INTERNATIONAL ISO

STANDARD 6974-2

Second edition
2012-05-15

Natural gas — Determination of

composition and associated uncertainty
by gas chromatography —
Part 2:
Uncertainty calculations
Gaz naturel — Détermination de la composition et de l’incertitude
associée par chromatographie en phase gazeuse — Partie 2: Calculs
d’incertitude

Reference number
ISO 6974-2:2012(E)

© ISO 2012
ISO 6974-2:2012(E)

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ii  © ISO 2012 – All rights reserved

 ISO 6974-2:2012(E)

Contents Page

Foreword............................................................................................................................................................................. iv
Introduction......................................................................................................................................................................... v
1 Scope....................................................................................................................................................................... 1
2 Normative references.......................................................................................................................................... 1
3 Terms and definitions.......................................................................................................................................... 1
4 Symbols.................................................................................................................................................................. 1
4.1 Symbols.................................................................................................................................................................. 1
4.2 Subscripts.............................................................................................................................................................. 2
5 Calculation of uncertainty.................................................................................................................................. 2
5.1 General considerations...................................................................................................................................... 2
5.2 Principles................................................................................................................................................................ 5
5.3 Step 9 — Calculation of uncertainty of mole fractions.............................................................................. 5
5.4 Step 10 — Calculation of the expanded uncertainty of mole fractions............................................... 11
Annex A (informative) Calculation of processed component uncertainties for
the methane‑by‑difference approach............................................................................................................ 12
Annex B (normative) Uncertainties of relative response factors......................................................................... 13
Annex C (informative) Alternative calculation of the uncertainty of the value of the unknown................... 14
Bibliography...................................................................................................................................................................... 16

© ISO 2012 – All rights reserved  iii

ISO 6974-2:2012(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 2.

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 document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.

ISO 6974‑2 was prepared by Technical Committee ISO/TC 193, Natural Gas, Subcommittee SC 1, Analysis
of natural gas.

This second edition of ISO 6974-2, together with ISO 6974-1:2012, cancels and replaces ISO 6974‑1:2000 and
ISO 6974‑2:2001, which have been technically revised.

ISO 6974 consists of the following parts, under the general title Natural Gas — Determination of composition
and associated uncertainty by gas chromatography:

— Part 1: General guidelines and calculation of composition

— Part 2: Uncertainty calculations

— Part 3: Determination of hydrogen, helium, oxygen, nitrogen, carbon dioxide and hydrocarbons up to C8
using two packed columns

— Part 4: Determination of nitrogen, carbon dioxide and C1 to C5 and C6+ hydrocarbons for a laboratory and
on-line measuring system using two columns

— Part 5: Determination of nitrogen, carbon dioxide and C1 to C5 and C6+ hydrocarbons for a laboratory and
on-line process application using three columns

— Part 6: Determination of hydrogen, helium, oxygen, nitrogen, carbon dioxide and C1 to C8 hydrocarbons
using three capillary columns

Future subsequent parts of ISO 6974 are planned.

iv © ISO 2012 – All rights reserved

 ISO 6974-2:2012(E)

Introduction

ISO 6974 describes methods of analysis of natural gas and methods for calculating component mole fractions and
uncertainties. ISO 6974 (all parts) is intended for the measurement of H2, He, O2, N2, CO2 and hydrocarbons, either
as individual components or as a group, for example all hydrocarbons above C5, defined as C6+. This approach is
suitable for a range of end applications, including calibrating gas mixtures and providing natural gas composition
and uncertainty data to be used in the calculation of calorific value and other additive physical properties of the
gas. Details of these end applications are provided in ISO 6974‑3 and subsequent parts of ISO 6974.

ISO 6974-1 gives guidelines for calculating the mole composition of natural gas, determined using one of the
gas chromatographic methods described in ISO 6974-3 and subsequent parts of ISO 6974. ISO 6974-1 also
describes all the essential steps for setting up an analysis, including outlining the structure of the analysis,
defining the working ranges and establishing the analytical procedure.

This part of ISO 6974 describes the steps required to calculate the uncertainty of the component mole fractions
of natural gas determined using gas chromatography.

