LINEAR REGRESSION FOR CALIBRATION: HOW CAN WE APPLY THIS UNDER THE GUM?

When the response of an apparatus is calibrated, linear regression is often the tool applied to put a value on the calibration factor. However, the standard deviation quoted by common statistics programs is based on the scatter of the points to the straight line. This value is not compatible with the Guide to the Expression of Uncertainties in Measurements (GUM). The reasons for this will be discussed in more detail in the paper. The GUM proscribes a „bottom-up? approach to estimating the uncertainty in measurements, with each contributing factor associated with its uncertainty. The question is how this approach can be applied to the common-place use of linear calibration curves. The problem is made easier if we accept that a linear curve fitting has two separate but often confused tasks: firstly to show clearly that a linear relationship is in fact correct and then to calculate the calibration factor(s). The first part can be done using a conventional linear regression. Two methods are presented here to calculate the calibration factor(s): firstly directly from the regression equations and secondly by following a procedure analogous to the measurement of mass on an analytical balance by weight substitution. In the first case we have to be careful how the calibration curve was measured, i.e. what standard samples were used and how large their values are correlated. Often the second option will be the better method. The problem discussed here can be seen as the simplest form of a generalised problem where a factor has to be calculated from a series of standard measured points. Higher polynomials can also be approached using adaptations of methods used for linear curves. However destructive analysis techniques in particular use fits to complex curves to derive the parameters. Making these common techniques usable for a GUM approach will require further work.