ISO 28037:2026

• Abstract:

  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).

• Status: Published
• Publication Date: 23-Jul-2026
• ICS: 03.120.30 - Application of statistical methods
• Technical Committee: ISO/TC 69/SC 6 - Measurement methods and results
• Drafting Committee: ISO/TC 69/SC 6 - Measurement methods and results
• Current Stage: 6060 - International Standard published
• Start Date: 24-Jul-2026
• Due Date: 15-Sep-2025
• Completion Date: 24-Jul-2026

• Revises: ISO/TS 28037:2010 - Determination and use of straight-line calibration functions
• Effective Date: 06-Jun-2022

ISO 28037:2026 is an ISO International Standard for determining and using straight-line calibration functions. It specifies how to fit a line of the form Y = A + BX to measured data when the stimulus and response values have uncertainties, and when relevant, covariances. The standard is aimed at calibration and measurement work where the uncertainty structure has to be respected, not treated as a generic least-squares exercise.

What does ISO 28037:2026 specify?

ISO 28037:2026 specifies methods for estimating the intercept A and slope B of a straight-line calibration function from measured pairs (x, y), while taking account of the associated uncertainties and covariances. It also specifies how to validate the fitted model and how to use the fitted line to predict X from a measured Y, or Y from a given X.

The scope covers straight-line calibration functions and uses their simple form wherever possible, even though some principles apply more broadly. It does not assume equal variance in the y-values, and it also handles x-value uncertainty when x is not known exactly. The document notes that it does not provide a general treatment of outliers, although the validation tests can flag discrepant data.

The document is organized into clauses 1 to 10 and informative annexes A to G.

Annex	What it covers
Annex A	Matrix operations and factorization tools used in the document
Annex B	Orthogonal factorization approach to the generalized Gauss-Markov problem
Annex C	How to obtain uncertainties and covariances for measured stimulus and response values
Annex D	Uncertainties known only up to a scale factor
Annex E	Worked examples for all uncertainty structures
Annex F	GUM-consistent straight-line parameter uncertainties in the errors-in-variables case
Annex G	Non-GUM-consistent calculation procedures

What are the key requirements of ISO 28037:2026?

ISO 28037:2026 requires the user to start with measured data and a matching uncertainty description, then choose the fitting method that matches that uncertainty structure. The practical point is that the standard is not one single formula - it is a set of calculation routes for different measurement situations.

Inputs, model and uncertainty description

Clause 5 treats the measured data as the basis of the calibration, with X as the independent variable and Y as the dependent variable. Clause 5.2 says the input information includes the measured values and their associated uncertainties and covariances, represented in matrix form when needed. In practice, this means the user has to assemble the uncertainty budget before fitting the line.

Clause 5.3 distinguishes several uncertainty structures, and Clause 5.4 notes that the more general cases require matrix calculations. The simplest cases can be handled with elementary calculations, while the most general case needs more advanced numerical methods. For a buyer, engineer or quality manager, this matters because spreadsheet methods are not always enough when both x and y are uncertain or correlated.

Estimating the straight line

Clause 5.8 gives the ordinary least-squares straight-line fit as a special case. It uses translated data to simplify the calculation and improve numerical stability. That matters when a quick fit is needed and the y-uncertainties are treated as equal.

Clauses 6 to 9 then cover the four main uncertainty structures:

• response data only and no covariance
• stimulus and response data and no covariance
• response data only with covariance
• stimulus and response data with covariance

These clauses each provide a fitting procedure, uncertainty calculation, and model validation. The practical difference is how much of the uncertainty matrix has to be used in the fit.

Validation of the fitted model

Clauses 6.3, 7.3, 8.3 and 9.3 use a chi-squared check when m > 2. The observed chi-squared value is compared with the 95 % quantile of the chi-squared distribution, and if it is exceeded the straight-line model is rejected. In practice, this is the document’s consistency check: it tests whether the scatter of the data is compatible with the stated uncertainty structure and a straight-line relationship.

Clause 5.6 also frames validation in two ways - the data should support the straight-line form, and the spread should agree with the uncertainties. That matters because a fit can look numerically acceptable but still be inconsistent with the measurement model.

