiple-Choice Items 89Introduction 89Advantages and Limitations of Multiple-Choice Items 89

    Advantages 89Limitations 91

    Attributes Desired in Multiple-Choice Items 92Evaluating Multiple-Choice Items 98Various Types of Multiple-Choice Item 100

    Matching Items 100Range-of-Value Items 100Ranking Options 101Interpretive Exercises 101

    Number of Alternatives 102Summary 103Exercises 103Spotlight on the Classroom 103

    Chapter 9 Truefalse Items (and Variations) 105Introduction 105Advantages and Limitations of Truefalse Items 105

    Advantages 105Limitations 106

    Attributes of Good Truefalse Items 107Evaluating Truefalse Items 111Variations in the Truefalse Format 112

    Truefalse with Correction 112

    x Contents

  • Embedded Truefalse Items 112Sequential Truefalse Items 113Checklists 114

    Summary 114Exercises 114Spotlight on the Classroom 115

    Chapter 10 Producing and Administering Tests 116Introduction 116Designing a Test 116

    Dening the Purpose of the Test 116Choosing the Types of Items to Use 117Choosing the Number of Items to be Used 117Choosing the Diculty Level of the Items 117Assuring Sucient Accuracy 118

    Producing a Test 120Preparing the Items 120Ordering the Items 120Formatting the Test 120Preparing Instructions 121Proofreading 121

    Administering the Test 122Setting Up an Appropriate Testing Environment 122

    Summary 122Exercises 123Spotlight on the Classroom 124

    Chapter 11 Analyzing Tests 125Introduction 125Test Analysis 125Item Analysis 126

    Item Diculty 127Item Discrimination 127Distractor Analysis 129Item Analysis Practice 130The Stability of Item Analyses 132

    Summary 133Exercises 133Spotlight on the Classroom 134

    Contents xi

  • Section III Alternative Assessment Techniques 135

    Chapter 12 Informal Assessment 137Introduction 137

    What is Informal Assessment? 137Types of Informal Assessment 138

    Informal Observations 138Questions 138

    Characteristics of Informal Assessment 139Planning for Observations and Questions 142

    Choosing Behaviors to Observe 142The Validity Question 142The Reliability Question 143

    Techniques for Eective Informal Assessment 143Planning Informal Assessment 143Use Informal Assessment Frequently 144Maintain Positive Interactions with your Students 144Use the Results of Informal Assessment to Alter

    Instruction 144Summary 145Exercises 145Spotlight on the Classroom 146

    Chapter 13 Performance Assessments 147Introduction 147What are Performance Assessments? 147Types of Performance Assessment 148

    Process vs. Product 148Simulated vs. Real Settings 148Natural vs. Structured Settings 149

    When Are Performance Assessments Appropriate To Use? 149Advantages and Limitations of Performance Assessment 150

    Advantages 150Limitations 150

    Planning and Developing Performance Assessments 151Tie Assessment to Objectives 151Measure Important Skills 151Establish Precise Skills to Measure 152Focus on Process or Product Only 152Dene the Tasks for the Students 152

    Scoring Performance Assessments 152Checklists 152

    xii Contents

  • Rating Scales 153Rubrics 154

    Summary 156Exercises 157Spotlight on the Classroom 157

    Chapter 14 Portfolios 159Introduction 159What Makes Portfolios Distinctive? 159Advantages and Limitations 160

    Advantages 160Limitations 160

    Components of Portfolios 161The List of Goals 161Work Samples 162Annotations 162

    When is Portfolio Assessment the Most Eective? 162Helping Students Develop Their Portfolios 163Scoring Portfolios 163The Future of Portfolio Assessment 164Summary 164Exercises 165Spotlight on the Classroom 165

    Section IV Additional Measurement Issues 167

    Chapter 15 Teaching Students Test-Taking Skills 169Introduction 169General Test-Taking Strategies 170

    Budgeting Time 170Reading Directions 170Reading Items Carefully 171Checking Tests before Turning Them In 172

    Test-Taking Strategies for Specic Test Formats 172Strategies for Short-Answer Tests 172Strategies for Essay Tests 173Strategies for Multiple-Choice Tests 173Strategies for Truefalse Tests 174

    Familiarity with Testing Approaches 175Approaches to Teaching Test-Taking Skills 175Summary 176Exercises 176Spotlight on the Classroom 176

    Contents xiii

  • Chapter 16 Standardized Tests 178Introduction 178General Characteristics of Standardized Tests 178

    A Case Study in Developing a Standardized Test 179Steps in Building a Standardized Test 180Setting Interpretation Standards 180Standardized Test Administration 181

