ISO 5168:2005

Measurement of fluid flow - Procedures for the evaluation of uncertainties

ISO 5168:2005 establishes general principles and describes procedures for evaluating the uncertainty of a fluid flow-rate or quantity. A step-by-step procedure for calculating uncertainty is given.

Mesure de débit des fluides — Procédures pour le calcul de l'incertitude

General Information

• Status: Published
• Publication Date: 14-Jun-2005
• ICS: 17.120.10 - Flow in closed conduits
• Technical Committee: ISO/TC 30 - Measurement of fluid flow in closed conduits
• Drafting Committee: ISO/TC 30 - Measurement of fluid flow in closed conduits
• Current Stage: 9092 - International Standard to be revised
• Start Date: 17-Oct-2023
• Completion Date: 13-Dec-2025

Relations

• Consolidated By: ISO 15254:2009 - Ophthalmic optics and instruments - Electro-optical devices for enhancing low vision
• Effective Date: 06-Jun-2022
• Revises: ISO/TR 5168:1998 - Measurement of fluid flow - Evaluation of uncertainties
• Effective Date: 15-Apr-2008

Overview

ISO 5168:2005 - Measurement of fluid flow: Procedures for the evaluation of uncertainties provides a standardized framework for quantifying the uncertainty in measured fluid flow-rate or flow quantity. It establishes general principles and a clear, step‑by‑step procedure (Annex A) for calculating combined standard uncertainty and expanded uncertainty, helping engineers and metrologists report flow measurements with a defined confidence level.

Key topics and requirements

• Uncertainty concepts and terminology: adopts terms from GUM and VIM (e.g., standard uncertainty u(x), combined standard uncertainty uc(y), expanded uncertainty U and coverage factor k).
• Type A and Type B evaluations: statistical (Type A) and non‑statistical (Type B) methods to assess individual uncertainty components.
• Probability distributions: guidance on rectangular, normal, triangular and bimodal distributions for Type B evaluations (Annex B).
• Sensitivity coefficients: analytical and numerical methods to determine how input quantity variations affect the measurand.
• Combination of uncertainties: rules to combine variances and covariances to obtain the combined uncertainty.
• Expression of results: computing expanded uncertainty (U = k·uc) and preparing an uncertainty budget for transparent reporting.
• Special topics and annexes: calibration of flow meters (Annex H), correlated input variables (Annex F), practical examples (Annex G), coverage factors (Annex C), and recommended procedures for multiple meters in parallel (Annex J).
• Reference to GUM: advises using the ISO Guide to the expression of uncertainty in measurement for additional depth.

Applications and who uses it

ISO 5168:2005 is used by:

• Flow metering and calibration laboratories for documenting meter performance and uncertainty on calibration rigs.
• Process and instrumentation engineers who need to quantify and report measurement uncertainty in flow‑rate results.
• Quality and compliance teams in water utilities, chemical, oil & gas, HVAC and manufacturing sectors to support measurement traceability and decision thresholds.
• Research and development groups evaluating new flow meter designs or experimental flow measurements.

Practical uses include preparing an uncertainty budget, selecting appropriate coverage factors (typically k ≈ 2–3 for common confidence levels), performing Type A/B analyses, and documenting uncertainty for reporting or contractual purposes.

Related standards

• ISO Guide to the expression of uncertainty in measurement (GUM, 1995)
• International vocabulary of basic and general terms in metrology (VIM)
• ISO 9300 (measurement of gas flow by critical flow Venturi nozzles)

Keywords: ISO 5168:2005, measurement of fluid flow, flow-rate uncertainty, uncertainty evaluation, combined standard uncertainty, expanded uncertainty, Type A, Type B, uncertainty budget, flow meter calibration.

Standards Content (Sample)

INTERNATIONAL ISO
STANDARD 5168
Second edition
2005-06-15
Measurement of fluid flow — Procedures
for the evaluation of uncertainties
Mesure de débit des fluides — Procédures pour le calcul de l'incertitude

Reference number
©
ISO 2005
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ii © ISO 2005 – All rights reserved

Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions. 1
4 Symbols and abbreviated terms . 3
4.1 Symbols . 3
4.2 Subscripts . 7
5 Evaluation of the uncertainty in a measurement process. 8
6 Type A evaluations of uncertainty . 9
6.1 General considerations. 9
6.2 Calculation procedure. 9
7 Type B evaluation of uncertainties . 10
7.1 General considerations. 10
7.2 Calculation procedure. 10
7.3 Rectangular probability distribution. 10
7.4 Normal probability distribution . 11
7.5 Triangular probability distribution. 11
7.6 Bimodal probability distribution . 11
7.7 Assigning a probability distribution . 11
7.8 Asymmetric probability distributions. 11
8 Sensitivity coefficients. 12
8.1 General. 12
8.2 Analytical solution. 12
8.3 Numerical solution. 12
9 Combination of uncertainties . 13
10 Expression of results . 14
10.1 Expanded uncertainty . 14
10.2 Uncertainty budget . 15
Annex A (normative) Step-by-step procedure for calculating uncertainty . 17
Annex B (normative) Probability distributions . 20
Annex C (normative) Coverage factors. 22
Annex D (informative) Basic statistical concepts for use in Type A assessments of uncertainty. 24
Annex E (informative) Measurement uncertainty sources. 36
Annex F (informative) Correlated input variables . 38
Annex G (informative) Examples . 40
Annex H (informative) The calibration of a flow meter on a calibration rig. 58
Annex I (informative) Type A and Type B uncertainties in relation to contributions to uncertainty
from “random” and “systematic” sources of uncertainty. 61
Annex J (informative) Special situations using two or more meters in parallel. 62
Annex K (informative) Alternative technique for uncertainty analysis. 64
Bibliography . 65

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 5168 was prepared by Technical Committee ISO/TC 30, Measurement of fluid flow in closed conduits,
Subcommittee SC 9, General topics.
This second edition of ISO 5168 cancels and replaces ISO/TR 5168:1998, which has been technically revised
(see Annex I).
iv © ISO 2005 – All rights reserved

Introduction
Whenever a measurement of fluid flow (discharge) is made, the value obtained is simply the best estimate
that can be obtained of the flow-rate or quantity. In practice, the flow-rate or quantity could be slightly greater
or less than this value, the uncertainty characterizing the range of values within which the flow-rate or quantity
is expected to lie, with a specified confidence level.
GUM is the authoritative document on all aspects of terminology and evaluation of uncertainty and should be
referred to in any situation where this International Standard does not provide enough depth or detail. In
particular, GUM (1995), Annex F, gives guidance on evaluating uncertainty components.

INTERNATIONAL STANDARD ISO 5168:2005(E)

Measurement of fluid flow — Procedures for the evaluation
of uncertainties
1 Scope
This International Standard establishes general principles and describes procedures for evaluating the
uncertainty of a fluid flow-rate or quantity.
A step-by-step procedure for calculating uncertainty is given in Annex A.
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 9300, Measurement of gas flow by means of critical flow Venturi nozzles
ISO Guide to the expression of uncertainty in measurement (GUM), 1995
International vocabulary of basic and general terms in metrology (VIM), 1993
3 Terms and definitions
For the purposes of this document, the terms and definitions given in VIM (1993), GUM (1995) and the
following apply.
3.1
uncertainty
parameter, associated with the results of a measurement, that characterizes the dispersion of the values that
could reasonably be attributed to the measurand
NOTE Uncertainties are expressed as an absolute value and do not take a positive or negative sign.
3.2
standard uncertainty
u(x)
uncertainty of the result of a measurement expressed as a standard deviation
3.3
relative uncertainty
*
u (x)
standard uncertainty divided by the best estimate
*
NOTE 1 u (x) = u(x)/x.
*
NOTE 2 u (x) can be expressed either as a percentage or in parts per million.
NOTE 3 Relative uncertainty is sometimes referred to as dimensionless uncertainty.
NOTE 4 The best estimate is in most cases the arithmetic mean of the related uncertainty interval.
3.4
combined standard uncertainty
u (y)
c
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 or
covariances of these other quantities weighted according to how the measurement result varies with changes
in these quantities
3.5
relative combined uncertainty
*
u (y)
c
combined standard uncertainty divided by the best estimate
*
NOTE 1 u (y) can be expressed as a percentage or parts per million.
c
*
NOTE 2 u (y) = u (y)/y.
c c
NOTE 3 Relative combined uncertainty is sometimes referred to as dimensionless combined uncertainty.
NOTE 4 The best estimate is in most cases the arithmetic mean of the related uncertainty interval.
3.6
expanded uncertainty
U
quantity defining an interval about the result of a measurement that can be expected to encompass a large
fraction of the distribution of values that could reasonably be attributed to the measurand
NOTE 1 The fraction can be viewed as the coverage probability or the confidence level of the interval.
NOTE 2 U = ku (y)
c
3.7
relative expanded uncertainty
*
U
expanded uncertainty divided by the best estimate
*
NOTE 1 U can be expressed as a percentage or in parts per million.
* *
NOTE 2 U = ku (y).
c
NOTE 3 Relative expanded uncertainty is sometimes referred to as dimensionless expanded uncertainty.
NOTE 4 The best estimate is in most cases the arithmetic mean of the related uncertainty interval.
3.8
coverage factor
k
numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded
uncertainty
NOTE A coverage factor is typically in the range 2 to 3.
3.9
Type A evaluation
〈uncertainty〉 method of evaluation of uncertainty by the statistical analysis of a series of observations
2 © ISO 2005 – All rights reserved

