ISO 17123-1:2014
(Main)Optics and optical instruments — Field procedures for testing geodetic and surveying instruments — Part 1: Theory
Optics and optical instruments — Field procedures for testing geodetic and surveying instruments — Part 1: Theory
ISO 17123-1:2014 gives guidance to provide general rules for evaluating and expressing uncertainty in measurement for use in the specifications of the test procedures of ISO 17123‑2, ISO 17123‑3, ISO 17123‑4, ISO 17123‑5, ISO 17123‑6, ISO 17123‑7 and ISO 17123‑8. ISO 17123-1:2014 is a simplified version based on ISO/IEC Guide 98‑3 and deals with the problems related to the specific field of geodetic test measurements.
Optique et instruments d'optique — Méthodes d'essai sur site pour les instruments géodésiques et d'observation — Partie 1: Théorie
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INTERNATIONAL ISO
STANDARD 17123-1
Third edition
2014-08-15
Optics and optical instruments —
Field procedures for testing geodetic
and surveying instruments —
Part 1:
Theory
Optique et instruments d’optique — Méthodes d’essai sur site pour les
instruments géodésiques et d’observation —
Partie 1: Théorie
Reference number
©
ISO 2014
© ISO 2014
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ii © ISO 2014 – All rights reserved
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
3.1 General metrological terms . 1
3.2 Terms specific to this part of ISO 17123 . 3
3.3 The term “uncertainty” . 5
3.4 Symbols . 6
4 Evaluating uncertainty of measurement . 8
4.1 General . 8
4.2 Type A evaluation of standard uncertainty . 9
4.3 Type B evaluation of standard uncertainty .18
4.4 Law of propagation of uncertainty and combined standard uncertainty .19
4.5 Expanded uncertainty .21
5 Reporting uncertainty .22
6 Summarized concept of uncertainty evaluation .22
7 Statistical tests .23
7.1 General .23
7.2 Question a): is the experimental standard deviation, s, smaller than or equal to a given
value σ? .23
7.3 Question b): Do two samples belong to the same population? .24
7.4 Question c) [respectively question d)]:Testing the significance of a parameter y .
k 24
Annex A (informative) Probability distributions .26
Annex B (normative) χ distribution, Fisher’s distribution and Student’s t-distribution .27
Annex C (informative) Examples .28
Bibliography .39
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.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
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. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers
to Trade (TBT) see the following URL: Foreword - Supplementary information
The committee responsible for this document is ISO/TC 172, Optics and photonics, Subcommittee SC 6,
Geodetic and surveying instruments.
This third edition cancels and replaces the second edition (ISO 17123-1:2010).
ISO 17123 consists of the following parts, under the general title Optics and optical instruments — Field
procedures for testing geodetic and surveying instruments:
— Part 1: Theory
— Part 2: Levels
— Part 3: Theodolites
— Part 4: Electro-optical distance meters (EDM measurements to reflectors)
— Part 5: Total stations
— Part 6: Rotating lasers
— Part 7: Optical plumbing instruments
— Part 8: GNSS field measurement systems in real-time kinematic (RTK)
iv © ISO 2014 – All rights reserved
Introduction
This part of ISO 17123 specifies field procedures for adoption when determining and evaluating the
uncertainty of measurement results obtained by geodetic instruments and their ancillary equipment,
when used in building and surveying measuring tasks. Primarily, these tests are intended to be field
verifications of suitability of a particular instrument for the immediate task. They are not proposed as
tests for acceptance or performance evaluations that are more comprehensive in nature.
The definition and concept of uncertainty as a quantitative attribute to the final result of measurement
was developed mainly in the last two decades, even though error analysis has already long been a part
of all measurement sciences. After several stages, the CIPM (Comité Internationale des Poids et Mesures)
referred the task of developing a detailed guide to ISO. Under the responsibility of the ISO Technical
Advisory Group on Metrology (TAG 4), and in conjunction with six worldwide metrology organizations,
a guidance document on the expression of measurement uncertainty was compiled with the objective
of providing rules for use within standardization, calibration, laboratory, accreditation and metrology
services. ISO/IEC Guide 98-3 was first published as an International Standard (ISO document) in 1995.
With the introduction of uncertainty in measurement in ISO 17123 (all parts), it is intended to finally
provide a uniform, quantitative expression of measurement uncertainty in geodetic metrology with the
aim of meeting the requirements of customers.
