The Elements of Graphing Data

DATA A Wlllnam S. Cleveland AT&T Bell Laboratories l lller D lgl lo llmlwéli WADSWORTH ADVANCED BOOKS AND SOFTWARE Monterey, California \ V

Wadsworth Advanced Book Program I A Division of Wadsworth, Inc. I Copyright © 1985 Bell Telephone Laboratories, Inc., Murray Hill, New Jersey. { All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without the prior written permission of the publisher, I Wadsworth Advanced Book Program, Monterey, California, 93940, a » division of Wadsworth, Inc. A V Printed in the United States of America ` 12345678910—8988878685 Library of Congress Cataloging in Publication Data Cleveland, William S., 1943- I The elements of graphing data. o Bibliography: p. Includes index. 1. Graphic methods. I. Title. V ‘QA90.C54 1985 r511’.5 85-10603 I § issm m—s2u-mavau-5 PAPER Q

To Lisa, Robert, and Scott

To Iohn Tukey, for ingenious inventions and applications of graphical data analysis. To many colleagues at Bell Labs, for creating an optimal environment to study graphical data analysis. To Marylyn McGill, for relentlessly pursuing perfection in experimenting with graphical displays and in managing the production of this book. A To Bob McGill, for our experiments in graphical perception and our many experiments with graphical inventions. A To Elsie Edelman, for the considerable word processing skills that were needed to produce the text. To Lisa Cleveland, for days of proofreading in Summit and Abcoude. To Iohn Kimmel, for a near perfect author—editor relationship. To many who commented on the manuscript, for helping greatly to steer the revisions - Paul Anderson, ]'on Bentley, Iohn Chambers, Lisa Cleveland, Arnold Court, Mary Donnelly, Bob Futrelle, Colin Mallows, Bob McGill, Brad Murphy, Richard Nuccitelli, Iames Palmer, Arno Penzias, and _]ohn Tukey.

M PREFACE .................................. 1 Chapter 1. INTRODUCTION ............................ 3 1.1 The Contents of the Book ................... 3 Chapter 2: Principles of Graph Construction . . . 3 Chapter 3: Graphical Methods .............. 4 Chapter 4: Graphical Perception ............ 7 1.2 The Power of Graphical Data Display .......... 9 l 1.3 The Challenge of Graphical Data Display ........ 12 Aerosol Concentrations ................... 12 Brain Masses and Body Masses of Animal SPECIES ....................... 13 W 1.4 Sources and Goals ......................... 17 Pr1nc1ples of Graph Construction ........... 17 Graph1cal Methods ...................... 18 Graphical Perception .................... 18

x comrsms g Chapter 2. PRINCIPLES OF GRAPH CONSTRUCTION ....... 21 2.1 Termirwlegy ....»......,..........__ , ____ 21 2.2 Clear Vision ............................. 24 2.3 Clear Understanding ....................... 56 2.4 Scales ...........................,...... 68 2.5 General Strategy .......................... 89 W 2.6 A Listing of the Principles of Graph 3 Construction ..............r...... I ........ 100 Chapter 3. GRAPHICAL METHODS ...................... 103 . 3.1 General Methods: Logarithms and Residuals ..... 104 3 Logarithms ............................ 104 a Leg Base 2 and Log Base e ................. 106 Graphing Percent Change ................. 111 Residuals ...........,................. 1 1 4 The Tukey Sum-Difference Graph ........... 118 3.2 One or More Categories of Measurements of One Quantitative Variable: Graphing Distributions ...........,................. 123 Point Graphs and Histograms .............. 124 Percentile Graphs ......... ji ............. 127 Box Graphs ............................ 129 Percentile Graphs with Summaries .......... 134 Percentile Comparison Graphs ............. 135 g A 3.3 One Quantitative Variable With Labels: 3 Dot Charts ............................... 144 Ordinary Dot Charts ..................... 144 ‘ Two-Way, Grouped, and Multi-Valued V Dot Charts .......................... 151 3 3.4 Two Quantitative Variables __________________ 154 M Overlap: Logarithms, Residuals, Moving, Sunflowers, littering, and Circles .......... 154