3.10
Type B evaluation
〈uncertainty〉 method of evaluation of uncertainty by means other than the statistical analysis of a series of
observations
3.11
sensitivity coefficient
c
i
change in the output estimate, y, divided by the corresponding change in the input estimate, x
i
3.12
relative sensitivity coefficient
*
c
i
relative change in the output estimate, y, divided by the corresponding relative change in the input estimate, x
i
4 Symbols and abbreviated terms
4.1 Symbols
a estimated semi-range of a component of uncertainty associated with input estimate, x ,
i i
as defined in Annex B
A area of the throat
t
b breadth associated with a vertical i
i
b′ upper bound of an asymmetric uncertainty distribution as defined in Annex B
i
c sensitivity coefficient used to multiply the uncertainty in the input estimate, x , to obtain
i i
the effect of a change in the input quantity on the uncertainty of the output estimate, y
*
c relative sensitivity coefficient used to multiply the relative uncertainty in input estimate, x ,
i i
to obtain the effect of a relative change in the input quantity on the relative uncertainty of
the output estimate, y
C calibration coefficient
c
C discharge coefficient
C coefficient of variation
V
d depth associated with a vertical i
i
d orifice diameter
o
d orifice diameter measured at temperature T
o,0 0,x
d pipe diameter
p
d pipe diameter measured at temperature T
p,0 0,x
E mean meter error, expressed as a fraction
E jth meter error, expressed as a fraction
j
f functional relationship between estimates of the measurand, y, and the input estimates,
x , on which y depends
i
∂f
partial derivative with respect to input quantity, x, of the functional relationship, f,
i
∂x
i
between the measurand and the input quantities
q
F flow factor, equal to
∆p
r
F flow factor for a new design
exp
0,8
F 19 000 ⋅ β Re
()
dp
Redp
F reference flow factor
ref
F factor, assumed to be unity, that relates the discrete sum over the finite number of
s
verticals to the integral of the continuous function over the cross-section
k coverage factor used to calculate the expanded uncertainty, U
k coverage factor derived from a table; see D.12
t
K meter factor
K mean meter factor
K jth K-factor;
j
l length of crest
b
l gauged head
h
l distance from the upstream tapping to the upstream face
L l divided by the pipe diameter, d
1 1 p
′
l distance from the downstream tapping to the downstream face
′ ′
L l divided by the pipe diameter, d
2 2
p
m particular item in a set of data
m′ number of data sets to be pooled
m″ number of verticals
′ ′
M 21L − β
( )
2 2
n number of repeat readings or observations
n′ exponent of l , usually 1,5 for a rectangular weir and 2,5 for a V-notch
h
4 © ISO 2005 – All rights reserved

