The uncertainty of a result from a linear calibration

Abstract

Calibration of a measuring system is at the heart of many chemical measurements. It has direct relevance to the traceability of the measurement and contributes to the measurement uncertainty. A measurement can be seen as a two-step process in which an instrument is calibrated using one or more standards, followed by presentation of a sample to the instrument and the assignment of the value of the measurand. Instrumental analytical methods, particularly chromatographic, spectroscopic and electrochemical methods, are usually calibrated over a range of concentrations of the analyte. Often the calibrations are assumed (or arranged to be) linear and in the past, a graph was prepared by drawing the best straight line by eye through the points. Having obtained a response from the instrument from the sample to be analysed, the concentration of this sample was read off the graph, going from the instrument response on the y-axis to the concentration on the x-axis. While drawing a graph for the purpose of calibration is no longer done in practice, with a spreadsheet performing a least squares regression to obtain the equation of the best straight line, the calibration function is often still referred to as a ‘calibration line’ or ‘calibration curve’. In this paper the commonly used expression for the standard error of a result obtained from a straight line calibration is extended to a quadratic calibration, and the case where weighted regression is necessary. Spreadsheet recipes are given to accomplish these calculations.