    Achievement Tests 181Single-Subject-Area Achievement Tests 181Survey Batteries 182

    Diagnostic Tests 183Reading Readiness Tests 184

    Aptitude Tests 185Individual Aptitude Tests 186Group Aptitude Tests 187

    Other Types of Standardized Test 188Using Standardized Tests Eectively in the Schools 188

    Selecting Standardized Tests 188Making Predictions Based on Test Scores 189Using Standardized Tests Appropriately 189

    The Eects of No Child Left Behind and OtherFederal Mandates 190

    Summary 191Exercises 191Spotlight on the Classroom 192

    Chapter 17 Alternative Ways to Report Test Scores 193Introduction 193Percentile Ranks 193Standardized Scores 195

    z-Scores 196T-Scores 197SAT Scores 198

    Normalized Standard Scores 199Normal Curve Equivalent Scores 200Stanines 201

    Grade Equivalent Scores 201Building Condence Intervals 202

    Error Variance 202Standard Error of Measurement 203Using the SEM to Build Condence Intervals 205Factors Aecting the Width of Condence Intervals 205

    xiv Contents

  • Summary 206Exercises 207Spotlight on the Classroom 208


    Chapter 18 The Language and Logic of Statistics 211Introduction 211Basic Language and Logic 211

    Constants and Variables 211Populations and Samples 212Parameters and Statistics 214

    Measurement Scales 216Categorical Data 216Ranked Data 217Numerical Data 218Discrete Data vs. Continuous Data 219

    Summary 219Exercises 220Spotlight on the Classroom 221Study Tips: How to Read a Measurement and Statistics Text 221

    Chapter 19 Frequency Distributions and Graphs 223Introduction 223Frequency Distributions 223

    Frequencies 223Proportion and Percentages 225Grouped Frequency Distributions 227

    Graphing Frequency Distributions 229Bar Charts 229Pie Charts 230Histograms 230Frequency Polygons 231

    Forms of Frequency Distribution 232Cumulative Frequency Distributions 234Summary 235Exercises 235Spotlight on the Classroom 237Study Tips: How to Take Lecture Notes in a Measurement and

    Statistics Class 237

    Chapter 20 Central Tendency: What is a Typical Score? 239Introduction 239

    Contents xv

  • Measures of Central Tendency 239Mode 239Median 240Mean 245

    Deviation Scores 247Characteristics of Central Tendency Measures 248

    Stability of Central Tendency Measures 248Uses of Central Tendency Measures 249Central Tendency and Form 250

    Summary 250Exercises 250Spotlight on the Classroom 251Study Tips: Learning Technical Terminology 252

    Chapter 21 Variability: How Spread Out Are the Scores? 254Introduction 254The Variability Question 254

    Ranges 255Variance and Standard Deviation 256

    Summary 262Exercises 262Spotlight on the Classroom 264Study Tips: How to Prepare for an Exam 264

    Chapter 22 Correlation 266Introduction 266Bivariate Statistics 266z-Scores 268Pearson Product-Moment Correlation Coecient 270

    Computational Formula for the PPMC 274Correlation and Prediction 275Summary 276Exercises 277Spotlight on the Classroom 278Study Tips: How to Learn from an Exam 278

    References 281Index 283

    xvi Contents


    This text is designed to give prospective and practicing teachers those skills requiredthat will allow them to make intelligent decisions about testing and grading. My per-sonal experiences have led me to conclude that many teachers feel that they are out ontheir own when it comes to testing and grading, having often had very little formaltraining in measurement and statistics. This text is not designed to turn teachers intoeither statisticians or psychometricians. However, it is designed to give you a basicunderstanding of both statistical and measurement principles and specic skills thatwill allow you to make intelligent choices about testing and related issues.

    This text will be partly theoretical and partly practical. It will be theoretical in thatyou will learn about the basic principles of measurement and statistics. This will allowyou to know when it will be appropriate and useful to use certain techniques. Anunderstanding of statistics is also necessary for an understanding of measurement. Thistext will also be practical in that you will learn how to calculate means, standarddeviations, correlation, and other statistics, and will learn how to develop frequencydistributions and graphs. In addition, you will learn how to develop good test items anduse a variety of measurement techniques.

    I have had several goals in mind as I planned and wrote this book. First and foremost,I want teachers to be able to make intelligent choices about using testing and grading intheir classrooms. What type of test would be the most appropriate given the materialthat was covered, your goals, and the age of your students? I also want teachers to beable to defend their methods to colleagues, supervisors, parents, and students. You mustbe able to explain logically why you use those methods, and why they are better than thealternatives. Finally, I hope that the readers of this...


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