ISO 17123 (all parts) provides not only a means of evaluating the precision (experimental standard
deviation) of an instrument, but also a tool for defining an uncertainty budget, which allows for the
summation of all uncertainty components, whether they are random or systematic, to a representative
measure of accuracy, i.e. the combined standard uncertainty.
ISO 17123 (all parts) therefore provides, for defining for each instrument investigated by the procedures,
a proposal for additional, typical influence quantities, which can be expected during practical use. The
customer can estimate, for a specific application, the relevant standard uncertainty components in
order to derive and state the uncertainty of the measuring result.
INTERNATIONAL STANDARD ISO 17123-1:2014(E)
Optics and optical instruments — Field procedures for
testing geodetic and surveying instruments —
Part 1:
Theory
1 Scope
This part of ISO 17123 gives guidance to provide general rules for evaluating and expressing uncertainty
in measurement for use in the specifications of the test procedures of ISO 17123-2, ISO 17123-3,
ISO 17123-4, ISO 17123-5, ISO 17123-6, ISO 17123-7 and ISO 17123-8.
ISO 17123-2, ISO 17123-3, ISO 17123-4, ISO 17123-5, ISO 17123-6, ISO 17123-7 and ISO 17123-8 specify
only field test procedures for geodetic instruments without ensuring traceability in accordance with
ISO/IEC Guide 99. For the purpose of ensuring traceability, it is intended that the instrument be calibrated
in the testing laboratory in advance.
This part of ISO 17123 is a simplified version based on ISO/IEC Guide 98-3 and deals with the problems
related to the specific field of geodetic test measurements.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO/IEC Guide 99, International vocabulary of metrology — Basic and general concepts and associated
terms (VIM)
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO/IEC Guide 99 and the following
apply.
3.1 General metrological terms
3.1.1
(measurable) quantity
property of a phenomenon, body or substance, where the property has a magnitude that can be expressed
as a number and a reference
EXAMPLE 1 Quantities in a general sense: length, time, temperature.
EXAMPLE 2 Quantities in a particular sense: length of a rod.
3.1.2
value
value of a quantity
quantity value
number and reference together expressing the magnitude of a quantity
EXAMPLE Length of a rod: 3,24 m.
3.1.3
true value
true value of a quantity
true quantity value
value consistent with the definition of a given quantity
Note 1 to entry: This is a value that would be obtained by perfect measurement. However, this value is in principle
and in practice unknowable.
3.1.4
reference value
reference quantity value
quantity value used as a basis for comparison with values of quantities of the same kind
Note 1 to entry: A reference quantity value can be a true quantity value of the measurand, in which case it is
normally unknown. A reference quantity value with associated measurement uncertainty is usually provided by
a reference measurement procedure.
3.1.5
measurement
process of experimentally obtaining one or more quantity values that can reasonably be attributed to a
quantity
Note 1 to entry: Measurement implies comparison of quantities and includes counting of entities.
3.1.6
measurement principle
phenomenon serving as the basis of a measurement (scientific basis of measurement)
Note 1 to entry: The measurement principle can be a physical phenomenon like the Doppler effect applied for
length measurements.
3.1.7
measurement method
generic description of a logical organization of operations used in a measurement
Note 1 to entry: Methods of measurement can be qualified in various ways, such as “differential method” and
“direct measurement method”.
3.1.8
measurand
quantity intended to be measured
EXAMPLE Coordinate x determined by an electronic tacheometer.
3.1.9
indication
quantity value provided by a measuring instrument or measuring system
Note 1 to entry: An indication and a corresponding value of the quantity being measured are not necessarily
values of quantities of the same kind.
3.1.10
measurement result
result of measurement
set of quantity values attributed to a measurand together with any other available relevant information
Note 1 to entry: A measuring result can refer to
— the indication,
— the uncorrected result, or
2 © ISO 2014 – All rights reserved
— the corrected result.
A measurement result is generally expressed as a single measured quantity value and a measurement uncertainty.
3.1.11
measured quantity value
quantity value representing a measurement result
3.1.12
error
error of measurement
measurement error
measured quantity value minus a reference quantity value
3.1.13
random measurement error
random error
component of measurement error that in replicate measurements varies in an unpredictable manner
Note 1 to entry: Random measurement errors of a set of replicate measurements form a distribution that can be
summarized by its expectation, which is generally assumed to be zero, and its variance.
