SIST ISO 11222:2004
(Main)Air quality -- Determination of the uncertainty of the time average of air quality measurements
Air quality -- Determination of the uncertainty of the time average of air quality measurements
ISO 11222:2002 provides a method for the quantification of the uncertainty of a time average of a set of air quality data obtained at a specified location over a defined averaging time period. The method is applicable to air quality data obtained by continuous or intermittent monitoring by means of a specified measuring system. The uncertainty of the time average depends on both the uncertainty of the measurement results and the uncertainty due to incomplete time coverage of the data set.
ISO 11222:2002 is only applicable if
the set of air quality data used to calculate the time average is representative of the temporal structure of the measurand over the defined time period,
appropriate information on the uncertainty of the measurement results is available, and
the measurement results have all been obtained at the same location.
ISO 11222:2002 implements recommendations of the Guide to the Expression of Uncertainty in Measurement (GUM).
Qualité de l'air -- Détermination de l'incertitude de mesure de la moyenne temporelle de mesurages de la qualité de l'air
L'ISO 11222:2002 fournit une méthode permettant de quantifier l'incertitude d'une moyenne temporelle d'un ensemble de données relatif à la qualité de l'air, obtenu à partir d'un point de mesurage spécifié, sur une période moyenne de temps définie. La méthode est applicable aux données relatives à la qualité de l'air obtenues par surveillance continue ou intermittente, au moyen d'un système de mesure spécifié. L'incertitude de la moyenne temporelle dépend à la fois de l'incertitude des résultats de mesure et de l'incertitude due à une couverture incomplète de l'ensemble de données.
L'ISO 11222:2002 est applicable uniquement si
l'ensemble de données relatif à la qualité de l'air utilisé pour calculer la moyenne temporelle est représentatif de la structure temporelle du mesurande sur la période de temps définie,
les informations appropriées relatives à l'incertitude des résultats de mesure sont disponibles, et
les mesurages ont été effectués au même endroit.
L'ISO 11222:2002 met en oeuvre les recommandations du Guide pour l'expression de l'incertitude de mesure (GUM).
Kakovost zraka – Določanje negotovosti časovnih povprečij pri meritvah kakovosti zraka
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INTERNATIONAL ISO
STANDARD 11222
First edition
2002-07-01
Air quality — Determination of the
uncertainty of the time average of air
quality measurements
Qualité de l'air — Détermination de l'incertitude de mesure de la moyenne
temporelle de mesurages de la qualité de l'air
Reference number
ISO 11222:2002(E)
©
ISO 2002
---------------------- Page: 1 ----------------------
ISO 11222:2002(E)
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ii © ISO 2002 – All rights reserved
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ISO 11222:2002(E)
Contents Page
Foreword.iv
Introduction.v
1 Scope.1
2 Normative reference.1
3 Terms and definitions .1
4 Symbols and abbreviated terms .4
5 Requirements on the input data.5
5.1 General.5
5.2 Specific requirements on input data .6
6 Procedure.8
6.1 General.8
6.2 Standard uncertainty induced by the measuring system .8
6.3 Standard uncertainty due to incomplete time coverage .10
6.4 Combined standard uncertainty .11
6.5 Expanded uncertainty.11
7 Reporting uncertainty.12
Annex A (informative) Example — Quantification of the uncertainty of a monthly average of nitrogen
dioxide in ambient air.13
A.1 Input.13
A.2 Uncertainty estimation of the monthly average .17
A.3 Discussion.19
Bibliography.20
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ISO 11222:2002(E)
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 carried out through ISO technical
committees. Each member body interested in a subject for which a technical committee has been established has
the right to be represented on that committee. International organizations, governmental and non-governmental, in
liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
The main task of technical committees is to prepare International Standards. Draft International Standards adopted
by the technical committees are circulated to the member bodies for voting. Publication as an International
Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this International Standard may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 11222 was prepared by Technical Committee ISO/TC 146, Air quality, Subcommittee SC 4, General aspects.
Annex A of this International Standard is for information only.
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ISO 11222:2002(E)
Introduction
Measurands in the field of air quality monitoring can be highly varying functions of time. Special considerations
are-required when estimating measurement uncertainties of time averages of air quality monitoring data. The
approach [3], using the standard deviation of the recorded measurement results divided by the square root of the
number of measurement, is applicable only to measurands that do not change with time and to measuring systems
that do not exhibit systematic uncertainties.
The statistical treatment of random and systematic deviations of measurement results has been harmonized by the
concept of measurement uncertainty introduced by the Guide to the expression of uncertainty in measurement in
1993 (GUM). This approach is based on the general application of the rule of uncertainty propagation. Although not
addressed explicitly by the GUM, the concept of uncertainty propagation and measurement uncertainty can also be
applied to measurands exhibiting distinct time structure.
Standard uncertainty may be required as a measure of data quality to be provided when reporting a time average
of air quality monitoring data. If appropriate, data quality objectives can be defined separately for:
a) the uncertainty of the time average induced by the measuring system,
b) the uncertainty of the time average induced by incomplete time coverage of the monitoring data,
c) the uncertainty of the time average due to limited spatial coverage of monitoring data.
These influences make up independent contributions to the mean square uncertainty of a time average. In this
International Standard, a time average of measured air quality data is intended to describe the air quality at a
specified location or within a specified stack within a given time period. The uncertainty of the time average due to
spatial coverage of monitoring data is not considered.
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INTERNATIONAL STANDARD ISO 11222:2002(E)
Air quality — Determination of the uncertainty of the time average
of air quality measurements
1 Scope
This International Standard provides a method for the quantification of the uncertainty of a time average of a set of
air quality data obtained at a specified location over a defined averaging time period. The method is applicable to
air quality data obtained by continuous or intermittent monitoring by means of a specified measuring system. The
uncertainty of the time average depends on both the uncertainty of the measurement results and the uncertainty
due to incomplete time coverage of the data set.
This International Standard is only applicable if:
a) the set of air quality data used to calculate the time average is representative of the temporal structure of the
measurand over the defined time period,
b) appropriate information on the uncertainty of the measurement results is available, and
c) the measurement results have all been obtained at the same location.
This International Standard implements recommendations of the Guide to the expression of uncertainty in
measurement (GUM).
2 Normative reference
The following normative document contains provisions which, through reference in this text, constitute provisions of
this International Standard. For dated references, subsequent amendments to, or revisions of, any of these
publications do not apply. However, parties to agreements based on this International Standard are encouraged to
investigate the possibility of applying the most recent edition of the normative document indicated below. For
undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC
maintain registers of currently valid International Standards.
GUM:1995, Guide to the expression of uncertainty in measurement, First edition, BIPM/IEC/IFCC/ISO/IUPAC/
IUPAP/OIML
3 Terms and definitions
For the purposes of this International Standard, the following terms and definitions apply.
3.1
arithmetic mean
average
sum of values divided by the number of values
[ISO 3534-1:1993, 2.26]
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ISO 11222:2002(E)
3.2
combined standard uncertainty
standard uncertainty of the result of a measurement when that result is obtained from the values of a number of
other quantities, equal to the positive square root of a sum of terms, the terms being the variances and covariances
of these quantities weighted according to how the measurement result varies with changes in these quantities
[GUM:1995, 2.3.4]
NOTE The (combined) standard uncertainty of the result of a measurement is the positive square root of its mean square
uncertainty.
3.3
covariance
measure of the statistical dependence of two observable quantities which may be considered as random variables
NOTE Two observable quantities have a non-zero covariance if they are correlated, i.e. a change in one quantity results in
a change in the other quantity.
3.4
coverage factor
numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded
uncertainty
[GUM:1995, 2.3.6]
3.5
expanded uncertainty
quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction
of the distribution of values that could reasonably be attributed to the measurand
[GUM:1995, 2.3.5]
NOTE If the expanded uncertainty of a result X of measurement on the level of confidence p is given by U (X), the
p
unknown true value of X is expected with probability p to be located within the interval [X − U (X); X + U (X)].
p p
3.6
influence quantity
quantity that is not the measurand but that affects the result of the measurement
[GUM:1995, B.2.10]
3.7
mean square uncertainty
〈of a result of measurement〉 square of the combined standard uncertainty of a measurement result
NOTE The mean square uncertainty of a measurement result may also be estimated by the mean square deviation of the
measurement result from material measures of the “true” value.
