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STATISTICS with Applications to HIGHWAY TRAFFIC ANALYSES

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STATISTICS with Applications to HIGHWAY TRAFFIC ANALYSES STATISTICS with applications to HIGHWAY TRAFFIC ANALYSES BRUCE DOUGLAS GREENSHIELDS, C.E., Ph.D. Professor of Civil Engineering The George Washington University FRANK MARK WEIDA, Ph.D. Professor of Statistics The George Washington University THE ENO FOUNDATION FOR HIGHWAY TRAFFIC CONTROL SAUGATUCK . 1952 ' CONNECTICUT Eno Foundation Publications are provided through an endowment by the late William P. Eno I I Copyright.I952,bytheEnoFoundationforHighwayTrafficControllnc.Reproduc­ tioriofthispublicationinwholeorpartwithoutpermissionisprohibited.Publislied by the Eno Foundation at Saugatuck, Connecticut, October, 1952. Copies of this book are not to be sold. FOREWORD Realizing the need for a publication to encourage further scientific approach to the solution of many traffic problems, the Eno Foundation is pleased to present this methodical discussion of some statistical theories and their application in the analysis of traffic data. The Foundation was fortunate in acquiring the services of Dr. Bruce D. Greenshields, Professor and Executive Officer, Civil Engineering Department, and Dr. Frank M. Weida, Executive Officer, Departmentof StatisticsTheGeorgeWashingtonUniversi­ ty, as co-authors.Byknowledge and experiencethey are eminently qualified. They have been guided by a practical insight and have shown an unusual and necessary discernment of the subject. In some quarters, thinking on traffic as a national problem has reached a degree of desperation. This is due partly to confusion. It is hoped this study will provide some clarification by em­ phasizing the importance of an analytical basis for initiating logical improvements. Such procedure shouldtend to create better understanding and much-needed uniform basic methods. It has been a privilege for the Eno Foundation to provide the preparation and publication of this monograph. Publication has resulted from considerable time and effort by both authors and the Foundation Staff. Tiu& ENo FoUNDATION PREFACE The engineer, and particularly the traffic engineer working in a comparatively new field, faces constantly the need for new, more precise information. To obtain this information, he collects and analyzes data. The theory and procedures to be followed in such analyses have long been known to the statistician, but not always to the enameer. Mathematics he learns forhis engineering is of the classical type­ algebra, trigonometry, calculus - in which exact answers are obtained. In statistics no answer is exact for there is always a range of variability within which the true answer lies. Variance, the measure of this variability, may in some cases be so small that the result for practical purposes may be considered exact. But usually it is not. In traffic behavior, a phase of humanbehavior, it is well to employ the "mathematics of human welfare." Traffic research carried on at various times over a period of years by one of the writers has served to confirm the fact that traffic behavior tendsto follow definitestatistical patterns. The difficulty of solving the problems encounteredin analyzing thedata collected during that research pointed to the need for someone to gather together and explain the statistical methods most pertinent to traffic analyses. In response to this need, this monograph is written. Desired in­ formation, it was felt, could be assembled, developed, and presented most effectively, by a traffic engineer and a statistician working together. The one would know the viewpoint of the engineer and the limitation of his statistical training and vocabulary. The other would provide that knowledge and skill in his own field that can be obtained only after years of work and study. The authors, despite the work involved, have enjoyed what seemed to them a very worth while undertaking. This monograph is not in any sense the last word on the subject. It is merely an intro­ duction, which they hope will assist the engineer in determining the type and amount of data he needs to obtain sufficiently vi PREFACE accurate answers to his problems and save him time and effort. They trust that if it is a new tool to him it will be to his liking. In the first four chapters the authors have attempted to explain this mathematicaltool, and in the last one they have attempted to show how to use it. The authors wish to thank the Eno Foundation and staff for its kindly criticism, good counsel, encouragement and sponsorship. They are indebted to Professor Herman Betz of the Department of Mathematics at the Universityof Missouri for his careful review of the manuscript. WashingtonD. C. BR ucE D. GREENSHIELDS June 1, 1952 FRANK M. WEIDA ACKNOWLEDGEMENTS Professor RONALD A. FISHER, Cambridge, Dr. FRANK