ISO 6974-3 and subsequent parts of ISO 6974 describe different gas chromatographic methods. These
methods cover both daily practice in the laboratory and on-line field applications. ISO 6974‑1:2012, Annex A,
provides a comparison of the characteristics of the analytical methods described in ISO 6974-3 and subsequent
parts of ISO 6974.

It is intended that this part of ISO 6974 be used in conjunction with ISO 6974-1 and a method of analysis, e.g.
ISO 6974-3 or subsequent parts of ISO 6974.

ISO 6974-1:2012, 5.5, describes the conventional normalization approach for calculating processed mole
fractions from raw mole fractions. When conventional normalization is used for multiple operation methods
without bridging, the uncertainties of the calculated mole fractions will be conservative. If a more accurate
assessment of uncertainty is required, an alternative approach to normalization, using the generalized least
squares (GLS) method, can be used; this is described in ISO 6974-1:2012, Annex B. Further alternative
approaches are available for calculating processed mole fractions, including methane-by-difference (see
ISO 6974-1:2012, Annex C) and data harmonization (see Reference [1]).

© ISO 2012 – All rights reserved  v

INTERNATIONAL STANDARD ISO 6974-2:2012(E)

Natural gas — Determination of composition and associated

uncertainty by gas chromatography —

Part 2:
Uncertainty calculations

1 Scope
This part of ISO 6974 describes the process required to determine the uncertainty associated with the mole
fraction for each component from a natural gas analysis in accordance with ISO 6974‑1.

2 Normative references
The following referenced documents are indispensable for the application 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 6974-1:2012, Natural gas — Determination of composition and associated uncertainty by gas
chromatography — Part 1: General guidelines and calculation of composition

ISO/IEC Guide 98-3, Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in
measurement (GUM:1995)

3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO 6974-1 apply.

4 Symbols

4.1 Symbols
bz parameters of the regression function (z = 0, 1, 2 or 3)

bz mean parameters of the regression function (in “mean normalization” method)

Ci sensitivity coefficient

k coverage factor

K relative response factor with respect to the reference component

ni total number of components (direct plus indirect, but excluding “other components”)

nj total number of gas standards or mixtures

nl total number of injections (and therefore total number of responses)

s standard deviation

T total mole fraction of all raw components

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ISO 6974-2:2012(E)

u(…) standard uncertainty (of the quantity in parentheses)

U(…) expanded uncertainty (of the quantity in parentheses)

x normalized mole fraction

x* raw mole fraction

x′ mole fraction calculated using the methane-by-difference approach [Annex A]

x̂ adjusted mole fraction [Annex C]

y instrumental response of the specified analyte

y mean instrumental response (in “mean normalization” method)

ŷ adjusted instrumental response [Annex C]

Y instrumental response [Annex C]

α intercept of a first-order calibration function [Annex C]

β gradient of a first-order calibration function [Annex C]

γ gradient of the calibration curve [Annex C]

δ mean of the distribution of non-linearity errors

4.2 Subscripts
cal calibration [Annex A]

i component

ind components or groups of components to be analysed by indirect measurement

j gas standard/mixture

l injection

oc other components

p, q indices defining a regression coefficient

ref reference (component or pressure)

s index defining a component

wms working measurement standard

5 Calculation of uncertainty

5.1 General considerations

The process of setting up a gas chromatograph for the analysis of natural gas consists of the steps outlined in
the flowcharts in Figures 1 and 2.

Steps 1 to 8 are covered in ISO 6974-1. This part of ISO 6974 covers steps 9 and 10.

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 ISO 6974-2:2012(E)

START

Step 1: Define the working range

[ISO 6974-1:2012, 6.2]

Step 2: Define the requirements of

the analytical method
[ISO 6974-1:2012, 6.3]

Step 3: Select equipment and

working conditions
[ISO 6974-1:2012, 6.4]

Step 4 [ISO 6974-1:2012, 6.5]

Is primary calibration or
performance evaluation Yes Type 1 or Type 2 Type 2
required? analysis?