3.8
measurand
particular quantity subject to measurement
[VIM:1993, 2.6]
NOTE In the field of air quality monitoring, the measurand can be a highly varying function of time.
3.9
measuring system
complete set of measuring instruments and other equipment with operating procedures for carrying out specified air
quality measurements
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ISO 11222:2002(E)
NOTE The operating procedure includes or refers to a specification of the calibration routine, if calibration of the measuring
system is needed for its proper operation.
3.10
model equation
mathematical model of the measurement that transforms the set of (repeated) observations performed into the
measurement result
3.11
number of degrees of freedom
in general, the number of terms in a sum minus the number of constraints on the terms of the sum
[GUM:1995, C.2.31]
3.12
random variable
a variable that may take any of the values of a specified set of values and with which is associated a probability
distribution
[GUM:1995, C.2.2]
3.13
reference material
material or substance one or more of whose property values are sufficiently homogeneous and well established to
be used for the calibration of an apparatus, the assessment of a measurement method, or for assigning values to
materials
[VIM:1993, 6.13]
3.14
reference standard
standard, generally having the highest metrological quality available at a given location or in a given organization,
from which measurements made there are derived
[VIM:1993, 6.6]
3.15
result of a measurement
value attributed to a measurand, obtained by measurement
[VIM:1993, 3.1]
3.16
standard
material measure, measuring instrument, reference material or measuring system intended to define, realize,
conserve or reproduce a unit or one or more values of a quantity to serve as a reference
[VIM:1993, 6.1]
3.17
standard deviation
positive square root of the variance of the random variable considered
NOTE Adapted from the GUM:1993, C.2.12.
3.18
standard uncertainty
uncertainty of the result of a measurement expressed as a standard deviation
[GUM:1995, 2.3.1]
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ISO 11222:2002(E)
3.19
time average
mean value of a set of measurement results (air quality data) recorded within a defined time period
3.20
uncertainty
parameter, associated with the result of a measurement, that characterises the dispersion of the values that could
reasonably be attributed to the measurand
[VIM:1993, 3.9]
NOTE The uncertainty of a result of a measurement may be described by the (combined) standard uncertainty or by an
expanded uncertainty on a stated level of confidence.
3.21
variance
〈of a random variable or of a probability distribution〉 central moment of order 2
NOTE The variance of a random variable may be defined equivalently as the expected value of the quadratic deviation of
the random variable about its expected value.
4 Symbols and abbreviated terms
C individual measurement result recorded in the time period T
i
C time average of air quality monitoring data C
T
i
f number of degrees of freedom
f effective number of degrees of freedom
eff
f number of degrees of freedom assigned to the standard uncertainty uC() induced by the
M M T
measuring system applied
f number of degrees of freedom assigned to the standard uncertainty uC() due to incomplete
S
S T
time coverage
f uj() number of degrees of freedom when assessing the standard uncertainty u(j)
( )
f uj() number of degrees of freedom when assessing the standard uncertainty u (j)
( )
r
r
fuj() number of degrees of freedom when assessing the standard uncertainty u (j)
( )
nr nr
fu number of of degrees of freedom when assessing the standard uncertainty u
( )
nr
nr
f uC() number of of degrees of freedom when assessing the standard uncertainty u (C )
( )
r i r i
k (f) coverage factor for confidence level p and number of degrees of freedom f
p
M number of time intervals T(j) covering the time period T
Max maximum of set of values
N number of measurement results C recorded in the time period T
i
N number of measurement results C necessary for complete coverage of the time period T
max i
n(j) number of observed measurement results in time interval T(j)
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ISO 11222:2002(E)
s(C ) standard deviation of the set of N individual measurement results C used to calculate the time
i i
average C
T
T time period allocated to the time average C
T
T time period allocated to an individual measurement result C
s i
T(j) sub-interval of time period T
u(C ) standard uncertainty of C
i i
u (C ) random part of the standard uncertainty of C
r i i
u constant random part of the standard uncertainty of C
r i
u non-random part of the standard uncertainty of C
nr i
u(j) standard uncertainty of C in time interval T(j)
i
u (j) random part of the standard uncertainty of C in time interval T(j)
r i
u (j) non-random part of the standard uncertainty of C in time interval T(j)
nr i
uC() (combined) standard uncertainty of the time average C
T T
uC() standard uncertainty of the time average C induced by the measuring system
M T T
uC() standard uncertainty of the time average C due to incomplete coverage of the time period T
S T T
by the data set used to calculate the time average
uC() random part of uC()
r T M T
uC() non-random part of uC()
nr T M T
UC() expanded uncertainty of C on the stated level of confidence p
pT T
v constant relative standard uncertainty of C
r i
5 Requirements on the input data
5.1 General
This International Standard provides methods to estimate the uncertainty of the time average of a set of scalar
measurement results quantifying a time series of an air quality measurand within a defined time period. The
measurand may exhibit significant time structure. The approach [3], using the standard deviation of the
measurement results divided by the square root of the number of available measurement results, is applicable only
to measurands not exhibiting significant temporal structure and to measuring systems that are only influenced by
random uncertainties. In the field of air quality monitoring, measurands often exhibit significant temporal structure
and distinct non-random uncertainties. Therefore, a different approach is needed to quantify the uncertainty of time
averages in the field of air quality monitoring.
The set of N measurement results C of air quality recorded within a defined averaging time period T used to
i
calculate the time average C is given by formula (1):
T
Ci:1=toN (1)
{ }
i
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ISO 11222:2002(E)
The index i indicates sequential time intervals of equal length T , which may be interspersed with intervals not
S
monitored (missing values). The measurement results C may have been recorded by continuous monitoring or
i
intermittent sampling by means of a specified air quality measuring system with sampling time T .
S
The measurement results C shall have been measured at the same location. The sampling time T of the
i S
individual measurement result C is generally shorter than the averaging time period T. The coverage of the
i
averaging time period T by N measurement results C is given by N/N u 1 with N = T/T . The N measurement
i max max S
results C are used to calculate the time average C (see 6.1).
i T
In order to quantify the uncertainty of the time average C , information is required on the uncertainty of the
T
measurement results C and on the coverage of the averaging time period T by the data set. Appropriate
i
information on the measurement uncertainty may be provided in accordance with the recommendations of the
GUM.
For the purposes of this International Standard, the mean square uncertainty of a measurement result C is
i
described by equation (2):
22 2
uC( )=+uC() u ()C (2)
iirnri
2 2
The term uC() designates the random and uC() the non-random part of the mean square uncertainty of the
r i nr i
2
measurement result C . The non-random part uC() describes uncorrected systematic deviation. In the field of air
i nr i
quality monitoring, the non-random part often exceeds the random part of the mean square uncertainty of the
measurement result C .
i
Splitting the mean square uncertainty into a random and a non-random part simplifies the quantification of
uncertainties of the resulting time average as described in clause 6. For identification of random and non-random
parts of the mean square uncertainty of the measurement result C , the following rule applies.
i
2
The random part uC() is due to random changes in the measuring process and to random variations of influence
r i
quantities of the measuring process, which take place under monitoring conditions. It may be assessed by the
variance of the response of the measuring system to repeated application of check standards under monitoring
2
conditions, e.g. by zero and span checks. The random part uC() is not influenced by the temporal structure of
r i
the measurand, but it may be a function of the measurement result C .
i
2 2
Furthermore, the mean square uncertainty uC() may exhibit a non-random part uC(). A non-random part
i nr i
may be induced by uncertainties of fixed influence quantities of the measuring process or by uncorrected
systematic deviations within the measuring process.
5.2 Specific requirements on input data
The available set of measurement results C used to calculate the time average C shall be representative of the
T
i
temporal structure of the measurand over the averaging time period T.
NOTE 1 In order to meet this requirement it is necessary to have knowledge of the expected temporal structure of the
measurand.
Missing values shall not have been replaced, e.g. by using interpolation technique. The individual measurement
results C shall have been generated independently.
i
NOTE 2 It is usually assumed that the measurement results are generated independently, if the sampling time is at least four
times the response time T of the measuring system.
R
Information on the uncertainty of the measurement results C used to calculate the time average C shall be
T
i
provided in line with the recommendations of the GUM.
