One Parameter Chi-squared Analysis

The chi-squared method for parameter estimation and model testing is developed for the one-parameter case of a line with a slope but no intercept. Curve fitting is motivated, and several methods for curve fitting are introduced. The chi-squared method is shown to be the optimal curve fitting method whenever Gaussian distributed measurement uncertainties and a model are present. The central limit theorem, which assures Gaussian distributed measurement uncertainties for a wide range of physical experiments, is introduced. End-of-chapter problems are included (with solutions in an appendix).

Advanced Topics

A variety of advanced topics are introduced to offer greater challenge for beginners and to answer thorny questions often asked by early researchers who are just starting to use chi-squared analysis. Topics covered include probability density functions, p-values, the derivation of the chi-squared probability density function and its uses, reduced chi-squared, the Poisson distribution, and advanced techniques for maximum likelihood estimation in cases where uncertainties are not Gaussian or the model is nonlinear. Problems are included (with solutions in an appendix).

Case Study 1: Falling Chains

The one-parameter chi-squared methodology for parameter estimation and model testing is applied to an introductory physics lab experiment involving a falling chain. Data collection details along with sample data are provided and the results of the chi-squared analysis are interpreted. Problems are included (with solutions in an appendix).

Case Study 2: Modeling Air Resistance on Falling Coffee Filters

The two-parameter chi-squared methodology for parameter estimation and model testing is applied to an introductory physics lab experiment to model air resistance on falling coffee filters. Data collection details along with sample data are provided and the results of a chi-squared analysis are interpreted. Problems are included (with solutions in an appendix).

Two–Parameter Chi-squared Analysis

The chi-squared parameter estimation and model testing methodology is extended to the two-parameter case of a line with a slope and an intercept. Chi-squared is shown to be a paraboloid and is displayed in a contour plot. Techniques for extracting parameter correlations and uncertainties are developed. End-of-chapter problems are included (with solutions in an appendix).

Statistical Toolkit

Chi-squared analysis requires familiarity with basic statistical concepts like the mean, standard deviation and standard error, and uncertainty propagation. Interesting aspects are presented to challenge even those familiar with these standard topics. End-of-chapter problems are included (with solutions in an appendix).

Introduction

Model testing and parameter estimation are inextricably linked but often done separately. Chi-squared analysis combines model testing with parameter estimation, improving on both and providing answers to five important questions about the results of an experiment on a system for which a model is available: 1. What are the best fit parameter values? 2. Is the best fit a good fit? 3. What are the uncertainties on the best fit parameters? 4. Even if the fit is good, should the model still be rejected? 5. Is the revised model an improvement?