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

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.