Concerning the uncertainty of the measurement results C , the following three cases shall be distinguished.
i
a) The whole data set has a single associated uncertainty statement, which separates random and non-random
uncertainty parts.
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ISO 11222:2002(E)
In this case, the following information shall be available:
2 2
1) variances uC() , u
r i nr
2) number of degrees of freedom f uC() , fu
( ) ( )
r i nr
2 2
Both the random part uC() and the non-random part u of the standard uncertainty shall be applicable
r i nr
2
over the averaging time period T. The random part uC() may depend on the measurement result C . The
r i i
2
non-random part u is considered to be the same for all measurement results C recorded in the averaging
nr
i
2
time period. The number of degrees of freedom assigned to uC() is f uC() . The number of degrees of
( )
r i r i
2 2 2
freedom assigned to u is fu( ). If uC() and u were assessed from the same set of data, then
nr nr r i nr
f uC() is equal to fu .
( ) ( )
r i nr
b) The set of data is split into a number M (M > 1) of sub-sets, each of which has an accociated uncertainty
statement, that separates random and non-random uncertainty parts.
In this case, the following information shall be available for the blocks of measurement j = 1 to M:
2 2
1) variances uj() , uj()
r nr
2) number of measurements n(j) and time period T(j)
3) number of degrees of freedom f uj() , fuj()
( ) ( )
r nr
M M
where TT= ()j and Nn= ()j
∑ ∑
j =1 j =1
The standard uncertainty of the measurement results C shall have been assessed independently for each time
i
interval T(j). The sum of the time intervals T(j) shall completely cover the averaging time period T. The random
2 2
part uj() and the non-random part uj() of the standard uncertainty shall be applicable over the time
r nr
interval T(j). The uncertainty statements provided for the time intervals T(j) shall not be based on the same set
2
of reference standards. The number of degrees of freedom assigned to uj() is f uj() . The number of
( )
r r
2
degrees of freedom assigned to uj() is fuj() . If uj() and uj() were assessed from the same set of
( )
nr nr r nr
data, then f uj() is equal to fuj() .
( ) ( )
r nr
c) The data set has one or more associated uncertainty statement (M W 1) , that does not separate random and
non-random uncertainty parts.
In this case, the following information shall be available for j = 1 to M:
1) uncertainty u(j)
2) number of measurements n(j) and time period T(j)
3) number of degrees of freedom f(j)
M M
where TT= ()j and Nn= ()j
∑ ∑
j =1 j =1
The standard uncertainty of the measurement results C shall have been assessed independently for each time
i
interval T(j). The sum of the time intervals T(j) shall completely cover the time period T. The standard
uncertainty u(j) shall be applicable to the time interval T(j). The standard uncertainty u(j) is considered to be
constant within the time interval T(j). The uncertainty statements provided for the time intervals T(j) shall not be
based on the same set of reference standards. The number of degrees of freedom assigned to u(j) is f(j).
If the available uncertainty statements do not allow separation of random and non-random parts of the mean
²
square uncertainty u (C ), the provided standard uncertainty u(C ) shall be considered as non-random.
i i
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ISO 11222:2002(E)
6 Procedure
6.1 General
Based on the set of measurement results C (i = 1 to N) provided in accordance with clause 5, the time average C
T
i
shall be calculated by means of equation (3):
N
1
CC= (3)
Ti∑
N
i =1
This International Standard takes into account the following contributions to the uncertainty of a time average C :
T
a) the uncertainty of the individual measurement results C of sampling time T used to calculate the time average
i S
C ;
T
b) the uncertainty due to incomplete coverage of the time period T by the measurement results C used to
i
.
calculate the time average C , if N T < T.
T S
2
Since both contributions are not correlated, the mean square uncertainty uC() of the time average C is given
T T
by equation (4):
22 2
uC()=+u()C u()C (4)
TTMST
where
2
uC() is the mean square uncertainty of the time average C due to the measuring system used to
M T T
record the set of measurement results C ;
i
2
uC() is the mean square uncertainty of the time average C due to incomplete coverage of the time
S T T
period T by the set of measurement results C .
i
In case of complete coverage of the averaging time period by measurement results C , the uncertainty of the time
i
average is completely determined by the measuring system.
The uncertainty of the time average due to the measuring system is quantified in 6.2 in line with the
recommendations of the GUM, based on information to be provided on the uncertainty of the measurement results
used to calculate the time average. The uncertainty of the time average due to incomplete coverage of the time
period T by measurement results is not addressed explicitly by the GUM. The solution of this problem is described
in 6.3.
6.2 Standard uncertainty induced by the measuring system
The mean square uncertainty of the time average C due to the measuring system used to record the
T
measurement results C is given by equation (5):
i
22 2
uC()=+u()C u ()C (5)
MrTT nrT
where
2
uC() is the random part of the mean square uncertainty of the time average C due to the measuring
r T T
system;
2
uC() is the non-random part of the mean square uncertainty of the time average C due to the
nr T T
measuring system.
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ISO 11222:2002(E)
2 2
The random part uC() is quantified based on the random parts uC() of the mean square uncertainty of the
r T r i
2 2
measurement result C . The non-random part uC() is calculated based on the non-random parts uC() of the
nr T nr i
i
mean square uncertainty of the measurement results C .
i
2
Based on uncertainty statements provided in accordance with 5.2, the standard uncertainty uC(
...
SLOVENSKI STANDARD
SIST ISO 11222:2004
01-junij-2004
.DNRYRVW]UDND±'RORþDQMHQHJRWRYRVWLþDVRYQLKSRYSUHþLMSULPHULWYDKNDNRYRVWL
]UDND
Air quality -- Determination of the uncertainty of the time average of air quality
measurements
Qualité de l'air -- Détermination de l'incertitude de mesure de la moyenne temporelle de
mesurages de la qualité de l'air
Ta slovenski standard je istoveten z: ISO 11222:2002
ICS:
13.040.01 Kakovost zraka na splošno Air quality in general
SIST ISO 11222:2004 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST ISO 11222:2004
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SIST ISO 11222:2004
INTERNATIONAL ISO
STANDARD 11222
First edition
2002-07-01
Air quality — Determination of the
uncertainty of the time average of air
quality measurements
Qualité de l'air — Détermination de l'incertitude de mesure de la moyenne
temporelle de mesurages de la qualité de l'air
Reference number
ISO 11222:2002(E)
©
ISO 2002
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SIST ISO 11222:2004
ISO 11222:2002(E)
PDF disclaimer
This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but shall not
be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In downloading this
file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat accepts no liability in this
area.
Adobe is a trademark of Adobe Systems Incorporated.
Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation parameters
were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In the unlikely event
that a problem relating to it is found, please inform the Central Secretariat at the address given below.
© ISO 2002
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic
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SIST ISO 11222:2004
ISO 11222:2002(E)
Contents Page
Foreword.iv
Introduction.v
1 Scope.1
2 Normative reference.1
3 Terms and definitions .1
4 Symbols and abbreviated terms .4
5 Requirements on the input data.5
5.1 General.5
5.2 Specific requirements on input data .6
6 Procedure.8
6.1 General.8
6.2 Standard uncertainty induced by the measuring system .8
6.3 Standard uncertainty due to incomplete time coverage .10
6.4 Combined standard uncertainty .11
6.5 Expanded uncertainty.11
7 Reporting uncertainty.12
Annex A (informative) Example — Quantification of the uncertainty of a monthly average of nitrogen
dioxide in ambient air.13
A.1 Input.13
A.2 Uncertainty estimation of the monthly average .17
A.3 Discussion.19
Bibliography.20
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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 carried out through ISO technical
committees. Each member body interested in a subject for which a technical committee has been established has
the right to be represented on that committee. International organizations, governmental and non-governmental, in
liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
The main task of technical committees is to prepare International Standards. Draft International Standards adopted
by the technical committees are circulated to the member bodies for voting. Publication as an International
Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this International Standard may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 11222 was prepared by Technical Committee ISO/TC 146, Air quality, Subcommittee SC 4, General aspects.
Annex A of this International Standard is for information only.
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Introduction
Measurands in the field of air quality monitoring can be highly varying functions of time. Special considerations
are-required when estimating measurement uncertainties of time averages of air quality monitoring data. The
approach [3], using the standard deviation of the recorded measurement results divided by the square root of the
number of measurement, is applicable only to measurands that do not change with time and to measuring systems
that do not exhibit systematic uncertainties.