General least-squares treatment

Clause 9 gives the most general treatment, including the full covariance matrix for measured x and y values. It uses iterative least-squares calculations with Jacobians and factorization methods to solve the generalized Gauss-Markov problem. This is the route to use when the data have correlated uncertainties or when both stimulus and response uncertainties matter.

Annex B gives a more general algorithm for valid positive semi-definite covariance matrices, using orthogonal factorizations. That is useful when the covariance structure is broader than the main iterative route in Clause 9.

Using the fitted calibration function

Clause 10 explains how to use the fitted line after the parameters have been estimated. For inverse evaluation, it gives the formula for estimating X from a measured Y and propagating the uncertainty from the fitted parameters and the Y measurement. For forward evaluation, it gives the same kind of treatment in the opposite direction. This is the part that turns the calibration line into a working measurement result.

Annex D covers the special case where the uncertainties are known only up to a scale factor. The document describes how to estimate that scale factor from the dispersion of the data and how to scale the parameter uncertainties accordingly. In practice, that is relevant when the shape of the uncertainty pattern is known but its overall magnitude is not.

What terms does ISO 28037:2026 define?

ISO 28037:2026 defines the statistical and matrix terms needed to apply the calibration methods correctly.

• Correlation matrix - A symmetric positive-definite matrix that contains the correlations between components of an estimated vector quantity.
• Normal distribution - The familiar Gaussian probability distribution used to describe random variables with a mean and standard deviation.
• t-distribution - A probability distribution with degrees of freedom, used here in the treatment of uncertainty and scale estimation.
• Chi-squared distribution - The distribution used for the model-validation test based on weighted residuals.
• Positive definite matrix - A matrix for which the quadratic form is strictly positive for all non-zero vectors.
• Positive semi-definite matrix - A matrix for which the quadratic form is non-negative for all non-zero vectors.
• Quotient - The numerical result of one real number divided by another.

Who uses ISO 28037:2026?

ISO 28037:2026 is used by people who calibrate measuring systems and evaluate measurement uncertainty. That includes metrology specialists, laboratory staff, test engineers, quality managers, and technical buyers who need to judge whether a calibration method matches the data they have.

The examples show applications in torque measurement, gas analysis, leak calibration, pressure transducers, and mass-related calibration data. So the standard is relevant in physics, chemistry, instrumentation, and other measurement-based fields where a calibration curve is used to convert indications into assigned values.

What changed in ISO 28037:2026 from the previous edition?

ISO 28037:2026 replaces ISO/TS 28037:2010 and is described as a technically revised first edition. The foreword lists these main changes:

• improved structure throughout, with more user-friendly information
• some definitions removed and others added, including correlation matrix and quotient
• notation improved to align more closely with the normative references
• the clause on non-zero cross-covariances was deleted because that case was considered not to arise in calibration problems
• GUM-consistent solutions are given in the main text for all uncertainty structures, with a calculation procedure added as an annex
• non-GUM-consistent solutions from the withdrawn technical specification are retained as annexes for users who continue to use them
• real-life examples from physics and chemistry were added
• some references were added
• the annex on the Gauss-Newton algorithm was removed because the information can be obtained elsewhere

Which standards are used with ISO 28037:2026?

ISO 28037:2026 cites several standards and guides that support its terminology and uncertainty treatment.

• ISO 3534-1 - provides statistical vocabulary and symbols used in the definitions.
• ISO/IEC Guide 98-3 - the Guide to the Expression of Uncertainty in Measurement (GUM), which underpins the uncertainty propagation and GUM-consistent solutions.
• ISO/IEC Guide 99 - the International Vocabulary of Metrology (VIM), which supplies metrology terms and concepts.
• ISO/IEC Guide 98-3/Suppl. 1 - supports propagation of distributions using a Monte Carlo method, listed in the bibliography.
• ISO/IEC Guide 98-3/Suppl. 2 - supports extension to any number of output quantities and is referenced in the treatment of covariance.
• ISO 16269-4 - suggested for guidance on outliers when discrepant data need follow-up.
• ISO 11095:1996 - the earlier linear calibration standard discussed for comparison with this document.
• ISO/TS 28038:2018 - a related standard on polynomial calibration functions, listed in the bibliography.