Type 1
No

Primary calibration Performance evaluation

Determine analysis function: Determine initial analysis function
(a) Selection of reference gases (a) Selection of reference gases
(b) Measurement of reference gases (b) Measurement of reference gases
(c) Regression analysis (c) Regression analysis
(d) Selection of regression function

Are there any

No indirect
components?

Yes

Step 5: Assign relative response

factors
[ISO 6974-1:2012, 6.6]

FLOWCHART B

Figure 1 — Procedure for determining mole fraction and uncertainty — Steps 1 to 5

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ISO 6974-2:2012(E)

START FROM
FLOWCHART A

Will the analyser be

No used for routine
operation?

Yes
Step 6
[ISO 6974-1:2012, 6.7]

Quality assurance
(QA) check or routine QA check
calibration (RC)?

RC

RC with a WMS QA with a WMS

[ISO 6974-1:2012, 6.7.3.2] [ISO 6974-1:2012, 6.7.3.3]

Step 7: Analysis of samples

[ISO 6974-1:2012, 6.8]

Step 8: Calculation of component mole fractions

[ISO 6974-1:2012, 6.9]

Are uncertainties of
component mole No FINISH
fractions required?

Yes

Step 9: Calculation of uncertainty in mole fractions

[ISO 6974-2:2012, 5.3]

Step 10: Calculation of the expanded

uncertainty in mole fractions
[ISO 6974-2:2012, 5.4]

FINISH

Figure 2 — Procedure for determining mole fraction and uncertainty — Steps 6 to 10

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 ISO 6974-2:2012(E)

5.2 Principles
Uncertainties associated with the component mole fractions shall be calculated in accordance with
ISO/IEC Guide 98-3.

For Type 1 analyses in accordance with ISO 6974-1, the uncertainty calculation includes random and systematic
uncertainties from three main sources: uncertainty of the certified reference mixtures, uncertainty of analysis
and uncertainty of the data fitting procedure.

For Type 2 analyses in accordance with ISO 6974-1, the uncertainty calculation includes both random elements
and systematic errors introduced by the assumption of a linear response through the origin, the systematic
errors being calculated from the results of the initial performance evaluation.

Subclause 5.3 describes methods for estimating the uncertainties of processed mole fractions calculated from
raw mole fractions using the conventional normalization method. Annex A provides a method for use when the
methane-by-difference approach (see ISO 6974-1:2012, Annex C) is employed.

ISO 6974-1 recommends the use of the generalized least squares (GLS) approach for calculation of the
processed mole fraction. However, in some circumstances, an alternative approach using ordinary least
squares may be acceptable and calculation of uncertainty in processed mole fractions in this situation is
described in Annex C.

5.3 Step 9 — Calculation of uncertainty of mole fractions

5.3.1 Determining the equations to be used

5.3.1.1 General considerations

The equations to be used in this step for calculating the uncertainty of mole fractions are given in 5.3.2 and
5.3.3. The equations to be used should be determined by following the three-stage process described in
5.3.1.2 to 5.3.1.4.

The following points should be taken into consideration when selecting the equations to be used.

a) When using the “mean normalization” method (see 5.3.2), the following are calculated in turn for each analyte:

1) mean peak analyser response from all runs;

2) raw mole fraction;

3) normalized mole fraction.

b) When using the “run-by-run normalization” method (see 5.3.3), the following are calculated in turn for
each analyte:

1) raw mole fraction for each run;

2) normalized mole fraction for each run;

3) mean normalized mole fraction.

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ISO 6974-2:2012(E)

5.3.1.2 Stage 1

Calculate the uncertainty of the raw mole fraction for components determined directly by using the appropriate
equation selected from Table 1.

Table 1 — Selecting the equation for calculating the uncertainty

of the raw mole fraction for components determined directly

Equation
Normalization Type 2 analysis
method Type 1 analysis Linearity errors Linearity errors
not corrected corrected
Mean Equation (1) Equation (2) Equation (3)
Run-by-run Equation (12) Equation (13) Equation (14)

5.3.1.3 Stage 2

Calculate the uncertainty of the raw mole fraction for any additional components determined indirectly by using
the appropriate equation selected from Table 2.

