SOME STATISTICAL TOOLS IN HYDROLOGY TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS OF THE UNITED STATES GEOLOGICAL SURVEY, BOOK 4, CHAPTER A1 by H.C Riggs Prepared by the U.S. Geological Survey 1968 TABLE OF CONTENTS Preface Abstract Introduction Distributions     Cumulative distributions Statistical inference Correlation and regression     Serial correlation Regression methods     Regression models     Transformations     Simple linear regression     Multiple linear regression     Regression computation using "c" multipliers     Regressions having various numbers of independent variables     Use of digital computers     Application of the regression method     Graphical regression     Graphical multiple regression     Graphical multiple regression when the independent variables are highly correlated between themselves     Choice of graphical or analytical method for multiple regression Determining equations of graphical relations     General methods     Definition of equations Other tools     Analysis of variance     Analysis of covariance     Multivariate analysis Chacteristics of hydrologic data     Effects of data characteristics on analysis     Outliers Selected references FIGURES 1. Histogram, or frequency distribution, of 1,000 tree-ring indices 2. Probability density curve of 1,000 tree-ring indices 3. Probability density curve and its cumulative form 4-8. Diagrams showing -     4. Normal distribution     5. Distribution of means of samples from a normal distribution     6. Distribution of variances of samples     7. Hypothetical sampling distribution of means     8. Normal distribution of plotted points about the regression line 9. Graph showing plot used in demonstrating the effect of sample range on computed correlation coefficient 10. Equations and graphs of some common regression models 11. Data plotted on natural and log scales showing the achievement of equal variance about the regression line by use of the log transformation 12. Plot of data from table 2 showing computed regression line 13. Equations and graphs of three models based on the plotted data 14. Graphs showing four possible outcomes of plotting Y against X 15. Plot showing the two regression lines and the structural line 16. Graph showing method of estimating the standard error of a graphical regression 17. Example of graphical multiple regression 18. Example of graphical multiple regression using arithmetic scales 19. Graphical regression using highly correlated independent variables 20. Graphical regression in which one variable is used twice 21. Graph showing multiple linear regression by the method of residuals 22. Graph showing coaxial graphical multiple regression 23. Diagrams showing two conditions for which analysis of covariance will produce conclusions different from those of analysis of variance 24-26. Graphs showing -     24. Plot of data from table 5     25. Spurious relation using nonhomogeneous data     26. Discharge relations for individual months, two Utah stations TABLES 1. Results of dice-tossing experiment 2. Data and computations for example of two-variable regression 3. Multiple regression example: Tennessee low-flow characteristics 4. Data for graphical regression using highly correlated independent variables 5. Annual precipitation index and annual runoff, for example of analysis of covariance ABSTRACT This chapter of "Techniques of Water-Resources Investigations" provides background material needed for understanding the statistical procedures most useful in hydrology; it furnishes detailed procedures, with examples, of regression analyses; it described analysis of variance and covariance and discusses the characteristics of hydrologic data. LizardTech's Djvu plug-in is needed to view these reports. CLICK HERE TO VIEW THE ENTIRE REPORT For questions or comments, contact K. Van Wilson.