n′′ number of depths in a vertical at which velocity measurements are made
N number of input estimates, x , on which the measurand depends
i
p upstream pressure
∆p pressure difference across the orifice meter
mt
∆p pressure difference across the radiator
r
P(a ) probability that an input estimate, x , has a value of a
i i i
q volume flow-rate
q mass flow;
ma
Q flow, expressed in cubic metres per second, at flowing conditions
R specific gas constant
Re Reynolds number related to d by the expression Vd ρ/µ
dp p p
s pooled experimental standard deviation of the orifice plate readings
mt,po
s standard deviation of a larger set of data used with a smaller data set
pe
s standard deviation pooled from several sets of data
po
s pooled experimental standard deviation for the radiator readings
r,po
s(x) experimental standard deviation of a random variable, x, determined from n repeated
observations
s x experimental standard deviation of the arithmetic mean, x
( )
t Student’s statistic
T upstream absolute temperature
T temperature at which measurement x is made
0,x
T operating temperature
op
u (y) combined uncertainty for those components for multiple meters that are correlated
c,corr
u (y) combined uncertainty for those components for multiple meters that are uncorrelated
c,uncorr
*
u instrument calibration uncertainty from all sources, formerly called systematic errors or
cal
biases
*
u relative uncertainty in point velocity at a particular depth in vertical i due to the variable
cri
responsiveness of the current meter
*
u relative standard uncertainty in the coefficient of discharge
d
*
u relative uncertainty in point velocity at a particular depth in vertical i due to velocity
ei
fluctuations (pulsations) in the stream
*
u relative standard uncertainty in the measurement of the crest length
lb
*
u relative standard uncertainty in the measurement of the gauged head
lh
*
u relative uncertainty due to the limited number of verticals
m″
*
u relative uncertainty in mean velocity, V , due to the limited number of depths at which
pi i
velocity measurements are made at vertical, i
*
u (Q) combined relative standard uncertainty in the discharge;
u standard uncertainty of a single value based on past experience
sm
u(x ) correlated components of uncertainty in a single meter
i,corr
u(x ) uncorrelated components of uncertainty in a single meter
i,uncorr
*
u (x ) standard uncertainty associated with the input estimate, x
i i
*
uy() combined standard uncertainty associated with the output estimate, y
c
*
u (x ) relative standard uncertainty associated with the input estimate x
i i
*
uy() combined relative standard uncertainty associated with the output estimate, y
c
*
U (y) relative expanded uncertainty associated with the output estimate
U(y) expanded uncertainty associated with the output estimate, y
U combined uncertainty of the calibration rig
CMC
U type A uncertainty in meter error
AS-overall-E
*
U type A uncertainty in the K-factor
AS-overall-K
V mean velocity in the pipe
V mean velocity associated with a vertical i
i
x estimate of the input quantity, X
i i
x mth observation of random quantity, x
m
x dimension at temperature T
0 0,x
x arithmetic mean or average of n repeated observations, x , of randomly varying
m
quantity, x
y estimate of the measurand, Y
∆x increment in x used for numerical determination of sensitivity coefficient
i i
6 © ISO 2005 – All rights reserved

∆y increment in y found in numerical determination of sensitivity coefficient
Z Grubbs test statistic for outliers
n
β orifice plate diameter ratio, equal to d /d
o p
ϕ critical flow function
cf
Φ ratio of the factor F for a new design compared to the old design
F
λ expansion coefficient
µ dynamic fluid viscosity
ρ fluid density
ν degrees of freedom
ν effective degrees of freedom
eff
v degrees of freedom associated with a pooled standard deviation
po
4.2 Subscripts
c combined
corr correlated
do orifice diameter
dp pipe diameter, effective
ex external
i of the ith input
j of the jth set
k = 2 obtained with a coverage factor of 2
m of the mth observation
n of the nth observation
N of the Nth input
nom nominal value of
op operating temperature
pe from past experience
po pooled
sm based on a single measurement
t tolerance interval
uncorr uncorrelated
x of x
x of the mean value of x
95 with a 95 % confidence level
5 Evaluation of the uncertainty in a measurement process
The first stage in an uncertainty evaluation is to define the measurement process. For the measurement of
flow-rate, it will normally be necessary to combine the values of a number of input quantities to obtain a value
for the output. The definition of the process should include the enumeration of all the relevant input quantities.
Annex E enumerates a number of categories of sources of uncertainty. This categorization can be of value
when defining all of the sources of uncertainty in the process. It is assumed in the following sections that the
sources of uncertainty are uncorrelated; correlated sources require different treatment (see Annex F).
Consideration should also be given to the time over which the measurement is to be made, taking into
account that flow-rate will vary over any period of time and that the calibration can also change with time.
If the functional relationship between the input quantities X , X , …, X , and output quantity Y in a flow
1 2 N
measurement process is specified in Equation (1):
Yf= X ,X ,.,X (1)
( )
12 N
then an estimate of Y, denoted by y, is obtained from Equation (1) using input estimates x , x , … x , as shown
1 2 N
in Equation (2):
y =fx ,x ,.,x (2)
( )
12 N
Provided the input quantities, X, are uncorrelated, the total uncertainty of the process can be found by
i
calculating and combining the uncertainty of each of the contributing factors in accordance with Equation (3):
N

