Mathematics for Social Scientists

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.

• Part I: ALGEBRA, PRECALCULUS, AND PROBABILITY1. Algebra ReviewNumbersFractionsExponentsRootsLogarithmsSummations and ProductsSolving Equations and Inequalities2. Sets and FunctionsSet NotationIntervalsVenn DiagramsFunctionsPolynomials3. ProbabilityEvents and Sample SpacesProperties and Probability FunctionsCounting TheorySampling ProblemsConditional ProbabilityBayes′ RulePART II: CALCULUS4. Limits and DerivativesWhat is a Limit?Continuity and AsymptotesSolving LimitsThe Number ePoint Estimates and Comparative StaticsDefinitions of the DerivativeNotationShortcuts for Finding DerivativesThe Chain Rule5. OptimizationTerminologyFinding Maxima and MinimaThe Newton-Raphson Method6. IntegrationInformal Definitions of an IntegralRiemann SumsIntegral NotationSolving IntegralsAdvanced Techniques for Solving IntegralsProbability Density FunctionsMoments7. Multivariate CalculusMultivariate FunctionsMultivariate LimitsPartial DerivativesMultiple IntegralsPART III: LINEAR ALGEBRA8. Matrix Notation and ArithmeticMatrix NotationTypes of MatricesMatrix ArithmeticMatrix MultiplicationGeometric Representation of Vectors and Transformation MatricesElementary Row and Column Operations9. Matrix Inverses, Singularity, and RankInverse of a (2 x 2) MatrixInverse of a Larger Square MatrixMultiple Regression and the Ordinary Least Squares EstimatorSingularity, Rank, and Linear Dependency10. Linear Systems of Equations and EigenvaluesNonsingular Coefficient MatricesSingular Coefficient MatricesHomogeneous SystemsEigenvalues and EigenvectorsStatistical Measurement Models

"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."