Table 2 — Selecting the equation for calculating the uncertainty

of the raw mole fraction for components determined indirectly

Equation
Normalization method (Type 1 or Type 2
analysis)
Mean Equation (4)
Run-by-run Equation (15)

5.3.1.4 Stage 3

Calculate the uncertainty of the normalized mole fraction for all components by using the appropriate equation
selected from Table 3.

Table 3 — Selecting the equation for calculating the uncertainty

of the normalized mole fraction for all components

Equation
Normalization method (Type 1 or Type 2
analysis)
Mean Equation (5)
Run-by-run Equation (16)

5.3.2 Calculation of uncertainty of component mole fractions — Mean normalization method

5.3.2.1 General considerations

The mean normalization method is used in 5.3.2.2 to 5.3.2.4 to calculate the uncertainty of the component
mole fractions determined in accordance with ISO 6974-1:2012, 6.9.2.

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 ISO 6974-2:2012(E)

5.3.2.2 Uncertainty of raw mole fractions

For Type 1 analyses in accordance with ISO 6974-1, calculate the uncertainty of raw mole fractions using
Equation (1):

p =3 q =3
( ) ∑ ∑ C ( x ,b
2
u 2 x i* = i
*
i
*
p,i )C i ( x i , bq,i )u (b p,i ,bq,i ) + Ci ( xi* , yi ) ( )
u 2 yi (1)
p =0 q =0

For Type 2 analyses in accordance with ISO 6974-1, calculate the uncertainty of raw mole fractions using
Equation (2):


u y  
2 2
2  u (b1,i )  ( )
u 2
( ) ( )
= x i*   x i*

  b1,i 
+

i
 
yi  
(2)

   
For Type 2 analyses in accordance with ISO 6974-1, if mean raw mole fractions are corrected for non-linearity
errors associated with the assumed analyser response (see ISO 6974-1:2012, 6.9.4), then additional terms are
included in Equation (2) to allow for uncertainty in correction term δ x i* , giving Equation (3). ( )
NOTE 1 This approach is consistent with ISO/IEC Guide 98-3:2008, F.2.4.5.


 u
2

(b1,i ) 
2
u y 
2
( ) ( )
2
u 2 δ x i*  + u 2 δ i + δ i  ( )
( ) ( )
u 2 x i* = x i*  
  b1,i 
+
 y 
i
i
 + 
 

n l



(3)

    
2
The final term in Equation (3), δ i , is included only if mean mole fractions remain uncorrected (see
ISO 6974‑1:2012, 6.9.4 ).

( )
u 2 δ x i*  is the mean variance of the correction terms over the analytical range of the analyser and u 2 δ i
 
( )
is the variance of the mean correction term.

NOTE 2 This approach is consistent with ISO/IEC Guide 98-3:2008, F.2.4.5.

Using Equation (4), calculate the uncertainty of raw mole fractions of any indirect components from the
uncertainty of the raw mole fraction of the reference component determined using Equations (1) to (3):

 u (Ki ) 
2 
 
( )
2 2 2
2
u x*  u y  u y ( ) ( )  
K i 


( ) ( )  + 
  ref  + ind,i ref
u 2 xind
*
= x *
  +  (4)
,i ind,i  x*   y   y 
  ref  nl 
   ind,i   ref  
 
5.3.2.3 Uncertainty of normalized mole fraction

Calculate the uncertainty of the normalized mole fraction using Equation (5):
s = ni
∑ Ci ( xi , x s* ) ( )
 2  2
u 2 ( xi ) = u 2 x s*  + C i ( x i , x oc )  u 2 ( x oc ) (5)
s =1 

5.3.2.4 Input data

a) ( ) are estimated from the standard deviation, s, of the nl responses to the unknown sample, using
u yi
Equation (6):

( )
s y i,l
( )
u yi =
nl
(6)

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ISO 6974-2:2012(E)

( )
If a multiple operation method with bridging is used, estimate u y i for each appropriate component from
the standard error of the mean of the set of nl responses derived using ISO 6974‑1:2012, Equation (8).

The use of the mean and standard error of the mean as estimators of the desired statistics is based on the
assumption that the observations of responses are uncorrelated with time. The use of very large numbers
of repeat measurements may render this assumption unjustified and should therefore be avoided (see
ISO/IEC Guide 98-3:2008, 4.2.7).