The statistical treatment of random and systematic deviations of measurement results has been harmonized by the
concept of measurement uncertainty introduced by the Guide to the expression of uncertainty in measurement in
1993 (GUM). This approach is based on the general application of the rule of uncertainty propagation. Although not
addressed explicitly by the GUM, the concept of uncertainty propagation and measurement uncertainty can also be
applied to measurands exhibiting distinct time structure.
Standard uncertainty may be required as a measure of data quality to be provided when reporting a time average
of air quality monitoring data. If appropriate, data quality objectives can be defined separately for:
a) the uncertainty of the time average induced by the measuring system,
b) the uncertainty of the time average induced by incomplete time coverage of the monitoring data,
c) the uncertainty of the time average due to limited spatial coverage of monitoring data.
These influences make up independent contributions to the mean square uncertainty of a time average. In this
International Standard, a time average of measured air quality data is intended to describe the air quality at a
specified location or within a specified stack within a given time period. The uncertainty of the time average due to
spatial coverage of monitoring data is not considered.
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SIST ISO 11222:2004
INTERNATIONAL STANDARD ISO 11222:2002(E)
Air quality — Determination of the uncertainty of the time average
of air quality measurements
1 Scope
This International Standard provides a method for the quantification of the uncertainty of a time average of a set of
air quality data obtained at a specified location over a defined averaging time period. The method is applicable to
air quality data obtained by continuous or intermittent monitoring by means of a specified measuring system. The
uncertainty of the time average depends on both the uncertainty of the measurement results and the uncertainty
due to incomplete time coverage of the data set.
This International Standard is only applicable if:
a) the set of air quality data used to calculate the time average is representative of the temporal structure of the
measurand over the defined time period,
b) appropriate information on the uncertainty of the measurement results is available, and
c) the measurement results have all been obtained at the same location.
This International Standard implements recommendations of the Guide to the expression of uncertainty in
measurement (GUM).
2 Normative reference
The following normative document contains provisions which, through reference in this text, constitute provisions of
this International Standard. For dated references, subsequent amendments to, or revisions of, any of these
publications do not apply. However, parties to agreements based on this International Standard are encouraged to
investigate the possibility of applying the most recent edition of the normative document indicated below. For
undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC
maintain registers of currently valid International Standards.
GUM:1995, Guide to the expression of uncertainty in measurement, First edition, BIPM/IEC/IFCC/ISO/IUPAC/
IUPAP/OIML
3 Terms and definitions
For the purposes of this International Standard, the following terms and definitions apply.
3.1
arithmetic mean
average
sum of values divided by the number of values
[ISO 3534-1:1993, 2.26]
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3.2
combined standard uncertainty
standard uncertainty of the result of a measurement when that result is obtained from the values of a number of
other quantities, equal to the positive square root of a sum of terms, the terms being the variances and covariances
of these quantities weighted according to how the measurement result varies with changes in these quantities
[GUM:1995, 2.3.4]
NOTE The (combined) standard uncertainty of the result of a measurement is the positive square root of its mean square
uncertainty.
3.3
covariance
measure of the statistical dependence of two observable quantities which may be considered as random variables
NOTE Two observable quantities have a non-zero covariance if they are correlated, i.e. a change in one quantity results in
a change in the other quantity.
3.4
coverage factor
numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded
uncertainty
[GUM:1995, 2.3.6]
3.5
expanded uncertainty
quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction
of the distribution of values that could reasonably be attributed to the measurand
[GUM:1995, 2.3.5]
NOTE If the expanded uncertainty of a result X of measurement on the level of confidence p is given by U (X), the
p
unknown true value of X is expected with probability p to be located within the interval [X − U (X); X + U (X)].
p p
3.6
influence quantity
quantity that is not the measurand but that affects the result of the measurement
[GUM:1995, B.2.10]
3.7
mean square uncertainty
〈of a result of measurement〉 square of the combined standard uncertainty of a measurement result
NOTE The mean square uncertainty of a measurement result may also be estimated by the mean square deviation of the
measurement result from material measures of the “true” value.
3.8
measurand
particular quantity subject to measurement
[VIM:1993, 2.6]
NOTE In the field of air quality monitoring, the measurand can be a highly varying function of time.
3.9
measuring system
complete set of measuring instruments and other equipment with operating procedures for carrying out specified air
quality measurements
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NOTE The operating procedure includes or refers to a specification of the calibration routine, if calibration of the measuring
system is needed for its proper operation.
3.10
model equation
mathematical model of the measurement that transforms the set of (repeated) observations performed into the
measurement result
3.11
number of degrees of freedom
in general, the number of terms in a sum minus the number of constraints on the terms of the sum
[GUM:1995, C.2.31]
3.12
random variable
a variable that may take any of the values of a specified set of values and with which is associated a probability
distribution
[GUM:1995, C.2.2]
3.13
reference material
material or substance one or more of whose property values are sufficiently homogeneous and well established to
be used for the calibration of an apparatus, the assessment of a measurement method, or for assigning values to
materials
[VIM:1993, 6.13]
3.14
reference standard
standard, generally having the highest metrological quality available at a given location or in a given organization,
from which measurements made there are derived
[VIM:1993, 6.6]
3.15
result of a measurement
value attributed to a measurand, obtained by measurement
[VIM:1993, 3.1]
3.16
standard
material measure, measuring instrument, reference material or measuring system intended to define, realize,
conserve or reproduce a unit or one or more values of a quantity to serve as a reference
[VIM:1993, 6.1]
3.17
standard deviation
positive square root of the variance of the random variable considered
NOTE Adapted from the GUM:1993, C.2.12.
3.18
standard uncertainty
uncertainty of the result of a measurement expressed as a standard deviation
[GUM:1995, 2.3.1]
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3.19
time average
mean value of a set of measurement results (air quality data) recorded within a defined time period
3.20
uncertainty
parameter, associated with the result of a measurement, that characterises the dispersion of the values that could
reasonably be attributed to the measurand
[VIM:1993, 3.9]
NOTE The uncertainty of a result of a measurement may be described by the (combined) standard uncertainty or by an
expanded uncertainty on a stated level of confidence.
3.21
variance
〈of a random variable or of a probability distribution〉 central moment of order 2
NOTE The variance of a random variable may be defined equivalently as the expected value of the quadratic deviation of
the random variable about its expected value.
4 Symbols and abbreviated terms
C individual measurement result recorded in the time period T
i
C time average of air quality monitoring data C
T
i
f number of degrees of freedom
f effective number of degrees of freedom
eff
f number of degrees of freedom assigned to the standard uncertainty uC() induced by the
M M T
measuring system applied
f number of degrees of freedom assigned to the standard uncertainty uC() due to incomplete
S
S T
time coverage
f uj() number of degrees of freedom when assessing the standard uncertainty u(j)
( )
f uj() number of degrees of freedom when assessing the standard uncertainty u (j)
( )
r
r
fuj() number of degrees of freedom when assessing the standard uncertainty u (j)
( )
nr nr
fu number of of degrees of freedom when assessing the standard uncertainty u
( )
nr
nr
f uC() number of of degrees of freedom when assessing the standard uncertainty u (C )
( )
r i r i
k (f) coverage factor for confidence level p and number of degrees of freedom f
p
M number of time intervals T(j) covering the time period T
Max maximum of set of values
N number of measurement results C recorded in the time period T
i
N number of measurement results C necessary for complete coverage of the time period T
max i
n(j) number of observed measurement results in time interval T(j)
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s(C ) standard deviation of the set of N individual measurement results C used to calculate the time
i i
average C
T
T time period allocated to the time average C
T
T time period allocated to an individual measurement result C
s i
T(j) sub-interval of time period T
u(C ) standard uncertainty of C
i i
u (C ) random part of the standard uncertainty of C
r i i
u constant random part of the standard uncertainty of C
r i
u non-random part of the standard uncertainty of C
nr i
u(j) standard uncertainty of C in time interval T(j)
i
u (j) random part of the standard uncertainty of C in time interval T(j)
r i
u (j) non-random part of the standard uncertainty of C in time interval T(j)
nr i
uC() (combined) standard uncertainty of the time average C
T T
uC() standard uncertainty of the time average C induced by the measuring system
M T T
uC() standard uncertainty of the time average C due to incomplete coverage of the time period T
S T T
by the data set used to calculate the time average
uC() random part of uC()
r T M T
uC() non-random part of uC()
nr T M T
UC() expanded uncertainty of C on the stated level of confidence p
pT T
v constant relative standard uncertainty of C
r i
5 Requirements on the input data
5.1 General
This International Standard provides methods to estimate the uncertainty of the time average of a set of scalar
measurement results quantifying a time series of an air quality measurand within a defined time period. The
measurand may exhibit significant time structure. The approach [3], using the standard deviation of the
measurement results divided by the square root of the number of available measurement results, is applicable only
to measurands not exhibiting significant temporal structure and to measuring systems that are only influenced by
random uncertainties. In the field of air quality monitoring, measurands often exhibit significant temporal structure
and distinct non-random uncertainties. Therefore, a different approach is needed to quantify the uncertainty of time
averages in the field of air quality monitoring.