uy = cux (3)
() ( )
c ii
∑

i=1
Where the extent of interdependence is known to be small, Equation (3) may be applied even though some of
[1]
the input quantities are correlated; ISO 5167-1:2003 provides an example of this.
Each of the individual components of uncertainty, u(x ), is evaluated using one of the following methods:
i
 Type A evaluation: calculated from a series of readings using statistical methods, as described in
Clause 6;
 Type B evaluation: calculated using other methods, such as engineering judgement, as described in
Clause 7.
Uncertainty sources are sometimes classified as “random” or “systematic” and the relationship between these
categorizations and Type A and Type B evaluations is given in Annex I.
The sensitivity coefficients, c , provide the links between uncertainty in each input and the resulting uncertainty
i
in the output. The methods of calculating the individual sensitivity coefficients, c , are described in detail in
i
Clause 8.
8 © ISO 2005 – All rights reserved

6 Type A evaluations of uncertainty
6.1 General considerations
Type A evaluations of uncertainty are those using statistical methods, specifically, those that use the spread of
a number of measurements.
Whilst no correction can be made to remove random components of uncertainty, their associated uncertainty
becomes progressively less as the number of measurements increases. In taking a series of measurements, it
should be recognized that, as the purpose is to define the random fluctuations in the process, the timescale
for the data collection should reflect the anticipated timescale for the fluctuations. Collecting readings at
millisecond intervals for a process that fluctuates over several minutes will not characterize those fluctuations
adequately.
In many measurement situations, it is not practical to make a large number of measurements. In this case, this
component of uncertainty may have to be assigned on the basis of an earlier Type A evaluation, based on a
larger number of readings carried out under similar conditions. Caution should be exercised in making these
estimates (see Annex D), as there will always be some uncertainty associated with the assumption that the
earlier measurements were taken under truly similar conditions.
The methods of calculating the uncertainty in a mean and in a single value reflect the reduction in uncertainty
obtained by averaging several readings [Equations (4) to (8)] and are explained in more detail in D.4 to D.6.
6.2 Calculation procedure
Further explanation of the equations given below can be found in Annex D.
The standard uncertainty of a measured value, x, is calculated from a sample of measurements, x , in
i i,m
accordance with Equations (4) to (8):
a) Calculate the average value of the measurements in accordance with Equation (4); see D.1:
n
xx= (4)
ii,m
∑
n
m=1
b) Calculate the standard deviation of the sample in accordance with Equation (5); see D.2:
n
1 2
sx=−x x (5)
() ()
ii∑ ,mi
n −1
()
m=1
The standard uncertainty of a single sample is the same as its standard deviation and is given by
Equation (6):
ux =s x (6)
( ) ( )
ii
c) Calculate the standard deviation of the mean value in accordance with Equation (7); see D.4:
s x
()
i
sx = (7)
()
i
n
The standard uncertainty of the mean value is then given by Equations (8):
ux =s x (8)
( ) ( )
ii
The use of the mean of several readings is a key technique for reducing uncertainty in readings subject to
[2]
random variations. For the derivation of Equation (7) see Dietrich .
NOTE The approach outlined here represents a simplification and, when the functional relationship defined by
Equation (1) is highly non-linear and uncertainties are large, the more rigorous approach described in the GUM (1995),
4.1.4, could yield a more robust answer.
7 Type B evaluation of uncertainties
7.1 General considerations
Type B evaluations of uncertainty are those carried out by means other than the statistical analysis of series of
observations.
As explained in D.9, Type A uncertainties result in a bandwidth of 1 standard deviation that would encompass
68 % of the possible values of the measured quantity. In making Type B assessments, it is necessary to
ensure that a similar confidence level is obtained such that the uncertainties obtained by different methods
can be compared and combined.
Type B assessments are not necessarily governed by the normal distribution and the limits assigned can
represent varying confidence levels. Thus, a calibration certificate could give the meter factor for a turbine
meter with 95 % confidence while an instrument resolution uncertainty defines with 100 % confidence the
range of values that will be represented by that number rather than the next higher or lower. The equations for
obtaining the standard uncertainty for various common distributions are given in 7.3 to 7.8.
7.2 Calculation procedure
Type B evaluations of uncertainty require a knowledge of the probability distribution associated with the
uncertainty. The most common probability distributions are presented in 7.3 to 7.8; the shapes of the
distributions are shown in Annex B.
7.3 Rectangular probability distribution
Typical examples of rectangular probability distributions include
 maximum instrument drift between calibrations,
 error due to limited resolution of an instrument’s display,
 manufacturers' tolerance limits.
The standard uncertainty of a measured value, x , is calculated from Equation (9):
i
a
i
ux = (9)
()
i
where the range of measured values lies between x − a and x + a . The derivation of Equation (9) is given by
i i i i
[2]
Dietrich .
10 © ISO 2005 – All rights reserved