YATES, Rothamstead, and Messrs. OLIVER AND BOYD LTD., Edinburgh, for permissionto reprint Appendix Tables II and IV from their book, "Statistical Tables forBiological, Agricultural,andXedicalResearch." GEORGE W. SNEDECOR and the IOWA STATE COLLEGE PRESS, Ames, Iowa for permission to reprintAppendix Table V from their book "Statistical Methods," 4th edition. BUREAU OF PUBLIC ROADS, Washington, D. C. for charts used from "Highway Capacity Manual." vii TABLE OF CONTENTS Page FOREWORD . iii PREFACE . V AcKNOWLEDGEMENTS . .Vii TABLE OF CONTENTS . .ix LIST OF FIGURES . xiv LIST OF TABLES . .XVii CHAPTER I - THE NATURE AND UTILITY OF STATISTICS General Remarks . Definition and Nature of Statistics . .3 Statistics and Mathematics . .3 Two General Types of Problems . .4 Types of Sampling . .5 The Variables to be Measured and Interpreted . .5 Means of Measuring the Variable and Precautions to be Taken . 6 The Size of the Sample . .7 The Validity and Reliability of Measurement . .8 Cost of the Project . .9 The Report . 9 Purpose of the Book . .10 References, Chapter I. .10 CHAPTER II - SummARiziNG OF DATA . .12 Objective . .12 Frequency Distribution . .12 Class Interval and Class Mark . .12 Frequency Rectangles . .15 Histogram . .16 Frequency Polygon . .17 Smoothed Frequency Polygon . .17 ix TABLE OF CONTENTS Page, Frequency Curve . .18 Cumulative Frequencies . .19 Average . 22 Arithmetic Mean . .22 Measure of Central Tendency . .27 Mathematical Expectation or Expected Value of a Variable . .27 Deviation from Arithmetic Mean . .27 The Deviations from Any Arbitrary Value . .33 Mean Values in General . .33 The Mode . 35 Median . 38 Quantiles . .40 Geometric Mean . .42 Harmonic Mean . .44 Root Mean Square . .45 Centra, Harmonic Mean . .51 Mean or Average Deviation . .51 Moments and Mathematical Expectation of Powers of a Variable . .. 54 Relation Between Means . .58 Desirable Properties of an Average . .58 References, Chapter II . .60 CHAPTERIII-STANDARDDiSTP.IB-UTIONSAND'fHEIRMATIIE- MATICAL PATTERNS . .61 Objective . .61 The Elements of a Distribution . .61 Bernoulli's Theorem . .65 Cantelli's Theorem. .68 The Bienaym6-TchebycheffCriterion . .70 Permutations and Combinations . .71 Theorem of Compound Probability . .74 The Binomial Theorem . .75 Modal Term of Binomial Distribution . .79 Arithmetic Mean of Binomial Distribution . .80 TABLE OF CONTENTS xi Page Variance of Binomial Distribution . 81 Size of Sample Required for Stability . 82 The Normal Distribution . 85 Interpretation of the Properties of Normal Distribution . 88 Poisson Distribution . 90 The Sum of the Terms of the Poisson Distribution . 93 The Arithmetic Mean of Poisson Distribution . 93 The Variance of Poisson Distribution . 94 Dispersion and Variance . 97 The Multinomial Distribution . 102 Hypergeometric Distribution. 104 Correlation . 106 The Correlation Coefficient r-Linear Regression or Linear Trend . 107 Basic Theory of Correlation . 113 Coefficient of Regression . 115 Standard Deviation of Arrays . 116 Correlation Ratio: Non-Linear Regression . 117 Multiple Correlation . 120 Partial Correlation. 125 Regression (Trend) Lines . 127 References, Chapter III . 137 CHAPTER IV - SAMPLING THEoRy . 138 Reliability and Significance . 138 Objective . 138 Random Sampling. 139 Distribution of Sample Arithmetic Means . 139 Inference Concerning Population Mean . 141 Confidence Limits . 142 Difference Between Sample Arithmetic Means . 143 Size of Sample for Arithmetic Mean . 145 Reliability of Sample Standard Deviation . 146 Significanceof Difference Between Sample Variances . 147 Significance of a Correlation Coefficient . 147 References, Chapter IV . 149 xii TABLE OF CONTENTS Page CHAPTER V - SomE APPLICATIONS OF STATISTICAL METHODS 150 Objective . 150 Confusion as to Meaning of Highway Capacity . 150 Theoretical Maximum Capacity (Volume) . 151 Stopping Distance and Minimum Spacing . 152 Interpretation of Minimum Spacing Formula . 154 Limiting Factors . 154 Additional Relationships of Spacing and Speed . 154 Volume and Speed . 158 The Nature of the Problems of Highway Traffic . .160 Spacing as a Random Series . 161 Test of Goodness of Fit of the Poisson Series . 163 Test of Goodness of Fit of the Poisson Series to the Distribution of Spacings between Vehicles . 163 Minimum Spacing . 169 The Minimum Spacing of Four-Lane Traffic . 172 Frequency Distribution of Speeds . 173 A Graphical Method of Determining Goodnessof Fit . .178 Estimating Speeds and Volumes. 181 Estimate of Size Gap Required for Weaving . 187 Physical Features of Highway: Effect on Traffic Flow .187 Crossing Streams of Traffic . 189 Mathematical Determinationof Vehicle Delay Time . .190 Graphical Method of Determining Proportion of Time Occupied by Time-Gaps of Given Size . 192 The Average Length of All Intervals . 194 The Signalized Intersection . 198 Calculating Delay at Signalized
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