What does the ISO 28037:2026 document contain?

ISO 28037:2026 contains the notation, formulas, calculation procedures, validation tests, and uncertainty evaluations needed for straight-line calibration. Clauses 4 to 10 cover conventions, principles, special cases by uncertainty structure, and the use of the fitted calibration function.

The annexes provide the computational support: matrix operations, factorization methods, how to build uncertainty and covariance inputs, treatment of scale-factor uncertainty, worked examples, and alternative procedures for users who need them. The examples include tables, figures, and step-by-step calculations for calibration and subsequent inverse or forward evaluation.

International
Standard
ISO 28037
First edition
Determination and use of straight-
2026-07
line calibration functions
Détermination et utilisation des fonctions d'étalonnage linéaire
Reference number
© ISO 2026
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on
the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below
or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: [phone]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii
Contents
Foreword . vi
Introduction . viii
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 2
4 Conventions and notation . 4
4.1 General . 4
4.2 Symbols . 5
5 Principles of straight-line calibration. 6
5.1 General . 6
5.2 Inputs to determining the calibration function . 7
5.2.1 Measured data . 7
5.2.2 Associated uncertainties and covariances . 7
5.3 Determining the calibration function . 8
5.3.1 Line parameters . 8
5.3.2 Input uncertainty information . 8
5.3.3 Output uncertainty information . 9
5.4 Numerical treatment . 9
5.5 Uncertainties and covariance associated with the calibration function parameters . 9
5.5.1 Line parameter uncertainties . 9
5.5.2 Primary outputs . 10
5.6 Validation of the model . 10
5.6.1 Departure of data points from fitted calibration function . 10
5.6.2 Testing under normality . 10
5.6.3 Relevance of prior uncertainty information . 11
5.6.4 Failure of model validation . 11
5.7 Use of the calibration function . 11
5.7.1 Inverse evaluation . 11
5.7.2 Forward evaluation . 11
5.8 Determining the ordinary least-squares’ straight-line fit to data . 11
5.8.1 Ordinary least-squares objective function . 11
5.8.2 Steps in the calculation . 12
5.8.3 Extension to other forms of input uncertainty information . 12
6 Uncertainties associated with the response data only and no covariance . 12
6.1 Model . 12
6.1.1 Input information . 12
6.1.2 Statistical model . 13
6.1.3 Objective function . 13
6.2 Calibration parameter estimates and associated standard uncertainties and
covariance . 13
6.2.1 Steps in the calculation . 13
6.2.2 Solution properties . 14
6.3 Validation of the model . 15
iii
6.4 Organization of the calculation . 15
6.5 Illustrative examples . 16
7 Uncertainties associated with the stimulus and the response data and no
covariance . 16
7.1 Model . 16
7.1.1 Input information . 16
7.1.2 Statistical model . 16
7.1.3 Objective function . 17
7.2 Calibration parameter estimates and associated standard uncertainties and
covariance . 19
7.2.1 Calculation procedure . 19
7.2.2 Solution properties . 20
7.3 Validation of the model . 21
7.4 Example . 21
8 Uncertainties and covariances associated only with the response data . 21
8.1 Model . 21
8.1.1 Input information . 21
8.1.2 Statistical model . 22
8.1.3 Objective function . 22
8.2 Calibration parameter estimates and associated standard uncertainties and
covariance . 22
8.2.1 General . 22
8.2.2 Calculation procedure . 22
8.2.3 Solution properties . 23
8.3 Validation of the model . 24
8.4 Organization of the calculations . 24
8.5 Example . 25
9 Uncertainties and covariances associated with the stimulus and the response data . 25
9.1 Model . 25
9.1.1 Input information . 25
9.1.2 Statistical model . 26
9.1.3 Line parameter estimates . 26
9.2 Calibration parameter estimates and associated standard uncertainties and
covariance . 26
9.2.1 Calculation procedure . 26
9.2.2 Solution properties . 30
9.3 Validation of the model . 30
9.4 Illustrative examples . 30
10 Use of the calibration function . 31
10.1 General . 31
10.2 Inverse evaluation . 31
10.2.1 Input information . 31
10.2.2 Computation . 31
10.3 Forward evaluation . 32
10.3.1 Input information . 32
10.3.2 Computation . 32
Annex A (informative) Matrix operations . 33
Annex B (informative) Orthogonal factorization approach to solving the generalized
Gauss-Markov problem . 39
iv
Annex C (informative) Provision of uncertainties and covariances associated with the
measured stimulus and response values . 42
Annex D (informative) Uncertainties known up to a scale factor . 46
Annex E (informative) Examples illustrating straight-line regression for all uncertainty
structures . 48
Annex F (informative) GUM-consistent straight-line parameter uncertainties in the
errors-in-variables' case . 63
Annex G (informative) Non-GUM-consistent calculation procedures . 65
Bibliography . 76
v
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national
standards bodies (ISO member bodies). The work of preparing International Standards is normally
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technical committee has been established has the right to be represented on that committee.
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the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all
matters of electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
ISO draws attention to the possibility that the implementation of this document may involve the use of
(a) patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed
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For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
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.
www.iso.org/iso/foreword.html
This document was prepared by Technical Committee ISO/TC 69, Application of statistical methods,
Subcommittee, Subcommittee SC 6, Measurement methods and results.
This first edition of ISO 28037 cancels and replaces the first edition (ISO/TS 28037:2010), which has
been technically revised.
The main changes are as follows:
— improved structuring has been implemented throughout and more user-friendly information
incorporated;
— some definitions have been removed (functional model, structural model, etc.) and others have
been added (correlation matrix, quotient);
— some notation has been improved giving closer alignment with the normative references.
xy,
— the clause concerned with non-zero cross-covariances ( ij≠ ) has been deleted since its
( )
ij
occurrence seems not to arise in calibration problems;
— GUM-consistent solutions are provided in the main text for all uncertainty structures considered,
with a calculation procedure added as an annex;
vi
— solutions in the withdrawn TS, which are strictly inconsistent with the GUM, are valid for
sufficiently small -uncertainties and retained, as annexes, for users who wish to continue to use
x
procedures based on the TS;
— real-life examples from physics and chemistry illustrating GUM-consistent solutions have been
added as an annex. Some existing examples in the TS are retained because of the preceding bullet;
— some references have been added to support the updated material;
— the annex concerned with the Gauss-Newton algorithm has been removed because the
information there can readily be obtained elsewhere (a reference is given).
Any feedback or questions on this document should be directed to the user’s national standards body.
.
A complete listing of these bodies can be found at www.iso.org/members.html
vii
Introduction
Calibration is an essential part of many measurement procedures and often involves fitting to
measured data a calibration function that describes the relationship of one variable to another. This
document considers straight-line calibration functions that describe a dependent variable Y as a
function of an independent variable X . The straight-line relationship depends on the intercept A and
the slope of the line referred to as the parameters of the line. A calibration procedure determines
B
estimates a and b of A and B for a particular measuring system under consideration based on
measured data ( xy, ) , i 1,…,m, provided by that system. The measured data have associated
ii
uncertainty, which means there will be uncertainty associated with a and b and covariance between
a
them. This document describes how and b can be estimated given the data and the associated
uncertainty information using least-squares’ regression methods accounting for that information. It
also provides a means for evaluating the uncertainties and covariance associated with these estimates.
The treatment of uncertainty in this document is carried out in a way consistent with
ISO/IEC Guide 98-3 and other guides in the ISO/IEC Guide 98 series. 'GUM-consistency’ in this
document is described as follows. A characterization of the solution is the set of nonlinear equations
given by equating to zero the partial derivatives of the least-squares objective function with respect to
the parameters of the regression model. These equations generally constitute an implicit multivariate
model: see ISO Guide 98-3:2008, Suppl.2., 6.3. Accordingly, estimates of the model parameters are
given by solving those equations and their associated covariance matrix by expression (4) in that
clause (also, see Reference [25]).
Given the uncertainty information associated with the measured data, an appropriate method is used
to estimate the calibration function parameters. This uncertainty information may include quantified
covariance effects, relating to dependencies among some or all the quantities involved.
Once the straight-line model has been fitted to the data, it is necessary to determine whether the
model and data are mutually consistent. In cases of consistency, the model so obtained can validly be
used to predict a value x of the variable X corresponding to a measured value y of the variable Y
provided by the same measuring system, and to evaluate the uncertainty associated with x .
The determination and use of a straight-line calibration function can be considered to consist of five
steps:
a) Obtaining measured data and associated uncertainty and covariance information.
b) Providing estimates of the straight-line parameters accounting for the information in Step a).
c) Validating the model, both in terms of the functional form (does the data reflect a straight-line
relationship?) and statistically (is the spread of the data consistent with their associated
uncertainties?).
d) Obtaining the standard uncertainties and covariance associated with the estimates of the straight-
line parameters.
e) Using the calibration function for inverse evaluation (also sometimes known as ‘prediction’), that
is, determining an estimate x of the X -variable and its associated uncertainty corresponding to a
measured value y of the Y -variable and its associated uncertainty. An estimate y of the 𝑌𝑌-
variable and its associated uncertainty given a value x of the 𝑋𝑋-variable, a process known as
forward evaluation, and its associated uncertainty can also be determined.
Examples are provided, some from various areas of measurement and some synthetic.
viii
=
Figure 1 shows a related dependency chart.