7.4 Normal probability distribution
Typical examples include calibration certificates quoting a confidence level or coverage factor with the
expanded uncertainty. Here, the standard uncertainty is calculated from Equation (10):
U
ux = (10)
()
i
k
where
U is the expanded uncertainty;
k is the quoted coverage factor; see Annex C.
Where a coverage factor has been applied to a quoted expanded uncertainty, care should be exercised to
ensure that the appropriate value of k is used to recover the underlying standard uncertainty. However, if the
coverage factor is not given and the 95 % confidence level is quoted, then k should be assumed to be 2.
7.5 Triangular probability distribution
Some uncertainties are given simply as maximum bounds within which all values of the quantity are assumed
to lie. There is often reason to believe that values close to the bounds are less likely than those near the
centre of the bounds, in which case the assumption of rectangular distribution could be too pessimistic. In this
case, the triangular distribution, as given by Equation (11), may be assumed as a prudent compromise
between the assumptions of a normal and a rectangular distribution.
a
i
ux() = (11)
i
7.6 Bimodal probability distribution
When the error is always at the extreme value, then a bimodal probability distribution is applicable and the
standard uncertainty is given by Equation (12):
ux =a (12)
( )
ii
Examples of this type of distribution are rare in flow measurement.
7.7 Assigning a probability distribution
When the source of the uncertainty information is well defined, such as a calibration certificate or a
manufacturer’s tolerance, the choice of probability distribution will be clear-cut. However, when the information
is less well defined, for example when assessing the impact of a difference between the conditions of
calibration and use, the choice of a distribution becomes a matter of the professional judgement of the
instrument engineer.
7.8 Asymmetric probability distributions
The above cases are for symmetrical distributions, however, it is sometimes the case that the upper and lower
bounds for an input quantity, X , are not symmetrical with respect to the best estimate, x . In the absence of
i i
information on the distribution, GUM recommends the assumption of a rectangular distribution with a full range
equal to the range from the upper to the lower bound. The standard uncertainty is then given by Equation (13):
′
ab+
ii
ux = (13)
()
i
where (x − a ) < X < (x + b′ ).
i i i i i
A more conservative approach would be to take a rectangular distribution based on the larger of two
asymmetric bounds.
a b′
i i
ux = the greater of or (14)
( )
i
3 3
If the asymmetric element of uncertainty represents a very significant proportion of the overall uncertainty, it
would be more appropriate to consider an alternative approach to the analysis such as a Monte Carlo
analysis; see Annex K.
A common example of an asymmetric distribution is seen in the drift of instruments due to mechanical
changes, for example, increasing friction in the bearings of a turbine meter or erosion of the edge of an orifice
plate.
8 Sensitivity coefficients
8.1 General
Before considering methods of combining uncertainties, it is essential to appreciate that it is insufficient to
consider only the magnitudes of component uncertainties in input quantities, it is also necessary to consider
the effect each input quantity has on the final result. For example, an uncertainty of 50 µm in a diameter or
5 % in a thermal expansion coefficient is meaningless in terms of the flow through an orifice plate without
knowledge of how the diameter or thermal expansion impact the measurement of flow-rate. It is, therefore,
convenient to introduce the concept of the sensitivity of an output quantity to an input quantity, i.e. the
sensitivity coefficient, sometimes referred to as the influence coefficient.
The sensitivity coefficient of each input quantity is obtained in one of two ways:
 analytically; or
 numerically.
8.2 Analytical solution
When the functional relationship is specified as in Equation (1), the sensitivity coefficient is defined as the rate
of change of the output quantity, y, with respect to the input quantity, x , and the value is obtained by partial
i
differentiation in accordance with Equation (15):
∂y
c = (15)
i
∂x
i
However, when non-dimensional uncertainties (for example percentage uncertainty) are used, non-
dimensional sensitivity coefficients shall also be used in accordance with Equation (16):
∂y x
* i
c = (16)
i
∂xy
i
In certain special cases where, for example, a calibration experiment has made the functional relationship
*
between the input and output simple, the value of c or c can be unity. Example 1 in Annex G gives an
i i
example for a calibrated nozzle.
8.3 Numerical solution
Where no mathematical relationship is available, or the functional relationship is complex, it is easier to obtain
the sensitivity coefficients numerically, by calculating the effect of a small change in the input variable, x , on
i
the output value, y.
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First calculate y using x , and then recalculate using (x + ∆x ), where ∆x is a small increment in x . The result
i i i i i
of the recalculation can be expressed as y + ∆y, where ∆y is the increment in y caused by ∆x .
i
The sensitivity coefficients are then calculated in accordance with Equation (17):
∆y
c ≈ (17)
i
∆x
i
They are calculated in non-dimensional, or relative, form in accordance with Equation (18):
x
∆y
*
i
c ≈ (18)
i
∆xy
i
Table 1 shows how a typical spreadsheet could be set up to calculate a specific sensitivity coefficient for any
function where y = f(x , x , . , x ).
1 2 N
Table 1 — Spreadsheet set-up for calculating sensitivity coefficients
*
Sensitivity Increment
x x . x x y c c
1 2 i N
coefficient
x x x x y = f(x , x ,…., x ) = y
— — . — —
1 2 i N 1 2 N nom
yy− x
( )
1nom
−6
c ⋅
c ∆x≈10 ⋅x x + ∆x x x x y = f(x + ∆x , x ,…., x)
... 1
1 i 1 i i 2 i N i 1 i 2
y
∆x
nom
The analytical solution calculates the gradient of y with respect to x at the nominal value, x , whereas the
i i
numerical solution obtains the average gradient over the interval x to (x + ∆x ). The increment used (∆x )
i i i i
should therefore be as small as practical and certainly no larger than the uncertainty in the parameter x .
i
However, a complication can arise if the increment is so small as to result in changes in the calculated result,
y, that are comparable with the resolution of the calculator or computer spreadsheet. In these circumstances
the calculation of c can become unstable. The problem can be avoided by starting with a value of ∆x equal to
i i
the uncertainty in x and progressively reducing ∆x until the value of c agrees with the previous result within a
i i i
suitable tolerance. This iteration process can, of course, be automated with a computer spreadsheet.
9 Combination of uncertainties
Once the standard uncertainties of the input quantities and their associated sensitivity coefficients have been
determined from either Type A or Type B evaluations, the overall uncertainty of the output quantity can be
determined in accordance with Equation (19):
N

