Fitting Curves to Experimental Data by Least Squares: An Examination of the Method of Plotting Experimental Results

Abstract

IF a number n of readings are taken of a dependent variable y for various values of the independent variable x we frequently need to determine a reasonably satisfactory curve to express the relation between the two variables; preferably we should be able to give an equation for this curve. We know that the observations will be subject to random errors due to various causes; we therefore expect the curve not to go through all the points representing the observations, but to lie evenly among these points. If the curve is expected to be a straight line, we may be able to determine it with sufficient accuracy by eye. The ‘least‐squares’ technique for finding the equation of the best‐fitting curve C for which y is a polynomial of degree m in x is well known. If m is small, the curve Cm will be smooth, but may not fit very well; if m=n−1, the fit is perfect, but Cm is likely to have several oscillations which do not correspond to reality. If m has an intermediate value, there will be some oscillations and a fair fit; increasing m rapidly increases the complexity of the equations determining Cm, and the unreliability of Cm from the statistical point of view.

Citation

Head, J.W. and Oulton, G.M. (1957), "Fitting Curves to Experimental Data by Least Squares: An Examination of the Method of Plotting Experimental Results", Aircraft Engineering and Aerospace Technology, Vol. 29 No. 9, pp. 268-270. https://doi.org/10.1108/eb032868