This document is concerned with linear, that is, straight-line, calibration functions that describe the relationship between two variables  and , namely, functions of the form . Although many of the principles apply to more general types of calibration function, the approaches described exploit the simple form of the straight-line calibration function wherever possible.

Values of the parameters  and  are estimated based on measured data points  ,  Various cases are considered relating to the nature of the uncertainties associated with these data. No assumption is made that the errors relating to the  are homoscedastic (having equal variance), and similarly for the  when the errors are not negligible.

Estimates of the parameters  and  are determined using least‑squares’ methods. The emphasis of this document is on using the method most appropriate for the type of measured data, that is, respecting the associated uncertainties. The most general type of covariance matrix associated with the measured data is treated, but important special cases that lead to simpler calculations are described in detail.

For all cases considered, methods for validating the use of the straight-line calibration functions and for evaluating the uncertainties and covariance associated with the parameter estimates are given.

The document also describes the use of the estimates of the calibration-function parameters and their associated uncertainties and covariance to predict a value of  and its associated standard uncertainty given a measured value of  and its associated standard uncertainty.

NOTE 1    The document does not give a general treatment of outliers in measured data, although the validation tests given can be used to indicate discrepant data. ISO 16269-4 can be consulted for guidance.

NOTE 2    The document describes a method to evaluate the uncertainties associated with the measured data when those uncertainties are known only up to a scale factor (see Annex D).