uy = cux (19)
() ∑ ( )
c ii

i=1
Where relative uncertainties have been used, relative sensitivity coefficients shall also be used, in accordance
with Equation (20):
N 2
***

uy = c u x (20)
() ∑ ( )
c ii

i=1
Equations (19) and (20) assume that the individual input quantities are uncorrelated; the treatment of
correlated uncertainties is discussed in C.6. Correlation arises where the same instrument is used to make
several measurements or where instruments are calibrated against the same reference.
In general, the choice of absolute or relative uncertainties is of little consequence. However, once the decision
has been made, care is needed to ensure that all uncertainties are expressed in the same terms.
Measurements with arbitrary zero points give rise to problems if uncertainties are expressed in relative terms.
For example, an uncertainty of 1 mm in a diameter of 500 mm gives a relative uncertainty of 0,2 % and, if
expressed in inches, the uncertainty becomes 0,039 4 in out of 19,69 in, leaving the relative uncertainty
unchanged. However, if the uncertainty in a temperature of 20 °C is 0,5 °C, the relative uncertainty is 2,5 %,
but by expressing the values in degrees Fahrenheit, the temperature becomes 68 °F and the uncertainty
becomes 0,9 °F, giving a relative uncertainty of 1,3 %. Relative uncertainties cannot be used in these
circumstances and absolute uncertainties should be used. A relative uncertainty can only be used when it is
based on a measurement that is used to calculate the end result.
10 Expression of results
10.1 Expanded uncertainty
In Equations (19) and (20), the overall result is obtained from a summation of the contributions of the standard
uncertainty of each input source to the uncertainty of the result. The resulting combined uncertainty is,
therefore, a standard uncertainty; by referring to Figure 1, it can be seen that, with an effective k factor of 1,
the bandwidth defined by a standard uncertainty will only have a confidence level of about 68 % associated
with it. There is, therefore, a 2:1 chance that the true value will lie within the band, or a 1 in 3 chance that it will
lie outside the band. Such odds are of little value in engineering terms and the normal requirement is to
provide an uncertainty statement with 90 % or 95 % confidence level; in some extreme cases, 99 % or higher
might be required. To obtain the desired confidence level, an expanded uncertainty, U, is used in accordance
with Equation (21):
Uk= u y (21)
( )
c
or, where relative uncertainties are being used, in accordance with Equation (22):
**
Uk=u ()y (22)
c
Key
X standard deviation
X coverage factor
Y percent of readings in bandwidth
Figure 1 — Coverage factors for different levels of confidence with the normal,
or Gaussian, distribution
It is recommended that for most applications a coverage factor, k = 2, be utilized to provide a confidence level
of approximately 95 %; the choice of coverage factor will depend on the requirement of the application. Values
of k for various levels of confidence are given in Table 2.
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Table 2 — Coverage factors for different levels of confidence with the normal, or Gaussian,
distribution
Confidence level, % 68,27 90,00 95,00 95,45 99,00 99,73
Coverage factor, k 1,000 1,645 1,960 2,000 2,576 3,000

