Chapter 8: Least-Squares Fitting

Using the principle of maximum likelihood and the assumption that uncertainties exist primarily in the dependent variable while the independent variable remains relatively certain, the derivation yields two normal equations whose simultaneous solution produces explicit formulas for the slope and y-intercept. The treatment then addresses how to estimate measurement uncertainty from the data itself by examining residual scatter around the best-fit line, introducing the concept of degrees of freedom to explain why the denominator uses N minus two rather than N, since two parameters must first be extracted from the data. Once the standard deviation of the measurements is known, error propagation techniques determine the individual uncertainties associated with both constants. The chapter extends beyond simple linear relationships to cover polynomial fitting, which generates additional simultaneous equations for each new parameter, and exponential fitting through linearization via logarithmic transformation, though this approach technically requires weighted methods to maintain equal uncertainty assumptions. Multiple regression analysis is introduced for situations where one variable depends linearly on several independent variables. Special cases receive particular attention, including fits constrained to pass through the origin where only the slope requires determination, and weighted least-squares fitting for datasets where measurements possess unequal uncertainties, in which each data point receives a weight inversely proportional to its variance. Throughout, the chapter emphasizes that least-squares fitting provides both optimal parameter estimates and quantifiable measures of their precision, making it indispensable for translating experimental observations into reliable physical relationships.