The set of N measurement results C of air quality recorded within a defined averaging time period T used to
i
calculate the time average C is given by formula (1):
T
Ci:1=toN (1)
{ }
i
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The index i indicates sequential time intervals of equal length T , which may be interspersed with intervals not
S
monitored (missing values). The measurement results C may have been recorded by continuous monitoring or
i
intermittent sampling by means of a specified air quality measuring system with sampling time T .
S
The measurement results C shall have been measured at the same location. The sampling time T of the
i S
individual measurement result C is generally shorter than the averaging time period T. The coverage of the
i
averaging time period T by N measurement results C is given by N/N u 1 with N = T/T . The N measurement
i max max S
results C are used to calculate the time average C (see 6.1).
i T
In order to quantify the uncertainty of the time average C , information is required on the uncertainty of the
T
measurement results C and on the coverage of the averaging time period T by the data set. Appropriate
i
information on the measurement uncertainty may be provided in accordance with the recommendations of the
GUM.
For the purposes of this International Standard, the mean square uncertainty of a measurement result C is
i
described by equation (2):
22 2
uC( )=+uC() u ()C (2)
iirnri
2 2
The term uC() designates the random and uC() the non-random part of the mean square uncertainty of the
r i nr i
2
measurement result C . The non-random part uC() describes uncorrected systematic deviation. In the field of air
i nr i
quality monitoring, the non-random part often exceeds the random part of the mean square uncertainty of the
measurement result C .
i
Splitting the mean square uncertainty into a random and a non-random part simplifies the quantification of
uncertainties of the resulting time average as described in clause 6. For identification of random and non-random
parts of the mean square uncertainty of the measurement result C , the following rule applies.
i
2
The random part uC() is due to random changes in the measuring process and to random variations of influence
r i
quantities of the measuring process, which take place under monitoring conditions. It may be assessed by the
variance of the response of the measuring system to repeated application of check standards under monitoring
2
conditions, e.g. by zero and span checks. The random part uC() is not influenced by the temporal structure of
r i
the measurand, but it may be a function of the measurement result C .
i
2 2
Furthermore, the mean square uncertainty uC() may exhibit a non-random part uC(). A non-random part
i nr i
may be induced by uncertainties of fixed influence quantities of the measuring process or by uncorrected
systematic deviations within the measuring process.
5.2 Specific requirements on input data
The available set of measurement results C used to calculate the time average C shall be representative of the
T
i
temporal structure of the measurand over the averaging time period T.
NOTE 1 In order to meet this requirement it is necessary to have knowledge of the expected temporal structure of the
measurand.
Missing values shall not have been replaced, e.g. by using interpolation technique. The individual measurement
results C shall have been generated independently.
i
NOTE 2 It is usually assumed that the measurement results are generated independently, if the sampling time is at least four
times the response time T of the measuring system.
R
Information on the uncertainty of the measurement results C used to calculate the time average C shall be
T
i
provided in line with the recommendations of the GUM.
Concerning the uncertainty of the measurement results C , the following three cases shall be distinguished.
i
a) The whole data set has a single associated uncertainty statement, which separates random and non-random
uncertainty parts.
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In this case, the following information shall be available:
2 2
1) variances uC() , u
r i nr
2) number of degrees of freedom f uC() , fu
( ) ( )
r i nr
2 2
Both the random part uC() and the non-random part u of the standard uncertainty shall be applicable
r i nr
2
over the averaging time period T. The random part uC() may depend on the measurement result C . The
r i i
2
non-random part u is considered to be the same for all measurement results C recorded in the averaging
nr
i
2
time period. The number of degrees of freedom assigned to uC() is f uC() . The number of degrees of
( )
r i r i
2 2 2
freedom assigned to u is fu( ). If uC() and u were assessed from the same set of data, then
nr nr r i nr
f uC() is equal to fu .
( ) ( )
r i nr
b) The set of data is split into a number M (M > 1) of sub-sets, each of which has an accociated uncertainty
statement, that separates random and non-random uncertainty parts.
In this case, the following information shall be available for the blocks of measurement j = 1 to M:
2 2
1) variances uj() , uj()
r nr
2) number of measurements n(j) and time period T(j)
3) number of degrees of freedom f uj() , fuj()
( ) ( )
r nr
M M
where TT= ()j and Nn= ()j
∑ ∑
j =1 j =1
The standard uncertainty of the measurement results C shall have been assessed independently for each time
i
interval T(j). The sum of the time intervals T(j) shall completely cover the averaging time period T. The random
2 2
part uj() and the non-random part uj() of the standard uncertainty shall be applicable over the time
r nr
interval T(j). The uncertainty statements provided for the time intervals T(j) shall not be based on the same set
2
of reference standards. The number of degrees of freedom assigned to uj() is f uj() . The number of
( )
r r
2
degrees of freedom assigned to uj() is fuj() . If uj() and uj() were assessed from the same set of
( )
nr nr r nr
data, then f uj() is equal to fuj() .
( ) ( )
r nr
c) The data set has one or more associated uncertainty statement (M W 1) , that does not separate random and
non-random uncertainty parts.
In this case, the following information shall be available for j = 1 to M:
1) uncertainty u(j)
2) number of measurements n(j) and time period T(j)
3) number of degrees of freedom f(j)
M M
where TT= ()j and Nn= ()j
∑ ∑
j =1 j =1
The standard uncertainty of the measurement results C shall have been assessed independently for each time
i
interval T(j). The sum of the time intervals T(j) shall completely cover the time period T. The standard
uncertainty u(j) shall be applicable to the time interval T(j). The standard uncertainty u(j) is considered to be
constant within the time interval T(j). The uncertainty statements provided for the time intervals T(j) shall not be
based on the same set of reference standards. The number of degrees of freedom assigned to u(j) is f(j).
If the available uncertainty statements do not allow separation of random and non-random parts of the mean
²
square uncertainty u (C ), the provided standard uncertainty u(C ) shall be considered as non-random.
i i
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6 Procedure
6.1 General
Based on the set of measurement results C (i = 1 to N) provided in accordance with clause 5, the time average C
T
i
shall be calculated by means of equation (3):
N
1
CC= (3)
Ti∑
N
i =1
This International Standard takes into account the following contributions to the uncertainty of a time average C :
T
a) the uncertainty of the individual measurement results C of sampling time T used to calculate the time average
i S
C ;
T
b) the uncertainty due to incomplete coverage of the time period T by the measurement results C used to
i
.
calculate the time average C , if N T < T.
T S
2
Since both contributions are not correlated, the mean square uncertainty uC() of the time average C is given
T T
by equation (4):
22 2
uC()=+u()C u()C (4)
TTMST
where
2
uC() is the mean square uncertainty of the time average C due to the measuring system used to
M T T
record the set of measurement results C ;
i
2
uC() is the mean square uncertainty of the time average C due to incomplete coverage of the time
S T T
period T by the set of measurement results C .
i
In case of complete coverage of the averaging time period by measurement results C , the uncertainty of the time
i
average is completely determined by the measuring system.
The uncertainty of the time average due to the measuring system is quantified in 6.2 in line with the
recommendations of the GUM, based on information to be provided on the uncertainty of the measurement results
used to calculate the time average. The uncertainty of the time average due to incomplete coverage of the time
pe
...
