A Global Least-Squares Fit for Absolute Zero

Abstract

A simple, nonlinear least-squares method is described that permits gas thermometry data to be fitted directly to absolute zero. This nonlinear method can be implemented using Solver in Excel, and unlike other linear methods previously reported, it is statistically sound. The Excel macro SolverAid can be used to compute the error in absolute zero. The method can be applied simultaneously to multiple sets of data, permitting a global value of absolute zero to be computed from different gas samples. Constant volume thermometry data for helium are used to illustrate the global fit to absolute zero using Solver in an Excel spreadsheet. The relationship between the global value of absolute zero and the values from the individual fits is analyzed.

Using Least Squares To Solve Systems of Equations

“…We can obtain multiple estimates of the common parameters from separate fits and then average, but the statistically optimal approach is to estimate them from a single, combined or global fit. 2 In the two-straight-lines example, a common parameter reduces p to 3, and we obtain a LS solution as long as both n 1 and n 2 ≥ 1 and n 1 + n 2 > 3. We obtain an exact solution when one of these n's is 1 and the other 2.…”

“…More interesting are cases where the multiple equations in the fit model share some parameters, for example, a common intercept or a common slope in the fit of data to two or more straight lines. We can obtain multiple estimates of the common parameters from separate fits and then average, but the statistically optimal approach is to estimate them from a single, combined or global fit . In the two-straight-lines example, a common parameter reduces p to 3, and we obtain a LS solution as long as both n 1 and n 2 ≥ 1 and n 1 + n 2 > 3.…”