Last year I had an experience that I imagine is common among mathematicians, especially those of us who teach at small colleges. Despite the fact that my training is in areas of mathematics as pure and abstract as one could imagine, I was asked to teach a course in probability and statistics. And while I felt that my knowledge of the theoretical aspects of the material was good enough to teach the class, the fact that I have never been a practitioner of statistics led to a few awkward moments, as I often did not have good examples at my fingertips of how my students would actually *use* the theory in their courses outside of the math department. I wish I had seen a copy of the new book *A Quantitative Tour of the Social Sciences*, edited by Andrew Gelman and Jeronimo Cortina and published this spring by Cambridge University Press.

The book originated as part of a course taught at Columbia University, where five professors from different social science disciplines — history, economics, sociology, political science, and psychology — spent three weeks apiece discussing quantitative methods in their disciplines. The five main parts of the book come from these disciplines, and they vary greatly in style and level of technical detail. While the book is sprinkled with exercises, ranging from discussion questions and essays to computational questions that one could use to practice some of the techniques discussed, it is definitely not written in a traditional textbook style: all of the authors write very casually and rather than describing the statistical methods they use in full generality they tend to focus on specific case studies. As is probably to be expected, there is a good deal of variation in other ways, and I found some of the chapters much more interesting than others, although I think this is largely a matter of taste.

The first part of the book is an introduction written by Andrew Gelman, who holds joint positions in the statistics department and the political science department at Columbia. He introduces the motivation for the book and gives a justification for using quantitative methods in the social sciences at all — something which I imagine will be uncontroversial among MAA members, but perhaps less so among a group of sociologists or historians.

He then discusses a few examples of the use and limitations of quantitative methods by social scientists that are of particular interest to him, including estimating the dollar value of a life and using game theory and the prisoners dilemma to model trench warfare. There is also a section on political representation and fairness, and eaders of Gelman’s recent book *Red State, Blue State, Rich State, Poor State* or his blog on things statistical will not be surprised to learn that these chapters are very well written, as he asks interesting questions and uses statistical reasoning to get new insights into their answers.

The second part of the book deals with statistical methods used by historians and is written by Herbert S. Klein and Charles Stockley. It begins by discussing the history of quantitative historical studies, and the authors give some examples of the history of quantitative thinking in their field. They then lament that the discipline of history has moved away from quantitative methods and modelling at the same time that many of the other social sciences have embraced statistical techniques, writing that “too many historians think like lawyers and pile on evidence to make their case while ignoring both the biases of their sources and data that do not fit their argument.” (In full disclosure, I recently brought this point up over lunch with several of my colleagues in the history department and they had a somewhat different take on the situation) Klein and Stockley go on to make some compelling arguments in favor of quantitative historical thinking, discussing case studies about international exchange rates over time, estimating the size of the slave trade from the historical records, and immigration. They give compelling arguments as to why historians have much to add to these questions rather than economists or demographers. Throughout, the authors give information about how historians can find sources of data and use them to pose and answer interesting questions.

The social science which has embraced statistical methods with the most vigor is probably economics, and the third part of this book consists of several chapters written by Richard Clarida and Marta Noguer about some case studies in economics. They state up front that there is no way to give a complete overview of quantitative methods in economics in a year, let alone a segment of a course, so they focus on a few issues related to time series analysis which hold special interest to them. In particular, they look at econometric forecasting as well as some studies of the interaction between monetary policy and interest rates. In this latter chapter, the authors discuss a variable which is important for economists to understand yet which cannot be directly measured, the expected real rate of interest. While it cannot be measured directly, one can attempt to estimate it and test various hypotheses about it, and this is what the authors use time-series analysis to do. This section of the book is perhaps the most technical mathematically, and readers who are not already familiar with a reasonable amount of statistics will likely need to go look things up in order to follow the authors’ work.

The fourth part of the book, written by Seymour Spilerman and Emanuele Gerratana, is about quantitative methods in sociology. They begin with a chapter which discusses the very notion of explanatory structures and the way that quantitative data can give feedback to help sociologists develop and refine their theories. The next chapter looks at racial disturbances in American cities in the 1960s, and walks the reader through various models used to explain why there were more disturbances in some cities than in others.

The authors do a particularly nice job looking at the differences between what would be predicted by models which would lead to Poisson or negative binomial distributions and how this can be used to help distinguish between the models. Another chapter looks at time-series analyses of lynchings in the American south at the end of the nineteenth century, spending time discussing how one can use statistics to even try to get an accurate count of the number of lynchings as well as their geographical distribution. A final chapter in this section looks at career advancement in a large insurance company, and some models to describe various phenomena about the topic. Spilerman and Gerratana do a very nice job of balancing mathematical exposition with sociological questions, and I found this section of the book to be the most approachable.

Charles Cameron contributes several chapters on political science, first giving a brief overview of several different subdisciplines of political science and their use of quantitative reasoning. He then goes on to focus on supreme court nominations, and looks at different models of how senators determine whether to approve a nominee to the supreme court as well as how the presence of scandals affects the chances of a given nominee. He goes on to consider questions about the hearings of such a nominee, after noticing that nominees who have long hearings are more likely to be rejected than those with short hearings. In particular, he constructs theories around the length of a hearing compared to the ideological distance between the chair of the judiciary committee and the nominee, and shows how various data can be used to analyze the situation.

In the sixth part of the book, E. Tory Higgins, Elke Weber, and Heidi Grant write about psychology. Their section of the book is perhaps the least quantitative of all of the sections, and they spend most of their pages discussing meta-questions about what makes a good theory and what is even meant by a theory in psychology. It is perhaps this less quantitative approach which made this section the least interesting to me, although I must admit that the authors may have colored my impression of their chapters early on when they write that “Math, like all other sciences, is not some heavenly light shining truth down upon is; rather it is a *social product*.” Whether you agree with this statement or not, I think that most MAA readers will find their example that “at some point it was simply decided that certain mathematical operations won’t work unless we all agree that you can’t divide by 0” to be a bit problematic.

While some of the stops on this quantitative tour are more interesting than others, I found it to be a tour worth taking. This book is not written for mathematicians, and I imagine that most MAA members will find parts of the book to be overly simplistic and other parts to be uninteresting. But I also think that most of the book is well written and engaging, and most of us will find parts of great interest. Moreover, anyone looking for some examples or readings to bring into a course on probability and statistics which is populated by social science students will find some good ones in this volume.

Darren Glass is an associate professor of mathematics at Gettysburg College whose training is in Galois theory, but is finding himself more drawn to mathematical questions from the social sciences. He can be reached at

[email protected].