(
u b p,i , bq,i )
b) (
u b p,i , bq,i ) are estimated for Type 1 analyses in accordance with ISO 6974-1 as the values of
nl
obtained during the determination of the analysis function, using the generalized least squares (GLS) method.

The use of a WMS to scale the calibration curve during a Type 1 analysis [see ISO 6974‑1:2012, Equation (5)]
is likely to result in additional uncertainty. Care should be taken to account for this.

c) ( ) are estimated for Type 2 analyses in accordance with ISO 6974-1 from the uncertainties of the
u 2 b1,i
mean responses to the WMS combined with the uncertainties of the mole fractions of the WMS, using
Equation (7):

 2 2

b1,i 
2  u ( yi,wms )  +
(
 u x i,wms ) 

  y i,wms 

 x i,wms

 

( )
u 2 b1,i = 
nl
 (7)

d) The sensitivity coefficients for the raw mole fraction with respect to the mean response to the unknown
sample C i ( x i* , y i ) are derived from ISO 6974‑1:2012, Equation (9):

p =3
∂x i*
∑  p ⋅ b p,i ( yi )
 p −1 
C i ( x i* , y i ) = =  (8)
∂ yi p =0 

e) The sensitivity coefficients for the raw mole fraction with respect to the coefficients of the analysis function
C i ( x i* , b p,i ) are derived from ISO 6974‑1:2012, Equation (9):

∂x i*
( )
p
C i ( x i* , b p,i ) = = yi (9)
∂b p,i

f) The sensitivity coefficients for mole fractions with respect to raw mole fractions are derived from
ISO 6974‑1:2012, Equation (11):

∂x i T − x i*
C i ( x i , x s* ) = = (1 − xoc )         (when i = s)
∂x s* T2

∂x i − x i*
C i ( x i , x s* ) = = (1 − xoc )             (when i ≠ s) (10)
∂x s* T2
ni
where T is the so-called “unnormalized total”, ∑ xi* .
i =1
g) The sensitivity coefficients for mole fractions with respect to mole fractions of the “other components” (see
ISO 6974‑2:2012, 3.4) are derived from ISO 6974‑1:2012, Equation (11):

∂x i x*
C i ( x i , x oc ) = =− i (11)
∂x oc T

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 ISO 6974-2:2012(E)

h) Values of u ( K i ) for flame ionization detectors (FIDs) and thermal conductivity detectors (TCDs) are
specified in Annex B.

5.3.3 Calculation of uncertainty of component mole fractions — Run-by-run normalization method

5.3.3.1 General considerations

The run-by-run normalization method is used in 5.3.3.2 to 5.3.3.4 to calculate the uncertainty of the component
mole fractions determined in accordance with ISO 6974‑1:2012, 6.9.3.

5.3.3.2 Uncertainty of raw mole fraction

For Type 1 analyses in accordance with ISO 6974-1, calculate the uncertainty of raw mole fractions using
Equation (12):

p =3 q =3
( ) ∑ ∑ C ( x
2
u 2 x i*,l = i
* *
i,l , b p,i,l )C i ( x i,l , bq,i,l )u (b p,i,l ,bq,i,l ) + Ci ( xi*,l , yi,l ) ( )
u 2 y i,l (12)
p =0 q =0

For Type 2 analyses in accordance with ISO 6974-1, calculate the uncertainty of raw mole fractions using
Equation (13):

2   u y i,l  
2
b1,i,l  2  u ( ) ( )
u 2
( ) ( )
= x i*,l
  x i*,l
  b1,i,l 
+  
 y i,l  
 (13)
   
As in 5.3.2.2, for Type 2 analyses in accordance with ISO 6974-1, if raw mole fractions are corrected for non-
linearity errors associated with the assumed analyser response (see ISO 6974‑1:2012, 6.9.4 ), then additional
terms are included in Equation (13) to allow for the uncertainty of correction term δ x i* , giving Equation (14): ( )
 2 2 
2  u (b1,i,l )   u y i,l( ) 
u 2
( )=( )
x i*,l x i*,l 
  b1,i,l 
+
 y i,l 
 * 
2 2
( )
2
+ u δ xi + u δ i + δ i 
 

( ) (14)
    
2
The final term in Equation (14), δ i , is included only if raw mole fractions are left uncorrected (see
ISO 6974‑1:2012, 6.9.4).