If the random contribution to uncertainty is large compared with the other contributions and the number of
readings is small, the above method provides an optimistic coverage factor. In this case, the procedure
outlined in Annex C should be used to estimate the actual coverage factor. A criterion that can be used to
determine whether the procedure described in Annex C should be applied is as follows.
Generally, if an uncertainty evaluation involves only one Type A evaluation and that Type A standard
uncertainty is less than half the combined standard uncertainty, there is no need to use the method described
in Annex C to determine a value for the coverage factor, provided that the number of observations used in the
Type A evaluation is greater than 2.
The uncertainty associated with an expanded uncertainty can be denoted using subscripts.
EXAMPLE U or U = .
95 k 2
10.2 Uncertainty budget
In reports providing an uncertainty estimate, an uncertainty budget table should be presented, (or referenced)
providing at least the information set out in Table 3.
Table 3 — Uncertainty budget
Contribution
Standard Sensitivity
Divisor
to overall
Source of Input Probability
uncertainty coefficient
Symbol [[see Equations (9)
[[ uncertainty
uncertainty uncertainty distribution
to (14)]]
]]
u(x ) c
[c u(x )]
i i
i i
e.g.
u(x )
5 Normal 2 2,5 0,5 1,56
calibration
e.g. output
u(x )
1 Rectangular 0,58 2,0 1,35
resolution
…
u(x )
i
u(x )
N
Combined
a
u
uy = Σ
— — — ( ) ← = cu()x
c c []
∑ii
uncertainty
Expanded
a
= k u (y) a
U k — —
←
c ↵
uncertainty
a
The arrows in the last two rows of the table indicate that, whereas in the upper rows the calculation proceeds from left to right, in
these rows the calculation of the final expanded uncertainty proceeds from right to left.
Table 3 is presented here in absolute terms and each input and corresponding standard uncertainty will have
the units of the appropriate input parameter. The table may, equally validly, be presented in relative terms, in
which case all input and resulting standard uncertainties will be in percentages or parts per million. Where the
inputs are all standard uncertainties, the columns headed “Input uncertainty,” “Probability distribution” and
“Divisor” may be omitted.
If the computation of a combined uncertainty is in response to a requirement for a test result to have a
specified level of uncertainty and the analysis shows that level to be exceeded, the budget table can be of
particular value in identifying the largest sources of uncertainty as an indicator of the problem areas which
should be addressed.
After the expanded uncertainty has been calculated for a minimum confidence level of 95 %, the
measurement result should be stated as follows.
 “The result of the measurement is [value].”
 “The uncertainty of the result is [value] (expressed in absolute or relative terms as appropriate).”
 “The reported uncertainty is based on a standard uncertainty multiplied by a coverage factor k = 2,
providing a confidence level of approximately 95 %.”
In cases where the procedure of Annex C has been followed, the actual value of the coverage factor should
be substituted for k = 2. In cases where a confidence level greater than 95 % has been used, the appropriate k
factor and confidence level should be substituted.
In reporting the result of any uncertainty analysis, it is important to make a clear statement of whether the
reported uncertainty is that of a single value, of a mean of a specified number of values, or of a curve fit based
on a specified number of values.
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Annex A
(normative)
Step-by-step procedure for calculating uncertainty
A.1 Dimensional and non-dimensional uncertainty
Decide whether dimensional or non-dimensional uncertainty estimates (for example parts per million or
per cent) will be used to prevent any confusion. In making this decision, the guidance of Clause 9 concerning
...