comr-mrs xi Box Graphs for Summarizing Distributions of Repeat Measurements of a " Dependent Variable .................... 163 Strip Summaries Using Box Graphs .......... 166 Smoothing: Lowess ..................... 167 Time Series: Connected, Symbol, Connected M Symbol, and Vertical Line Graphs ......... 178 Time Series: Seasonal Subseries Graphs ...... 186 A An Equally-Spaced Independent Variable with a Single-Valued Dependent Variable . . . 188 Wi Step Function Graphs ........... , ......... 189 3.5 Two or More Categories of Measurements of Two Quantitative Variables: Superposition 4 and Iuxtaposition ......................... 191 Superposed Plotting Symbols .............. 191 Superposed Curves in Black and White ....... 196 Iuxtaposition .....................; . Q, . . . 198 Color ................................ 205 3.6 Three or More Quantitative Variables .......... 208 Framed-Rectangle Graphs ....i . ............ 208 W 1 Scatterplot Matrices ..................... 210 A View of the Future: High—Interaction W Graphical Methods .................... 213 . 3.7 Statistical Variation ...................... .. . 218 Empirical Distribution of the Data ...7 . ....... 219 Sample-to—Sample Variation of a Statistic ...... 222 One¥Standard-Error Bars .......... A ........ 223 . Two-Tiered Error Bars .................... 226 Chapter 4. GRAPHICAL PERCEPTION .................... 229 W A 4;1 Cognitive Tasks and Perceptual Tasks .......... 230 4.2 The Elements of the Paradigm ................ 233 E , Elementary Graphical-Perception Tasks ....... 233

xii cowreurs Distance ...........,...... Q ........... 238 Detection ............................. 239 4.3 Theory and Experimentation ................. 241 Weber’s Law ........................... 241 Stevens' Law .......................... 243 Angle Iudgments ....................... 245 The Angle Contamination of Slope y Iudgments ............................ 245 Experiments in Graphical Perception ......... 247 summary and Discussion ................. 254 4.4 Application of the Paradigm to Data Display ..... 255.9 Slope judgments: Graphing Rate of Change . . . 255 Length Judgments: Divided Bar Charts ...... 259 Angle Judgments: Pie Charts .............. 264 Cognition ............................. 264 Distance and Detection ................... 265 _ 9 Detection: Superposed Curves ............. 271 4 Area ................................. 278 A Density and Length: Statistical Maps ........ 284 Residuals ..................... A ........ 288 Dot Charts and Bar Charts ................ 291 Summation ............... ’ ............. 294 W 2 REFERENCES ................. . ....... A ...... 295 GRAPH INDEX ............................. 307 rnxr INDEX ..................... A . .· ..... , . . ais

PREFACE This book is about graphing data in science and technology. It contains graphical methods and principles that are powerful. tools for iif Q° showing the structure of data. The material is relevant for data analysis, f when the analyst wants to study data, and for data communication, when the analyst wants to communicate data to others. Many of the methods and principles in the book are new; many others are old, but not widely known. The first few decades of the 20th r century were an exceptionally fertile time for the invention of numerical jjj]? statistical procedures. Statistical scientists invented methods and approaches to data analysis that eventually permeated all of science and technology. The period since about 1960 has been an exceptionally fertile time in statisticalscience for the invention of graphical procedures i s_..W. for data analysis. An infusion of this graphical methodology into » science and technology will raise the effectiveness of data analysis just . as confidence intervals and hypothesis tests did decades ago. The prerequisites for understanding the book are minimal. A few topics require a knowledge of the elementary concepts of probability Q and statistical science, but these topics can be skipped without affecting comprehension of the remainder of the book. The book was meant to be read from the beginning and to be enjoyed. However, it is possible to read here and there. Winding its way through the book is a summary of the material: the figures and their legends. Reading this summary can help readers direct themselves to specific items. E

2 pm-tr=AcE Except for one small section, there is nothing in this book about I computer graphics. The basic ideas, the methods, and the principles of ¥ the book transcend the medium used to implement them, but the reality is that the computer looms behind the book content because it is the medium of the present for many and of the future for almost all. The graphs of the book that are not copies of other peop1e’s graphs were computer generated. The software used was the S system for data analysis and graphics [9] developed by Richard Becker and John Y Chambers of AT&T Bell Laboratories, and CRAP [13], a very recent system developed by ]on Bentley and Brian Kernighan, also of Bell Labs. I a Many graphical methods are missing from this book. I included only those that had promise for application to the most commonly occurring types of data and that would be relevant for all areas of science and technology. Many specialized methods, important as they are, are omitted. The graphs in this book are communicating information about fascinating subjects, and I have not hesitated to describe the subjects in some detail when needed. In many cases some knowledge of the subject is required to understand the purpose of a graphical analysis or c why a graph is not doing what was intended or what a new graphical method can show us about data. I hope the reader will share with me the excitement of experiencing the increased insight that graphical data I V I display brings us about these subjects.

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