Figure 1 — Dependency chart for determining and using straight-line calibration functions
The main aim of this document is the consideration of Steps 2 to 5. Therefore, as part of Step 1, before
using this document, the user will need to provide standard uncertainties, and covariances if relevant,
associated with the measured Y -values and, as appropriate, those associated with the measured
X -values. Some guidance is given in Annex C. Account is taken of the principles of the GUM in
evaluating these uncertainties based on a measurement model that is specific to the input uncertainty
structure.
[20]
ISO 11095:1996 is concerned with linear calibration using reference materials. It differs from this
document in the ways given in Table 1. For example, the present document addresses data x -values
that may not be known exactly but their associated uncertainties and, when appropriate, covariances
ix
are provided. Moreover, the systematic errors in the data may be appreciable. Ways to account for
such effects are given.
Table 1 — Differences between ISO 11095:1996 and this document
Feature ISO 11095:1996 This document
Specifically addresses reference materials Yes More general
-values assumed to be known exactly Yes More general uncertainty
x
information
All measured values obtained independently Yes More general uncertainty
information
Terminology aligned with GUM Not totally Yes
Types of uncertainty structure treated Two Four, including the most
general case
Only uncertainty associated with random Yes More general uncertainty
errors information
Consistency test ANOVA Chi-squared
Uncertainty associated with inverse Ad hoc GUM-consistent
evaluation
The provisions of this document are supported by Annexes A to H:
x
INTERNATIONAL STANDARD ISO 28037:2026(en)

Determination and use of straight-line calibration
functions
1 Scope
This document is concerned with linear, that is, straight-line, calibration functions that describe the
relationship between two variables X and Y , namely, functions of the form Y A+ BX . 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 A and B are estimated based on measured data points xy, ,
( ) im1, … .
ii
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 y are homoscedastic (having equal variance), and
i
similarly for the x when the errors are not negligible.
i
Estimates of the parameters A and B 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 X and its associated standard
uncertainty given a measured value of Y 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).
2 Normative references
The following documents are referred to in the text in such a way that some or all their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments)
applies.
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: Probability and general statistical terms
ISO/IEC Guide 98-3, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in
measurement (GUM:1995)
ISO/IEC Guide 99, International vocabulary of metrology — Basic and general concepts and associated
terms (VIM)
=
=
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1, ISO/IEC Guide 98-3
and ISO/IEC Guide 99 and the following apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https://www.iso.org/obp
— IEC Electropedia: available at https://www.electropedia.org/
NOTE ISO/IEC Guide 99 uses the term ‘measured quantity value’. Here, the term ‘measured value’, which will
be used in future versions of ISO/IEC Guide 99, is adopted when there is no ambiguity. Similarly, ‘standard
uncertainty’; is used instead of ‘standard measurement uncertainty’ and ‘covariance matrix’ instead of
‘measurement covariance matrix’.
3.1
correlation matrix
symmetric positive-definite matrix of dimension NN× associated with an estimate of a vector
quantity of dimension N ×1, containing the correlations associated with pairs of components of the
estimate
Note 1 to entry: A correlation matrix R of dimension NN× associated with the estimate x of a vector quantity
x
X has the representation
r ( x ,,x )  r ( x x )

