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Mathematics for Social Scientists


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Table of Contents

Part I: ALGEBRA, PRECALCULUS, AND PROBABILITY 1. Algebra Review Numbers Fractions Exponents Roots Logarithms Summations and Products Solving Equations and Inequalities 2. Sets and Functions Set Notation Intervals Venn Diagrams Functions Polynomials 3. Probability Events and Sample Spaces Properties and Probability Functions Counting Theory Sampling Problems Conditional Probability Bayes' Rule PART II: CALCULUS 4. Limits and Derivatives What is a Limit? Continuity and Asymptotes Solving Limits The Number e Point Estimates and Comparative Statics Definitions of the Derivative Notation Shortcuts for Finding Derivatives The Chain Rule 5. Optimization Terminology Finding Maxima and Minima The Newton-Raphson Method 6. Integration Informal Definitions of an Integral Riemann Sums Integral Notation Solving Integrals Advanced Techniques for Solving Integrals Probability Density Functions Moments 7. Multivariate Calculus Multivariate Functions Multivariate Limits Partial Derivatives Multiple Integrals PART III: LINEAR ALGEBRA 8. Matrix Notation and Arithmetic Matrix Notation Types of Matrices Matrix Arithmetic Matrix Multiplication Geometric Representation of Vectors and Transformation Matrices Elementary Row and Column Operations 9. Matrix Inverses, Singularity, and Rank Inverse of a (2 x 2) Matrix Inverse of a Larger Square Matrix Multiple Regression and the Ordinary Least Squares Estimator Singularity, Rank, and Linear Dependency 10. Linear Systems of Equations and Eigenvalues Nonsingular Coefficient Matrices Singular Coefficient Matrices Homogeneous Systems Eigenvalues and Eigenvectors Statistical Measurement Models

About the Author

Jonathan Kropko is a professor in the Department of Politics at the University of Virginia, where he also serves on the steering committee of the Quantitative Collaborative, an interdisciplinary research initiative for applied statistics in the social sciences. Previously, he held a postdoctoral research fellowship at the Applied Statistics Center at Columbia University and was a statistics consultant at the H. W. Odum Institute for Research in the Social Sciences at the University of North Carolina. He holds degrees in mathematics (BS) and political science (BA) from Ohio State University, and earned a PhD in political science from the University of North Carolina in 2011. He is a specialist in political methodology, with a focus on missing data imputation, time series, and measurement methods.


"Students in the social and behavioral sciences increasingly need a solid foundation of mathematical knowledge to be able to contribute to the research literature and be able to keep themselves current on new methodology. Unfortunately, math department classes really are not tailored to their needs. Mathematics for Social Scientists, on the other hand, is clearly aimed at what students need to be able to advance in subsequent methodology courses and in their future careers. It is written in an inviting and clear manner, without ever sacrificing rigor." -- Jay Verkuilen
"Many students entering higher-level statistics classes have somehow forgotten their basic statistics or were never properly exposed to more than a cookbook explanation. More often than not, a student will leave the course without an understanding of probability, random variables, basic distribution theory and concepts etc. Without some background, it proves difficult for students to catch up with these ideas when they are introduced (or assumed to be known) in more advanced courses. This gap is especially pronounced between those students who were exposed to basic probability in a previous course and those who were not. Mathematics for Social Scientists will be a great resource for an instructor wishing to add this content to a basic statistics course as well as for the motivated self-learner." -- Dan Powers

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