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Uncertainty of a Least Squares Curve Fit through Points with Known Uncertainties

Uncertainties in least squares curve fits to data with uncertainties are examined. First, experimental data with nominal curve shapes, representing property profiles between boundaries, are simulated by adding known uncertainties to individual points. Next, curve fits to the simulated data are achieved and compared to the nominal curves. By using a large number of different sets of data, statistical differences between the two curves are quantified and, thus, the uncertainty of the curve fit is derived. Studies for linear, quadratic, and higher-order nominal curves with curve fits up to fourth order are presented herein. Typically, curve fits have uncertainties that are 50% or less than those of the individual data points. These uncertainties increase with increasing order of the least squares curve fit. The uncertainties decrease with increasing number of data points on the curves.