Using Equation (15), calculate the uncertainty of the raw mole fraction of any indirect components from the
uncertainty of the raw mole fraction of the reference component determined using Equations (12) to (14):


( x )  2
2 2 2
 u
2
*
(
 u y ind,i,l )  (
 u y ref ,l )  u (Ki )  
( )=( )
2 * * ref ,l
u xind,i xind,i  * +  + +   (15)
  y ind,i,l  y ref ,l
  xref ,l  

 

  K i  
  

5.3.3.3 Uncertainty of normalized mole fraction

Calculate the uncertainty of the normalized mole fraction using Equation (16):

l = nl  s = ni 

( ) 
2 2
∑  ∑ Ci ( xi,l , x s*,l ) u 2 x s*,l  + C i ( x i,l , x oc ) u 2 ( x oc ) 
 
u 2
( xi ) = l =1  s =1
nl2
 (16)

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ISO 6974-2:2012(E)

5.3.3.4 Additional input data

a) ( )
u y i,l are estimated from the standard deviations, s, of the l responses to the sample. If a multiple
( )
operation method with bridging is used, estimate u y i,l , for each appropriate component, from the
standard deviation of the set of nl responses derived using ISO 6974‑1:2012, Equation (12).

b) ( )
u b p,i,l , bq,i,l are estimated for Type 1 analyses in accordance with ISO 6974-1 as the values of u b p,i , bq,i ( )
obtained during the determination of the analysis function using the generalized least squares (GLS) method.

The use of a WMS to scale the calibration curve during a Type 1 analysis [see ISO 6974‑1:2012, Equation (5)]
is likely to result in additional uncertainty. Care should be taken to account for this.

c) (
u 2 b1,i,l ) are estimated for Type 2 analyses in accordance with ISO 6974-1 from the uncertainties of the
mean responses to the WMS combined with the uncertainties of the mole fractions of the WMS, using
Equation (17):

 2 2
(
 u y i,wms )  (
 u x i,wms ) 
u 2
(b1,i,l ) = b12,i,l  
+
 x i,wm

 
(17)
 y i,wms   ms 
 

d) The sensitivity coefficients for the raw mole fraction with respect to the response to the unknown sample
C i ( x i*,l , y i,l ) are derived from ISO 6974‑1:2012, Equation (13):

∂x i*,l p =3
p −1 
C i ( x i*,l , y i,l ) =
∂y i,l
= ∑  p ⋅ b p,i,l ( yi,l )  (18)
p =0

e) The sensitivity coefficients of the raw mole fraction with respect to the coefficients of the analysis function
C i ( x i*,l , b p,i,l ) are derived from ISO 6974‑1:2012, Equation (13):

∂x i*,l p
C i ( x i*,l , b p,i,l ) =
∂b p,i,l
= y i,l( ) (19)

f) The sensitivity coefficients for the mole fraction with respect to the raw mole fraction are derived from
ISO 6974‑1:2012, Equation (11):

∂x i,l Ti − x i*,l
C i ( x i,l , x s*,l ) = = (1 − xoc )          (when i = s)
∂x s*,l Tl2

∂x i,l − x i*,l
C i ( x i,l , x s*,l ) = = (1 − xoc )               (when i ≠ s) (20)
∂x s*,l Tl2

g) The sensitivity coefficients for the mole fractions with respect to the mole fractions of the “other components”
(see ISO 6974‑2:2012, 3.4) are derived from ISO 6974‑1:2012, Equation (15):

∂xi,l xi*,l
Ci ( xi,l , xoc ) = = (21)
∂xoc Tl

10  © ISO 2012 – All rights reserved

 ISO 6974-2:2012(E)

5.4 Step 10 — Calculation of the expanded uncertainty of mole fractions

Calculate the expanded uncertainty, U(xi), of the normalized components by multiplying u(xi) by an appropriate
coverage factor using Equation (22):

U ( xi ) = k × u ( xi ) (22)

NOTE A coverage factor of k = 2, providing a level of confidence of approximately 95 %, is usually used.