NORME ISO
INTERNATIONALE 11222
Première édition
2002-07-01
Qualité de l'air — Détermination de
l'incertitude de mesure de la moyenne
temporelle de mesurages de la qualité de
l'air
Air quality — Determination of the uncertainty of the time average of air
quality measurements
Numéro de référence
ISO 11222:2002(F)
©
ISO 2002
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ISO 11222:2002(F)
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© ISO 2002
Droits de reproduction réservés. Sauf prescription différente, aucune partie de cette publication ne peut être reproduite ni utilisée sous quelque
forme que ce soit et par aucun procédé, électronique ou mécanique, y compris la photocopie et les microfilms, sans l'accord écrit de l’ISO à
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Imprimé en Suisse
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ISO 11222:2002(F)
Sommaire Page
Avant-propos .iv
Introduction.v
1 Domaine d'application.1
2 Références normatives.1
3 Termes et définitions.1
4 Symboles et abréviations .4
5 Exigences relatives aux données d’entrée .5
5.1 Généralités.5
5.2 Exigences spécifiques relatives aux données d'entrée .6
6 Procédure.8
6.1 Généralités.8
6.2 Incertitude-type due au système de mesure.8
6.3 Incertitude-type due à une couverture temporelle incomplète.10
6.4 Incertitude-type composée.11
6.5 Incertitude élargie.11
7 Rapport sur l’incertitude.12
Annexe A (informative) Exemple — Quantification de l'incertitude d'une concentration moyenne
mensuelle en dioxyde d’azote présente dans l'air ambiant .13
A.1 Entrée.13
A.1.1 Système de mesure.13
A.1.2 Procédure de contrôle.13
A.1.3 Procédure d’estimation de l’incertitude des moyennes horaires .13
A.2 Estimation de l’incertitude de la moyenne mensuelle.17
A.2.1 Incertitude de mesure .17
A.2.2 Incertitude due aux valeurs manquantes.18
A.2.3 Déclarations d’incertitude de la moyenne temporelle .18
A.3 Discussion.19
Bibliographie.20
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ISO 11222:2002(F)
Avant-propos
L'ISO (Organisation internationale de normalisation) est une fédération mondiale d'organismes nationaux de
normalisation (comités membres de l'ISO). L'élaboration des Normes internationales est en général confiée aux
comités techniques de l'ISO. Chaque comité membre intéressé par une étude a le droit de faire partie du comité
technique créé à cet effet. Les organisations internationales, gouvernementales et non gouvernementales, en
liaison avec l'ISO participent également aux travaux. L'ISO collabore étroitement avec la Commission
électrotechnique internationale (CEI) en ce qui concerne la normalisation électrotechnique.
Les Normes internationales sont rédigées conformément aux règles données dans les Directives ISO/CEI,
Partie 3.
La tâche principale des comités techniques est d'élaborer les Normes internationales. Les projets de Normes
internationales adoptés par les comités techniques sont soumis aux comités membres pour vote. Leur publication
comme Normes internationales requiert l'approbation de 75 % au moins des comités membres votants.
L'attention est appelée sur le fait que certains des éléments de la présente Norme internationale peuvent faire
l'objet de droits de propriété intellectuelle ou de droits analogues. L'ISO ne saurait être tenue pour responsable de
ne pas avoir identifié de tels droits de propriété et averti de leur existence.
L'ISO 11222 a été élaborée par le comité technique ISO/TC 146, Qualité de l'air, sous-comité SC 4, Aspects
généraux.
L’annexe A de la présente Norme internationale est donnée uniquement à titre d’information.
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ISO 11222:2002(F)
Introduction
En matière de surveillance de la qualité de l’air, les mesurandes peuvent connaître d’importantes variations en
fonction du temps. Des précautions particulières sont requises lors de l’estimation des incertitudes de mesure des
moyennes temporelles des données relatives à la qualité de l’air. L’approche [3], qui utilise l’écart-type des
résultats de mesure relevés divisé par la racine carrée du nombre de mesurages, s’applique uniquement aux
mesurandes qui ne varient pas dans le temps et aux systèmes de mesure qui ne présentent pas d’incertitudes
systématiques.
Le traitement statistique des écarts aléatoire et systématique des résultats de mesure a été harmonisé par le
concept d’incertitude de mesure introduit par le Guide pour l’expression de l’incertitude de mesure en 1993 (GUM).
Cette approche est fondée sur l’application générale de la loi de propagation de l’incertitude. Bien que le GUM ne
traite pas explicitement de la propagation de l’incertitude ni de l’incertitude de mesure, ces concepts peuvent
également s’appliquer aux mesurandes présentant une structure temporelle distincte.
L’incertitude-type peut s’avérer nécessaire pour évaluer la qualité des données obtenues dans le domaine de la
surveillance de la qualité moyenne de l’air dans le temps. Le cas échéant, des objectifs de qualité des données
peuvent être définis séparément pour
a) l’incertitude de la moyenne temporelle due au système de mesure;
b) l’incertitude de la moyenne temporelle due à une couverture temporelle incomplète des données de
surveillance;
c) l’incertitude de la moyenne temporelle due à une couverture spatiale limitée des données de surveillance.
Chacune de ces grandeurs d’influence contribue de façon indépendante à l’incertitude quadratique moyenne d’une
moyenne temporelle. Dans la présente Norme internationale, une moyenne temporelle des données calculées
relatives à la qualité de l’air vise à décrire la qualité de l’air dans un endroit précis ou dans une cheminée spécifiée,
sur une période de temps donnée. La présente Norme internationale ne couvre pas l’incertitude de la moyenne
temporelle due à la couverture spatiale des données de surveillance.
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NORME INTERNATIONALE ISO 11222:2002(F)
Qualité de l'air — Détermination de l'incertitude de mesure de la
moyenne temporelle de mesurages de la qualité de l'air
1 Domaine d'application
La présente Norme internationale fournit une méthode permettant de quantifier l’incertitude d'une moyenne
temporelle d’un ensemble de données relatif à la qualité de l’air, obtenu à partir d’un point de mesurage spécifié,
sur une période moyenne de temps définie. La méthode est applicable aux données relatives à la qualité de l’air
obtenues par surveillance continue ou intermittente, au moyen d’un système de mesure spécifié. L’incertitude de la
moyenne temporelle dépend à la fois de l’incertitude des résultats de mesure et de l’incertitude due à une
couverture incomplète de l’ensemble de données.
La présente Norme internationale est applicable uniquement si
a) l’ensemble de données relatif à la qualité de l’air utilisé pour calculer la moyenne temporelle est représentatif
de la structure temporelle du mesurande sur la période de temps définie,
b) les informations appropriées relatives à l’incertitude des résultats de mesure sont disponibles, et
c) les mesurages ont été effectués au même endroit.
La présente Norme internationale met en œuvre les recommandations du Guide pour l’expression de l’incertitude
de mesure (GUM).
2 Référence normative
Le document normatif suivant contient des dispositions qui, par suite de la référence qui y est faite, constituent des
dispositions valables pour la présente Norme internationale. Pour les références datées, les amendements
ultérieurs ou les révisions de ces publications ne s'appliquent pas. Toutefois, les parties prenantes aux accords
fondés sur la présente Norme internationale sont invitées à rechercher la possibilité d'appliquer l’édition la plus
récente du document normatif indiqué ci-après. Pour les références non datées, la dernière édition du document
normatif en référence s'applique. Les membres de l'ISO et de la CEI possèdent le registre des Normes
internationales en vigueur.
GUM:1995, Guide pour l’expression de l’incertitude de mesure, première édition, BIPM/CEI/FICC/ISO/OIML/
UICPA/UIPPA
3 Termes et définitions
Pour les besoins de la présente Norme internationale, les termes et définitions suivants s'appliquent.
3.1
moyenne arithmétique
moyenne
somme des valeurs divisée par le nombre de valeurs
[ISO 3534-1:1993, 2.26]
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ISO 11222:2002(F)
3.2
incertitude-type composée
incertitude-type du résultat d’un mesurage, lorsque ce résultat est obtenu à partir des valeurs d’autres grandeurs,
égale à la racine carrée positive d’une somme de termes, ces termes étant les variances ou covariances de ces
autres grandeurs, pondérées selon la variation du résultat de mesure en fonction de celle de ces grandeurs
[GUM:1995, 2.3.4]
NOTE L’incertitude-type (composée) du résultat d’un mesurage est la racine carrée positive de son incertitude quadratique
moyenne.