3.1.14
systematic error
systematic error of measurement
component of measurement error that in replicate measurements remains constant or varies in a
predictable manner
Note 1 to entry: Systematic error, and its causes, can be known or unknown. A correction can be applied to
compensate for a known systematic measurement error.
3.2 Terms specific to this part of ISO 17123
3.2.1
accuracy of measurement
closeness of agreement between a measured quantity value and the true value of the measurand
Note 1 to entry: “Accuracy” is a qualitative concept and cannot be expressed in a numerical value.
Note 2 to entry: “Accuracy” is inversely related to both systematic error and random error.
3.2.2
experimental standard deviation
estimate of the standard deviation of the relevant distribution of the measurements
Note 1 to entry: The experimental standard deviation is a measure of the uncertainty due to random effects.
Note 2 to entry: The exact value arising in these effects cannot be known. The value of the experimental standard
deviation is normally estimated by statistical methods.
3.2.3
precision
measurement precision
closeness of agreement between measured quantity values obtained by replicate measurements on the
same or similar objects under specified conditions
Note 1 to entry: Measurement precision is usually expressed by measures of imprecision, such as experimental
standard deviation under specified conditions of measurement.
3.2.4
repeatability condition
repeatability condition of measurement
condition of measurement, out of a set of conditions
Note 1 to entry: Conditions of measurement include
— the same measurement procedure,
— the same observer(s),
— the same measuring system,
— the same meteorological conditions,
— the same location, and
— replicate measurements on the same or similar objects over a short period of time.
3.2.5
repeatability
measurement repeatability
measurement precision under a set of repeatability conditions of measurement
3.2.6
reproducibility conditions of measurement
condition of measurement, out of a set of conditions
Note 1 to entry: Conditions of measurement include
— different locations,
— different observers,
— different measuring systems, and
— replicate measurements on the same or similar objects.
3.2.7
reproducibility
measurement reproducibility
measurement precision under reproducibility conditions of measurement
3.2.8
influence quantity
quantity, which in a direct measurement does not affect the quantity that is actually measured, but
affects the relation between the indication of a measuring system and the measurement result
EXAMPLE Temperature during the length measurement by an electronic tacheometer.
4 © ISO 2014 – All rights reserved
3.3 The term “uncertainty”
3.3.1
uncertainty
uncertainty of measurement
measurement uncertainty
non-negative parameter characterizing the dispersion of quantity values attributed to a measurand,
based on the information used
Note 1 to entry: Measurement uncertainty comprises, in general, many components. Some of these components
can be evaluated by a Type A evaluation of measurement uncertainty from the statistical distribution of the
quantity values from series of measurements and can be characterized by an experimental standard deviation.
The other components, which can be evaluated by a Type B evaluation of measurement uncertainty, can also
be characterized by an approximation to the corresponding standard deviations, evaluated from assumed
probability distributions based on experience or other information.
3.3.2
Type A evaluation
Type A evaluation of measurement uncertainty
evaluation of a component of measurement uncertainty (standard uncertainty) by a statistical analysis
of quantity values obtained by measurements under defined measurement conditions
Note 1 to entry: For information about statistical analysis, see 4.1 and ISO/IEC Guide 98-3.
3.3.3
Type B evaluation of measurement uncertainty
evaluation of a component of measurement uncertainty (standard uncertainty) determined by means
other than a Type A evaluation of measurement uncertainty
EXAMPLE The component of measurement uncertainty can be based on
— previous measurement data,
— experience with, or general knowledge of, the behaviour and property of relevant instruments or
materials,
— manufacturer’s specifications,
— data provided in calibration and other reports,
— uncertainties assigned to reference data taken from handbooks, and
— limits deduced through personal experiences.
Note 1 to entry: For more information see 4.3 and ISO/IEC Guide 98-3.
3.3.4
standard uncertainty
standard uncertainty of measurement
standard measurement uncertainty
measurement uncertainty expressed as a standard deviation
Note 1 to entry: Standard uncertainty can be estimated either by a Type A evaluation or by a Type B evaluation.
3.3.5
combined standard uncertainty
combined standard measurement uncertainty
standard (measurement) uncertainty, obtained by using the individual standard uncertainties (and
covariances as appropriate), associated with the input quantities in a measurement model
Note 1 to entry: The procedure for combining standard uncertainties is often called the “law of propagation of
uncertainties” and in common parlance the “root-sum-of-squares” (RSS) method.