© ISO 2012 – All rights reserved 11

ISO 6974-2:2012(E)

Annex A
(informative)

Calculation of processed component uncertainties

for the methane‑by‑difference approach

A.1 General considerations

This annex describes a procedure for calculating the uncertainties of processed component mole fractions
determined using the methane-by-difference approach described in ISO 6974‑1:2012, Annex C.

A.2 Calculation of methane-by-difference component uncertainties

For all components including methane, the standard uncertainty of a mole fraction x′i is calculated using
Equation (A.1):

u ( x′i ) = s 2 ( x′i ) + u cal

2
( x′i ) (A.1)

where

s(x′i) is the standard deviation of replicate processed (methane-by-difference) mole fractions at level x′i;

ucal(x′i) is the calibration uncertainty at level x′i.

For components other than methane, the calibration uncertainty ucal(x′i) is determined in accordance with
ISO 6974‑1:2012, 6.5 (step 4; see Figure 1).

For methane, the calibration uncertainty is calculated using Equation (A.2):

ni
u cal ( x1′ ) = ∑ u cal
2
( x′i ) (A.2)
i=2

Equation (A.1) applies to a mole fraction obtained from a single analysis. If the mole fractions are mean values
of nl replicates, the standard uncertainty is given by Equation (A.3):

s 2 ( x′i )
u ( x′i ) = 2
+ u cal ( x′i ) (A.3)
nl

12  © ISO 2012 – All rights reserved

 ISO 6974-2:2012(E)

Annex B
(normative)

Uncertainties of relative response factors

B.1 Uncertainties of relative response factors for flame ionization detectors (FIDs)
The relative standard uncertainties of the relative response factors, as calculated for an FID and given in
ISO 6974‑1:2012, Table D.1, shall be taken to be equal to 2 %[2]. Alternative figures may be used if determined
by thoroughly validated experimental procedures.

NOTE The method for determining the relative response factors for an FID is given in ISO 6974‑1:2012, D.1.

B.2 Uncertainties of relative response factors for thermal conductivity

detectors (TCDs)
The relative standard uncertainties of the relative response factors, as calculated for a TCD and given in
ISO 6974‑1:2012, Table D.2, shall be taken to be equal to 10 %[2]. Alternative figures may be used if determined
by thoroughly validated experimental procedures.

NOTE The method for determining the relative response factors for a TCD is given in ISO 6974‑1:2012, D.2.

© ISO 2012 – All rights reserved 13

ISO 6974-2:2012(E)

Annex C
(informative)

Alternative calculation of the uncertainty of the value of the unknown

This annex provides an alternative procedure[3] to the generalized least squares approach (see ISO 6974‑1:2012,
6.5.5). The approach described in this annex has the benefit of being a more straightforward procedure with
which to carry out the calculations. In order to maintain the simplicity of this alternative approach, it can only be
applied when the analysis and calibration functions can be approximated in a first-order form.

Consider a set of data of points (xi, yi) forming a first-order calibration curve where xi is the mole fraction of
each standard and yi the instrumental response. The equation for a first-order calibration function is given by
Equation (C.1):

y i = β xi + α (C.1)

where α and β are the intercept and gradient of the line respectively. In this case, the intercept and the gradient
are highly correlated.

This alternative approach reduces the data to a set of points expressed with respect to the centroid ( x , y ),
where x and y are the mean values of x and y.

The calibration curve can now be expressed as given in Equation (C.2):

( y i − y ) = γ ( xi − x ) (C.2)

where γ is the gradient of the calibration function.

Consider the measurement of an “unknown” standard producing an instrumental response of Y. The mole
fraction of the unknown ( x̂ ) can be expressed as the linear first-order function given in Equation (C.3):

x̂ =
(Y − y ) + x (C.3)
γ

NOTE The parameters of the analysis function in ISO 6974-1:2012, Equation (2), b 0 and b1, are related to the
parameters in Equation (C.3) by b0 = x − y γ and b1 = 1 γ .