3.3
covariance
mesure de la dépendance statistique de deux grandeurs observables qui peuvent être considérées comme deux
variables aléatoires
NOTE Deux grandeurs observables ont une covariance non nulle si elles sont corrélées, c’est-à-dire si la variation de l’une
de ces deux grandeurs entraîne la variation de l’autre grandeur.
3.4
facteur d’élargissement
facteur numérique utilisé comme multiplicateur de l’incertitude-type composée pour obtenir l’incertitude élargie
[GUM:1995, 2.3.6]
3.5
incertitude élargie
grandeur définissant un intervalle, autour du résultat d’un mesurage, dont on puisse s’attendre à ce qu’il
comprenne une fraction élevée de la distribution des valeurs qui pourraient être attribuées raisonnablement au
mesurande
[GUM:1995, 2.3.5]
NOTE Si l’incertitude élargie d’un résultat X de mesurage au niveau de confiance p est donnée par U (X), la valeur vraie
p
inconnue de X est attendue avec une probabilité p dans l’intervalle [X − U (X); X + U (X)].
p p
3.6
grandeur d’influence
grandeur qui n’est pas le mesurande mais qui a un effet sur le résultat du mesurage
[GUM:1995, B.2.10]
3.7
incertitude quadratique moyenne
〈d'un résultat de mesure〉 carré de l’incertitude-type composée d’un résultat de mesure
NOTE L’incertitude quadratique moyenne d’un résultat de mesure peut également être estimée à l’aide de l’écart
quadratique moyen du résultat de mesure obtenu à partir des mesurages physiques de la valeur «vraie».
3.8
mesurande
grandeur particulière soumise à mesurage
[VIM:1993, 2.6]
NOTE En matière de surveillance de la qualité de l’air, le mesurande peut être une fonction du temps soumise à
d’importantes variations.
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ISO 11222:2002(F)
3.9
système de mesure
ensemble complet des instruments de mesure, des autres appareils et des modes opératoires utilisés pour
effectuer les mesurages de la qualité de l’air
NOTE Le mode opératoire inclut une spécification de la routine d’étalonnage ou y fait référence si l’étalonnage du système
de mesure est nécessaire à son bon fonctionnement.
3.10
équation modèle
modèle mathématique du mesurage qui transforme l’ensemble des observations (répétées) en résultat de mesure
3.11
degrés de liberté
en général, le nombre de termes de la somme moins le nombre de contraintes sur les termes de la somme
[GUM:1995, C.2.31]
3.12
variable aléatoire
variable pouvant prendre n’importe quelle valeur d’un ensemble déterminé de valeurs, et à laquelle est associée
une loi de probabilité
[GUM:1995, C.2.2]
3.13
matériau de référence
matériau ou substance dont une (ou plusieurs) valeur(s) de la (des) propriété(s) est (sont) suffisamment
homogène(s) et bien définie(s) pour permettre de l’utiliser pour l’étalonnage d’un appareil, l’évaluation d’une
méthode de mesurage ou l’attribution de valeurs aux matériaux
[VIM:1993, 6.13]
3.14
étalon de référence
étalon, en général de la plus haute qualité métrologique, disponible en un lieu donné ou dans une organisation
donnée dont dérivent les mesurages qui y sont faits
[VIM:1993, 6.6]
3.15
résultat d’un mesurage
valeur attribuée à un mesurande, obtenue par mesurage
[VIM:1993, 3.1]
3.16
étalon
mesure matérialisée, appareil de mesure, matériau de référence ou système de mesure destiné à définir, réaliser,
conserver ou reproduire une unité ou une ou plusieurs valeurs d’une grandeur pour servir de référence
[VIM:1993, 6.1]
3.17
écart-type
racine carrée positive de la variance de la variable aléatoire considérée
NOTE Adaptée du GUM:1993, C.2.12.
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ISO 11222:2002(F)
3.18
incertitude-type
incertitude du résultat d’un mesurage exprimée sous la forme d’un écart-type
[GUM:1995, 2.3.1]
3.19
moyenne temporelle
valeur moyenne d’un ensemble de résultats de mesure (données relatives à la qualité de l’air) relevés sur une
période de temps donnée
3.20
incertitude
paramètre, associé au résultat d’un mesurage, qui caractérise la dispersion des valeurs qui pourraient
raisonnablement être attribuées au mesurande
[VIM:1993, 3.9]
NOTE L’incertitude d’un résultat de mesure peut être décrite par l’incertitude-type (composée) ou une incertitude élargie
au niveau de confiance déterminé.
3.21
variance
〈d’une variable aléatoire ou d’une loi de probabilité〉 moment centré d’ordre 2
NOTE La variance d’une variable aléatoire peut également être définie comme la valeur espérée de l’écart de la moyenne
quadratique de la variable aléatoire par rapport à sa valeur espérée.
4 Symboles et abréviations
C résultat de mesure individuel relevé au cours de la période T
i
C moyenne temporelle des données C de surveillance de la qualité de l’air
T
i
f nombre de degrés de liberté
f nombre effectif de degrés de liberté
eff
f nombre de degrés de liberté attribués à l’incertitude-type u (C ) due au système de mesure appliqué
T
M
M
f nombre de degrés de liberté attribués à l’incertitude-type uC due à la couverture temporelle
( T )
S S
incomplète
f()u()j nombre de degrés de liberté lors de l’évaluation de l’incertitude-type u()j
f()u ()j nombre de degrés de liberté lors de l’évaluation de l’incertitude-type u ()j
r r
f()u ()j nombre de degrés de liberté lors de l’évaluation de l’incertitude-type u ()j
nr nr
f()u nombre de degrés de liberté lors de l’évaluation de l’incertitude-type u
nr
nr
f()u()C nombre de degrés de liberté lors de l’évaluation de l’incertitude-type u()C
r i r i
k ()f facteur d’élargissement pour le niveau de confiance p et le nombre de degrés de liberté f
p
M nombre d’intervalles de temps T()j couvrant la période de temps T
Max maximum d’un ensemble de valeurs
N nombre de résultats de mesure C relevés sur la période de temps T
i
N nombre de résultats de mesure C relevés nécessaires pour obtenir une couverture totale de la
max i
période de temps T
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ISO 11222:2002(F)
n( j) nombre de résultats de mesure observés dans l’intervalle de temps T()j
s()C écart-type de l’ensemble de N résultats de mesure individuels C utilisé pour calculer la moyenne
i
i
temporelle C
T
T période de temps allouée à la moyenne temporelle C
T
T période de temps allouée à un résultat de mesure individuel C
S i
T()j sous-intervalle de temps de la période T
u(C ) incertitude-type de C
i i
u()C partie aléatoire de l’incertitude-type de C
r i
i
u partie aléatoire constante de l’incertitude-type de C
r i
u partie non-aléatoire de l’incertitude-type de C
nr i
u( j) incertitude-type de C dans l’intervalle de temps T ( j)
i
u ()j partie aléatoire de l’incertitude-type de C dans l’intervalle de temps T ( j)
r
i
u ()j partie non-aléatoire de l’incertitude-type de C dans l’intervalle de temps T()j
nr i
u(C ) incertitude-type (composée) de la moyenne temporelle C
T T
u (C ) incertitude-type de la moyenne temporelle C due au système de mesure
T T
M
u (C ) incertitude-type de la moyenne temporelle C due à une couverture incomplète de la période de
T T
S
temps T par l’ensemble de données utilisé pour calculer la moyenne temporelle
u (C ) partie aléatoire de u (C )
T T
r M
u (C ) partie non-aléatoire de u (C )
T T
nr M
UC incertitude élargie de C au niveau de confiance établi p
T T
p ( )
v incertitude-type relative constante de C
r i
5 Exigences relatives aux données d’entrée
5.1 Généralités
La présente Norme internationale fournit des méthodes pour estimer l’incertitude de la moyenne temporelle d’un
ensemble de résultats de mesure scalaires qui quantifient une série temporelle d’un mesurande relatif à la qualité
de l’air sur une période de temps définie. Ce mesurande peut présenter une structure temporelle significative.
