Standards Development

Scope

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 parameter 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 16269-4:2010 Statistical interpretation of data – Part 4: Detection and treatment of outliers 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 (Annex D).Software to support this standard is available. See Annex H.