The covariances cov( y,x ) and cov(γ, x ) are both zero. In accordance with ISO/IEC Guide 98-3, the uncertainty
of the mole fraction given by Equation (C.3) is calculated using Equation (C.4):

1  2
() ( ) ( ) u (γ ) ( ) ( )
2 2 2 2 2
u xˆ =u x +  xˆ − x +u Y +u y  (C.4)
γ2  

The mole fraction x is defined as given by Equation (C.5):

1
x=
nj ∑ xi (C.5)
i

where nj is the number of standards measured.

14  © ISO 2012 – All rights reserved

 ISO 6974-2:2012(E)

If the uncertainties u(x) are uncorrelated, the standard error of the estimated mean is given by Equation (C.6):

1
∑ u ( xi )
2 2
u(x) = (C.6)
2
nj i

When all the uncertainties are equal to u(x), this simplifies to Equation (C.7):
2
2 u ( x)
u(x) = (C.7)
nj

Similar expressions can be generated for u ( y ) and u(Y), as given in Equations (C.8) and (C.9) respectively:
2
2 u ( y)
u(y) = (C.8)
nl

2
2 u ( y)
u (Y ) = (C.9)
nl

where nl is the number of measurements of each standard.

Incorporating Equations (C.7), (C.8) and (C.9) into (C.4) gives Equation (C.10):

() ( ) ( )
2 2 2
u x u y  1 1 u γ
() ( xˆ − x )
2 2
u xˆ = +  + + (C.10)
nj γ 2  n j nl  γ2
 
This equation can be used to evaluate the uncertainty of the calculated value of the unknown from knowledge
of the uncertainty of the standards [u(x)], the uncertainty of the analysis [u(y)] and the uncertainty of the gradient
of the calibration curve [u(γ)]. This can be calculated from an ordinary least squares fit of the analytical data that
follow the relationship given by Equation (C.2).

© ISO 2012 – All rights reserved 15

ISO 6974-2:2012(E)

Bibliography

[1] Vargha, G., Milton, M., Cox, M. and K amvissis, S., Harmonisation of Coupled Calibration Curves to
Reduce Correlated Effects in the Analysis of Natural gas by Gas Chromatography, J. Chromatogr. A.,
2005, 1062, pp. 239‑245

[2] Tong, H.Y. and K arasek, F.W., Flame Ionisation Detector Response Factors for Compound Classes in
Quantitative Analysis of Complex Organic Mixtures, Anal. Chem., 1984, 56, 2124-2128

[3] Draper, N.R. and Smith, H., Applied Regression Analysis, 3rd edition, Wiley, New York, 1998

[4] ISO Guide 31, Reference materials — Contents of certificates and labels

[5] ISO Guide 34, General requirements for the competence of reference material producers

[6] ISO Guide 35, Reference materials — General and statistical principles for certification

[7] ISO/IEC Guide 99:2007, International vocabulary of metrology — Basic and general concepts and
associated terms (VIM)

[8] ISO 6974-3, Natural gas — Determination of composition with defined uncertainty by gas
chromatography — Part 3: Determination of hydrogen, helium, oxygen, nitrogen, carbon dioxide and
hydrocarbons up to C8 using two packed columns

[9] ISO 6974-4, Natural gas — Determination of composition with defined uncertainty by gas
chromatography — Part 4: Determination of nitrogen, carbon dioxide and C1 to C5 and C6+ hydrocarbons
for a laboratory and on-line measuring system using two columns

[10] ISO 6974-5, Natural gas — Determination of composition with defined uncertainty by gas
chromatography — Part 5: Determination of nitrogen, carbon dioxide and C1 to C5 and C6+ hydrocarbons
for a laboratory and on-line process application using three columns

[11] ISO 6974-6, Natural gas — Determination of composition with defined uncertainty by gas
chromatography — Part 6: Determination of hydrogen, helium, oxygen, nitrogen, carbon dioxide and
C1 to C8 hydrocarbons using three capillary columns

[12] ISO 10715, Natural gas — Sampling guidelines

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ISO 6974-2:2012(E)

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