L’approche [3], qui utilise l’écart-type des résultats de mesure divisé par la racine carrée du nombre de résultats de
mesure disponibles, s’applique uniquement aux mesurandes qui ne présentent pas de structure temporelle
significative et aux systèmes de mesure uniquement influencés par des incertitudes aléatoires. Dans le domaine
de la surveillance de la qualité de l’air, les mesurandes présentent souvent une structure temporelle significative et
des incertitudes non-aléatoires distinctes. Par conséquent, une approche différente est nécessaire pour quantifier
l’incertitude des moyennes temporelles en matière de surveillance de la qualité de l’air.
L’ensemble des N résultats de mesure C de la qualité de l’air relevés sur une période de moyennage définie T,
i
utilisé pour calculer la moyenne temporelle C est donné par la formule (1):
T
Ci:1=àN (1)
{ }
i
L’indice i indique des intervalles de temps séquentiels de durée identique T , qui peuvent être intercalés avec des
S
intervalles non surveillés (valeurs manquantes). Les résultats de mesure C peuvent avoir été relevés par
i
surveillance continue ou par prélèvement intermittent, à l’aide d’un système de mesurage de la qualité de l’air
spécifié sur une durée de prélèvement T .
S
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ISO 11222:2002(F)
Les résultats de mesure C doivent avoir été mesurés au même endroit. La durée de prélèvement T du résultat de
i S
mesure individuel C est généralement inférieure à la période de moyennage T. La couverture de la période de
i
moyennage par N résultats de mesure C est donnée par N N u 1 avec NT=T . Les N résultats de
max max S
i
mesure C sont utilisés pour calculer la moyenne temporelle C (voir 6.1).
T
i
Pour quantifier l’incertitude de la moyenne temporelle C , il est nécessaire de disposer d’informations sur
T
l’incertitude des résultats de mesure C et la couverture de la période de moyennage T par l’ensemble des
i
données. Les informations appropriées sur l’incertitude de mesure peuvent être fournies conformément aux
recommandations du GUM.
Pour les besoins de la présente Norme internationale, l’incertitude quadratique moyenne d’un résultat de mesure
C est décrite par l’équation (2):
i
2 2 2
u()C = u()C + u ()C (2)
i r i nr i
2 2
Le terme u (C ) désigne la partie aléatoire et le terme u (C ) la partie non-aléatoire de l’incertitude quadratique
r i nr i
2
moyenne du résultat de mesure C . La partie non-aléatoire u (C ) correspond aux écarts systématiques bruts.
i nr i
Dans le domaine de la surveillance de la qualité de l’air, la partie non-aléatoire est souvent supérieure à la partie
aléatoire de l’incertitude quadratique moyenne du résultat de mesure C .
i
Le fait de décomposer l’incertitude quadratique moyenne en une partie aléatoire et une partie non-aléatoire
simplifie la quantification des incertitudes de la moyenne temporelle qui en résulte, comme décrit à l’article 6. Pour
l’identification des parties aléatoire et non-aléatoire de l’incertitude quadratique moyenne du résultat de mesure C ,
i
la règle suivante s’applique.
2
La partie aléatoire u ()C est due aux variations aléatoires du processus de mesurage et aux variations aléatoires
r i
des grandeurs d’influence du processus de mesurage qui se déroule dans les conditions de surveillance. Elle peut
être évaluée à l’aide de la variance de la réponse du système de mesure à l’application répétée des étalons de
contrôle dans des conditions de surveillance, par exemple un contrôle du point zéro et de l’ajustage du gain. La
2
partie aléatoire u ()C n’est pas affectée par la structure temporelle du mesurande, mais elle peut être fonction du
r i
résultat de mesure C .
i
2 2
En outre, l’incertitude quadratique moyenne u (C ) peut comporter une partie non-aléatoire u ()C . Cette partie
i nr i
non-aléatoire peut être due à des incertitudes de grandeurs d’influence fixes du processus de mesurage ou à des
écarts systématiques non corrigés dans le processus de mesure.
5.2 Exigences spécifiques relatives aux données d'entrée
L’ensemble de résultats de mesure C utilisé pour calculer la moyenne temporelle C doit être représentatif de la
T
i
structure temporelle du mesurande sur la période de moyennage T.
NOTE 1 Afin de satisfaire à la présente exigence, il est nécessaire d’avoir une connaissance de la structure temporelle
attendue du mesurande.
Les valeurs manquantes ne doivent pas avoir été remplacées, par exemple à l'aide d'une technique d'interpolation.
Les résultats de mesure individuels C doivent avoir été générés de manière indépendante.
i
NOTE 2 Il est généralement admis que les résultats de mesure sont générés de façon indépendante si la durée de
prélèvement est au moins quatre fois supérieure au temps de réponse T du système de mesure.
R
Les informations relatives à l'incertitude des résultats de mesure C utilisés pour calculer la moyenne temporelle
i
C doivent être données conformément aux recommandations du GUM.
T
Concernant l’incertitude des résultats de mesure C , les trois cas suivants doivent être distingués.
i
a) L’ensemble des données est associé à une seule déclaration d’incertitude qui sépare les parties aléatoire et
non-aléatoire.
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ISO 11222:2002(F)
Dans ce cas, les informations suivantes doivent être disponibles:
2 2
1) les variances u ()C , u ;
r i nr
2) le nombre de degrés de liberté f()u()C , f (u ).
r i nr
2 2
Les parties aléatoire u ()C et non-aléatoire u de l’incertitude-type doivent s’appliquer sur toute la période
r i nr
2
de moyennage T. La partie aléatoire u (C ) peut dépendre du résultat de mesure C . La partie non-aléatoire
r i i
2
u est considérée comme étant la même pour tous les résultats de mesure C relevés durant la période de
nr i
2
moyennage. Le nombre de degrés de liberté attribués à u ()C est f()u()C . Le nombre de degrés de liberté
r i r i
2 2 2
attribué à u est f()u . Lorsque u ()C et u sont évaluées à partir du même ensemble de données, alors
nr nr r i nr
f()u()C est égal à f()u .
r i nr
b) L’ensemble des données est divisé en un nombre M (M > 1) de sous-ensembles, chaque sous-ensemble étant
associé à une déclaration d’incertitude séparant les parties aléatoire et non-aléatoire de l’incertitude.
Dans ce cas, les informations suivantes doivent être disponibles pour l’ensemble des valeurs de mesurage
j = 1 à M:
2 2
1) les variances u ()j , u ()j ;
r nr
2) le nombre de mesurages n()j et la période T ( j);
3) le nombre de degrés de liberté f()u ()j , f (u ()j )
r nr
M M
où T = T()j et N = n()j
∑ ∑
j =1 j =1
L’incertitude-type des résultats de mesure C doit avoir été estimée de manière indépendante pour chaque
i
intervalle de temps T()j . La somme des intervalles de temps T()j doit couvrir complètement la période de
2 2
moyennage T. La partie aléatoire u ()j et la partie non-aléatoire u ( j) de l’incertitude-type doivent être
r nr
applicables sur l’intervalle de temps T()j . Les déclarations d’incertitude fournies pour les intervalles de temps
T()j ne doivent pas être basées sur le même ensemble d’étalons de référence. Le nombre de degrés de
2 2
liberté attribués à u ()j est f()u ()j . Le nombre de degrés de liberté attribués à u ( j) est f()u ()j . Si u ()j
r r nr nr r
et u ()j sont évalués à partir du même ensemble de données, alors f()u ( j) est égal à f (u ()j ).
nr r nr
c) L’ensemble des données est associé à une déclaration d’incertitude ou plus (M W 1) ne séparant pas les
parties aléatoire et non-aléatoire de l’incertitude.
Dans ce cas, les informations suivantes doivent être disponibles pour j = 1 à M:
1) l’incertitude u()j ;
2) le nombre de mesurages n()j et la période T ( j);
3) le nombre de degrés de liberté f()j
M M
où T = T()j et N = n()j
∑ ∑
j =1 j =1
L’incertitude-type des résultats de mesure C doit avoir été estimée de manière indépendante pour chaque
i
intervalle de temps T()j . La somme des intervalles de temps T()j doit couvrir complètement la période de
moyennage T. L’incertitude-type u( j) doit être applicable à l’intervalle de temps T()j . L’incertitude-type u()j
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ISO 11222:2002(F)
est considérée comme constante dans les lim
...
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