11 1 N

R =   ,
x


r ( x ,,x )  r ( x x )

N 1 NN
where r ( x ,1x ) = and rx , x is the correlation associated with estimates x and x . When elements X
( )
ii ij i j i
X rx ,0x =
and of X are uncorrelated, .
( )
j ij
Note 2 to entry: Correlations are also known as correlation coefficients.
Note 3 to entry: R is related to the covariance matrix V by
x x
R =DV D ,
x xx x
−1 −1
where D is a diagonal matrix of dimension NN× with diagonal elements u ( x ) ,,… u ( x ) and ux() is
x 1 N i
the standard uncertainty associated with x . Element (i , j) of R , with ux( , y ) the covariance associated
i x ii
x and x , is
with
i j
ux , x
( )
ij
rx ,,x =
( )
ij
u( x )ux
( )
ij

[SOURCE: ISO/IEC Guide 98-3:2008/Suppl.2:2011, 3.21, slightly modified, Notes 4 and 5 deleted.]
3.2
normal distribution
probability distribution of a continuous random variable 𝑋𝑋 having the probability density function

11 ξµ−

g ξ exp−
( )


x
2 σ

σπ2


for −∞ < ξ < +∞
Note 1 to entry: µ is the expectation and σ is the standard deviation of X .
Note 2 to entry: The normal distribution is also known as the Gaussian distribution.
Note 3 to entry: Definition and note 1 adapted from ISO 3534-1:1993, definition 1.37; note 2 adapted from
ISO/IEC Guide 98-3:2008, definition C.2.14.
3.3
t-distribution
probability distribution of a continuous random variable X having the probability density function
ν + 1 ν +1

−
Γ

 2
ξ
2
g (ξ ) 1,+

x

ν ν
 

πν Γ
 
 
for −∞ < ξ < +∞, with parameter ν , a positive integer, the degrees of freedom of the distribution,
where
∞
zt−1 −
Γ>z = t e dtz, 0,
( )
∫
is the gamma function
[SOURCE: ISO/IEC Guide 98-3:2008/Suppl.1:2008 3.5]
3.4
chi-squared distribution
probability distribution of a continuous random variable X having the χ probability density
function
ν
 
−1
 
 
ξξ

g (ξ ) exp− ,

x
ν
2
ν
 
2 Γ
 
 2 
for 0 ≤ ξ <∞ , with parameter ν , a positive integer, where Γ is the gamma function
Note 1 to entry: The sum of the squares of ν independent standardized normal variables is a χ random
variable with parameter ν , termed the degrees of freedom.
3.5
positive definite matrix
•
matrix M of dimension nn× having the property z Mz > 0 for all non-zero vectors z of dimension

n×1
=
=
=
3.6
positive semi-definite matrix
•
matrix M of dimension nn× having the property z Mz ≥ 0 for all non-zero vectors z of dimension
n×1
3.7
quotient
value of NN/ resulting from the division of the real number N by the real number N
12 1 2
Note 1 to entry: ‘Ratio’ is not a synonym for ‘quotient’, the former being a statement of the relative proportions of
two numbers in the form NN: . To illustrate, for N = 9 and N = 16, their quotient is 9/16 = 0,562 5 and
1 2 1 2
their ratio is 9:16.
4 Conventions and notation
4.1 General
The following conventions and notation are adopted in this document.
X is termed the independent variable and Y the dependent variable even when the knowledge of X
and Y is 'interchangeable', as in Clause 7, for example.
The quantities A and B are termed the parameters of the straight-line calibration function
Y A+ BX . A and B are also used to denote random variables in expressions involving the
calibration function parameters.
th
The quantities X and Y are used as random variables to denote the coordinates of the i data point,
i i
im1,… , .
th
x and y are the measured values of the coordinates of the i data point X and Y .
i i i i
* *
th
x and y are estimates of the coordinates of the i data point X and Y .
i i i i
a
and b are estimates of A and B that specify the straight-line calibration function for a particular
measuring system.
* * **
a , b , and the x and y satisfy y a+ bx , .
i = 1, ., m
i i ii
a and b specify the estimated (fitted) straight-line calibration function Y a+ bX for a particular
measuring system.
A vector of dimension m×1 and its transpose are generically denoted by

x

•

xx ,,  xx…

1 m


x

m
where • denotes transpose, and a matrix of dimension mn× by
   
a ……a aa
11 1nm11 1
   
•
AA   ,.    
   