3.3.6
coverage factor
numerical factor larger than one, used as a multiplier of the (combined) standard uncertainty in order
to obtain the expanded uncertainty
Note 1 to entry: The coverage factor, which is typically in the range of 2 to 3, is based on the coverage probability
or level of confidence required of the interval.
3.3.7
expanded uncertainty
expanded measurement uncertainty
half-width of a symmetric coverage interval, centred around the estimate of a quantity with a specific
coverage probability
Note 1 to entry: A fraction can be viewed as the coverage probability or level of confidence of the interval.
3.3.8
coverage interval
interval containing the set of true quantity values of a measurand with a stated probability, based on
the information available
Note 1 to entry: It is intended that a coverage interval not be termed “confidence interval” in order to avoid
confusion with the statistical concept. To associate an interval with a specific level of confidence requires explicit
or implicit assumptions regarding the probability distribution, characterized by the measurement result.
3.3.9
coverage probability
probability that the set of true quantity values of a measurand is contained within a specific coverage
interval
Note 1 to entry: The probability is sometimes termed “level of confidence” (see ISO/IEC Guide 98-3).
3.3.10
uncertainty budget
statement of a measurement uncertainty, of the components of that measurement uncertainty, and of
their calculation and combination
Note 1 to entry: It is intended that an uncertainty budget include the measurement model, estimates, measurement
uncertainties associated with the quantities in the measurement model, type of applied probability density
functions and type of evaluation of measurement uncertainty.
3.3.11
measurement model
mathematical relation among all quantities known to be involved in a measurement
3.4 Symbols
Table 1 — Symbols and definitions
a Half-width of a rectangular distribution of possible values of input quantity X :a = (a − a )/2
i + −
a Upper bound or upper limit of input quantity X
+ i
a Lower bound or lower limit of input quantity X
− i
A Design or Jacobian matrix (N × n)
∂f
c =
c
i
i
∂x
i
Partial derivates or sensitive coefficient: (i = 1, 2, ., N)
c Vector of sensitive coefficients c (i = 1, 2, ., N)
i
e Unit vector
6 © ISO 2014 – All rights reserved
Table 1 (continued)
Functional relationship between a measurand, Y , and the input quantity, X , and between output
k j
f
k
estimate, y , and input estimates, x
k j
T
f Vector with elements f (x ) (k = 1, 2, ., n)
k
Fisher’s F (or Fisher-Snedecor) distribution with degrees of freedom (v, v) and confidence level of
F (v, v)
1 −α/2
(1 − α) %
g Functional relationship between the estimate of input quantity, x , and the observables, l
j j i
Coverage factor used to calculate expanded uncertainty U = k × u ( y) of the output estimate y
c
k
from its combined uncertainty u ( y)
c
l Observables, random variables (i = 1, 2, ., m)
i
m Number of observations, l
i
M Number of input quantities, whose uncertainties can be estimated by a Type A evaluation
n Number of output quantities, measurands
N Number of input quantities
N − M Number of input quantities, whose uncertainties can be estimated by a Type B evaluation
N Normal equation matrix (n × n)
p Weight of the input estimates x ( j = 1, 2, ., N)
j j
P Weight matrix of p (N × N)
j
Q Cofactor of the output estimate, y
ykyk k
Q Cofactor matrix of the output estimates, y (n × n)
y k
r Residual of input estimates, x ( j = 1, 2, ., N)
j j
r Vector of residuals, r
j
r(x , x ) Correlation coefficient between the input estimates, x and x
i j i j
s Experimental standard deviation (general notation)
s( y ) Experimental standard deviation of the output estimate y
k k
t (v) Student’s t-distribution with the degree of freedom, v, and a confidence level of (1 − α) %
α
u Standard uncertainty (general notation)
u( y ) Standard uncertainty of the output estimate y
k k
u(x ) Standard uncertainty of the input estimate x
j j
u ( y ) Combined standard uncertainty of the output estimate y
c k k
U Expanded uncertainty (general notation)
x
Estimate of input quantity, input estimate ( j = 1, 2, ., N)
j
x Vector of the estimates of input quantities x
j
X jth input quantity on which the measurand Y depends
j k
X Vector of input quantities X
j
y Estimate of measurand Y , output estimate; (k = 1, 2, ., n)
k k
y Vector of output estimates of measurands y
k
Y kth measurand (k = 1, 2, ., n)
k
Y Vector of measurands Y
k
α Probability of error, as a percentage
(1 − α) