Statistics and Analysis of Scientific Data

Abstract

Across all sciences, a quantitative analysis of data is necessary to assess the significance of experiments, observations, and calculations. This book was written over a period of 10 years, as I developed an introductory graduate course on statistics and data analysis at the University of Alabama in Huntsville. My goal was to put together the material that a student needs for the analysis and statistical interpretation of data, including an extensive set of applications and problems that illustrate the practice of statistical data analysis. The literature offers a variety of books on statistical methods and probability theory. Some are primarily on the mathematical foundations of statistics, some are purely on the theory of probability, and others focus on advanced statistical methods for specific sciences. This textbook contains the foundations of probability, statistics, and data analysis methods that are applicable to a variety of fields— from astronomy to biology, business sciences, chemistry, engineering, physics, and more—with equal emphasis on mathematics and applications. The book is therefore not specific to a given discipline, nor does it attempt to describe every possible statistical method. Instead, it focuses on the fundamental methods that are used across the sciences and that are at the basis of more specific techniques that can be found in more specialized textbooks or research articles. This textbook covers probability theory and random variables, maximumlikelihood methods for single variables and two-variable datasets, andmore complex topics of data fitting, estimation of parameters, and confidence intervals. Among the topics that have recently become mainstream, Monte Carlo Markov chains occupy a special role. The last chapter of the book provides a comprehensive overview of Markov chains and Monte Carlo Markov chains, from theory to implementation. I believe that a description of the mathematical properties of statistical tests is necessary to understand their applicability. This book therefore contains mathematical derivations that I considered particularly useful for a thorough understanding of the subject; the book refers the reader to other sources in case of mathematics that goes beyond that of basic calculus. The reader who is not familiar with calculus may skip those derivations and continue with the applications.