   
a ……a aa
   
m11mn n mn
   
==
= =
=
=
=
=
The dimension of a vector or matrix is often specified to avoid possible confusion.
The zero matrix or vector is denoted by 0 and the unit vector by 1.
In matrices, a zero element is often indicated by a blank. A zero submatrix is often indicated by 0.
Some symbols have more than one meaning. The context clarifies the usage.
Numbers displayed in tables to a given number of decimal places are correctly rounded
representations of numbers stored to higher numerical precision, as would be the case in a
spreadsheet, for example. Therefore, minor inconsistencies may be perceived, for instance, between
displayed column sums and the column sums of the displayed numbers.
In some tables, a subclause number above a column or columns indicates where the formula is given
for determining the values below.
The mathematical symbols used in Annex A and Annex B have generally a different interpretation from
those in the bulk of this document,
In the examples, while data values are provided to a given numerical precision, the results of
calculations are sometimes provided to a higher precision to allow the user to compare results when
undertaking the calculations.
4.2 Symbols
A intercept of the straight-line calibration function
*
unknown value of A for a particular measuring system
a
a
estimate of A
•
a
vector (ab,) of parameter estimates
B slope of the straight-line calibration function
*
unknown value of B for a particular measuring system
b
b
estimate of B
* 2
d
i xX− ux
, a realization of a random variable with expectation zero and variance ( )
ii i
* 2
e
i
yY− , a realization of a random variable with expectation zero and variance uy( )
ii i
L
lower-triangular matrix
m
number of measured points
r weighted residual or weighted distance for the ith data point in terms of a and b
i
R weighted residual or weighted distance for the ith data point expressed in terms of A and
i
B
u standard deviation of random variable with distribution encoding knowledge of a random
R
effect
u as for u but for distribution encoding knowledge of a systematic effect
S
R
uz( ) standard uncertainty associated with denoting a, b, x , y , etc.
z
ii
u a, b
( ) covariance associated with a and b
V
covariance matrix of dimension 22mm× associated with the data ( xy, ), =i 1,…,m
ii
V covariance matrix of dimension 2×2 associated with
a
a
V
covariance matrix of dimension mm× associated with the data xi, 1,…,m
x i
V covariance matrix of dimension mm× associated with the data y , i 1,…,m
y i
v reciprocal of ux
( )
i i
w
reciprocal of uy( )
i
i
X independent (stimulus) variable
X i th independent (stimulus) variable
i
*
unknown value of i th dependent (response) variable provided by a measuring system
X
i
x unknown value of i th independent (stimulus) variable provided by a measuring system
x estimate of X (in the case of inverse evaluation) or measured value of X (forward
i
evaluation)
*
estimate of i th independent (stimulus) variable
x
i
Y
dependent (response) variable
Y i
th dependent (response) variable
i
*
i
unknown value of th dependent (response) variable provided by a measuring system
y
i
y
measured value of Y (in the case of inverse evaluation) or estimate of Y (forward
evaluation)
y
i th measured value of Y
i
*
estimate of i th dependent (response) variable
y
i
ν
degrees of freedom of a model, a chi-squared distribution or a t-distribution
σ
standard deviation of a random variable characterized by a probability distribution
σˆ
estimate of σ
observed chi-squared value
χ
obs
chi-squared distribution with ν degrees of freedom
χ
ν
5 Principles of straight-line calibration
5.1 General
This clause considers how a relationship Y A+ BX describing the dependent variable Y (also called
'response') as a function of the independent variable X (also called 'stimulus') can be determined
from measured data. In the context of calibration, the measured data arise when a measuring
a
instrument specified by (unknown) values and b of the calibration function parameters is
'stimulated' by artefacts with calibrated values x of X given in standard units, of a property of the
i i
artefacts, and the corresponding 'responses' or indications y of Y of the instrument are recorded.
i i
The relationship provides the response Y of the system given an artefact with calibrated quantity X .
This process is termed 'forward evaluation'. More useful in practice, the relationship allows a
measured response of Y to be converted to an estimate , in standard units, of the property X of
y x
an artefact. This process is termed 'inverse evaluation'.
=
=
=
The calibration of a measuring system should consider measurement uncertainties, and, if present,
covariances associated with the measured data. The output of a calibration procedure is a calibration
function to be used for inverse evaluation (and, if required, forward evaluation). The output also
includes the standard uncertainties and covariance associated with the estimates a and b of the
parameters A and B describing the calibration function, which are used to evaluate the standard
uncertainties associated with inverse and forward evaluation.
5.2 Inputs to determining the calibration function
5.2.1 Measured data
The information required to determine the straight-line calibration function is the measured data and
their associated standard uncertainties and covariances. In this document, the measured data are
denoted by ( xy, ) , i 1,…,m , that is, m pairs of measured values of X and Y . It is assumed that m
ii
is at least two and at least two of the x are distinct.
i
NOTE 1 The uncertainties associated with the estimates a and b generally decrease as m increases. Therefore,
calibration aims to use as many measured data points as is economically viable.
NOTE 2 It is beyond the scope of this document to consider strategies for selecting the measured points. For
[29]
this purpose, references on the theory of optimal experimental design such as can be consulted. Often,
however, the choice of measured points is dictated by the calibration experiment.
5.2.2 Associated uncertainties and covariances
The standard uncertainties associated with x and y are denoted by ux( ) and uy( ) respectively.
i i i i
x x ux , x
The covariance associated with and is denoted by . Similarly, the covariances
( )
ij
i j
associated with y and y , and with x and y , are denoted by uy , y and ux , y , respectively.
( ) ( )
i j i j ij ij
Annex C indicates how the uncertainties and covariances associated with the measured response and
stimulus variables can be evaluated and gives an interpretation of that uncertainty information. The
complete uncertainty information is represented by an array of elements (symmetric matrix) V of
dimension 2m × 2m holding the variances (squared standard uncertainties) u ( x ) = ux( , x ) and
i ii
u y = uy , y
( ) ( ) and the covariances:
i ii
For many applications, some or all covariances are zero or taken as such (see 5.3).
NOTE 1 This document is concerned with problems in which the ux( ) and the uy( ) can take any appropriate
i i
values.
NOTE 2 The data x and y and the associated uncertainties and covariances are assumed to be the only input
i i
information available.
=
5.3 Determining the calibration function
5.3.1 Line parameters
The inputs to determining the calibration function are the measured data and their associated
uncertainties and possibly covariances. The inputs can be used to provide a measure of the departure
th
of the i data point ( xy, ) from the line Y A+ BX . The estimates a and b of A and B are
ii
determined by minimizing a weighted sum of squares of these departures, or a more general measure
when any covariances are non-zero. The way this computation is carried out depends on the
'uncertainty structure' associated with the measured data. This uncertainty structure relates to the
answers to the following questions:
a) Are the uncertainties associated with the measured values x negligible?
i
b) Are the covariances associated with pairs of measured values negligible?
5.3.2 Input uncertainty information
The following cases, given in increasing order of complexity and depending on the answers to the
questions in 5.3.1, are considered in this document:
a) The only uncertainties are associated with the measured values y and all covariances associated
i
with the data are regarded as negligible (see Clause 6).
b) Uncertainties are associated with the measured values x and y and all covariances associated
i i
with the data are regarded as negligible (see Clause 7).
c) The only uncertainties are associated with the measured values y and the only covariances are
i
associated with the y and the y ij≠ (see Clause 8).
( )
i j
d) Uncertainties are associated with the measured value x and y and covariances associated with
i i
all pairs of values of the x and the x and with the y and the y (the most general case
i j k 
considered, see Clause 9).
The case of non-zero cross-covariances ux , y (i ≠ j) is not considered since its occurrence is
( )
ij
extremely rare in calibration problems. The covariance matrix
...