Statistical Methods in Water Resources Chapter 3 of Section A, Statistical Analysis Book 4, Hydrologic Analysis and Interpretation Discharge, in cubic meters per second 50 100 200 500 1,000 2,000 5,000 3 20 ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● 10 ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● 5 ● ● ● ● ● ● ● 1 ● ● ● ● ● 2 ●●● ● 0 ● 1 ● ● ● ● 0.5 −1 ● ● ● 0.2 Concentration, in milligrams per liter in milligrams Concentration, ln(Concentration, in milligrams per liter) in milligrams ln(Concentration, −2 0.1 ● Normal quantiles −3 0.05 −3 −2 −1 0 1 2 3 ● 3 4 5 6 7 8 9 ● 1.0 ● ● ● ● ● ● ● ● ● ● ln(Discharge, in cubic meters per second) ● ● ● ● ● ● ● ● ● ● ● ● ● 0.8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● Cumulative frequency Cumulative ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0 100 200 300 400 Annual mean discharge, in cubic meters per second Techniques and Methods 4–A3 Supersedes USGS Techniques of Water-Resources Investigations, book 4, chapter A3 U.S. Department of the Interior U.S. Geological Survey Cover: Top Left: A loess smooth curve of dissolved nitrate plus nitrite concentration as a function of discharge, Iowa River, at Wapello, Iowa, water years 1990–2008 for the months of June, July, August, and September. Top Right: U.S. Geological Survey scientists Frank Engel (left) and Aaron Walsh (right) sample suspended sediment at the North Fork Toutle River near Kid Valley, Washington, upstream of a U.S. Army Corps of Engineers sediment retention structure. This gage is operated in cooperation with the U.S. Army Corps of Engineers. Photograph by Molly Wood, U.S. Geological Survey, March 7, 2018. Bottom Left: Jackson Lake Dam and USGS streamgage 13011000, Snake River near Moran, Wyoming, within Grand Teton National Park. Photograph by Kenneth M. Nolan, U.S. Geological Survey. Bottom Right: Overlay of James River annual mean discharge (1900–2015) and standard normal distribution quantile plot. Statistical Methods in Water Resources By Dennis R. Helsel, Robert M. Hirsch, Karen R. Ryberg, Stacey A. Archfield, and Edward J. Gilroy Chapter 3 of Section A, Statistical Analysis Book 4, Hydrologic Analysis and Interpretation Techniques and Methods 4–A3 Supersedes USGS Techniques of Water-Resources Investigations, book 4, chapter A3 U.S. Department of the Interior U.S. Geological Survey U.S. Department of the Interior DAVID BERNHARDT, Secretary U.S. Geological Survey James F. Reilly II, Director U.S. Geological Survey, Reston, Virginia: 2020 First release: 1992 by Elsevier, in print Revised: September 2002 by the USGS, online as Techniques of Water-Resources Investigations (TWRI), book 4, chapter A3, version 1.1 Revised: May 2020, by the USGS, online and in print, as Techniques and Methods, book 4, chapter A3 Supersedes USGS Techniques of Water-Resources Investigations (TWRI), book 4, chapter A3, version 1.1 For more information on the USGS—the Federal source for science about the Earth, its natural and living resources, natural hazards, and the environment—visit https://www.usgs.gov or call 1–888–ASK–USGS. For an overview of USGS information products, including maps, imagery, and publications, visit https://store.usgs.gov. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. Although this information product, for the most part, is in the public domain, it also may contain copyrighted materials as noted in the text. Permission to reproduce copyrighted items must be secured from the copyright owner. Suggested citation: Helsel, D.R., Hirsch, R.M., Ryberg, K.R., Archfield, S.A., and Gilroy, E.J., 2020, Statistical methods in water resources: U.S. Geological Survey Techniques and Methods, book 4, chap. A3, 458 p., https://doi.org/10.3133/tm4a3. [Super- sedes USGS Techniques of Water-Resources Investigations, book 4, chap. A3, version 1.1.] Associated data for this publication: Helsel, D.R., Hirsch, R.M., Ryberg, K.R., Archfield, S.A., and Gilroy, E.J., 2020, Statistical methods in water resources—Supporting materials: U.S. Geological Survey data release, https://doi.org/10.5066/P9JWL6XR. ISSN 2328-7047 (print) ISSN 2328-7055 (online) iii Contents Chapter 1 Summarizing Univariate Data ..................................................................................................1 1.1 Characteristics of Water Resources Data .........................................................................................2 1.2 Measures of Central Tendency ............................................................................................................5 1.2.1 A Classical Measure of Central Tendency—The Arithmetic Mean...................................5 1.2.2 A Resistant Measure of Central Tendency—The Median ...................................................6 1.2.3 Other Measures of Central Tendency .....................................................................................7 1.3 Measures of Variability ..........................................................................................................................8 1.3.1 Classical Measures of Variability ............................................................................................8 1.3.2 Resistant Measures of Variability ............................................................................................9 1.3.3 The Coefficient of Variation—A Nondimensional Measure of Variability ......................10 1.4 Measures of Distribution Symmetry ..................................................................................................11 1.4.1 A Classical Measure of Symmetry—The Coefficient of Skewness .................................11 1.4.2 A Resistant Measure of Symmetry—The Quartile Skew ..................................................12 1.5 Other Resistant Measures of Symmetry ...........................................................................................12 1.6 Outliers ...................................................................................................................................................12 1.7 Transformations ....................................................................................................................................13 1.7.1 The Ladder of Powers ..............................................................................................................14 Chapter 2 Graphical Data Analysis .........................................................................................................17 2.1 Graphical Analysis of Single Datasets ..............................................................................................18 2.1.1 Histograms .................................................................................................................................18 2.1.2 Quantile Plots ............................................................................................................................20 2.1.3 Boxplots......................................................................................................................................22 2.1.4 Probability Plots ........................................................................................................................26 2.1.5 Q-Q plots as Exceedance Probability Plots ..........................................................................28 2.1.6 Deviations from a Linear Pattern on a Probability Plot ......................................................28 2.1.7 Probability Plots for Comparing Among Distributions ........................................................30 2.2 Graphical Comparisons of Two or More Datasets ..........................................................................30 2.2.1 Histograms .................................................................................................................................32 2.2.2 Dot-and-line Plots of Means and Standard Deviations .....................................................33 2.2.3 Side-by-side Boxplots ..............................................................................................................34 2.2.4 Q-Q Plots of Multiple Groups of Data ....................................................................................36 2.3 Scatterplots and Enhancements ........................................................................................................38 2.3.1 Evaluating Linearity ..................................................................................................................39 2.3.2 Evaluating Differences in Central Tendency on a Scatterplot ..........................................42 2.3.3 Evaluating Differences in Spread ..........................................................................................42 2.4 Graphs for Multivariate Data ..............................................................................................................45 2.4.1 Parallel Plots..............................................................................................................................46 2.4.2 Star Plots ....................................................................................................................................46 2.4.3 Trilinear and Piper Diagrams ..................................................................................................48 2.4.4 Scatterplot Matrix.....................................................................................................................50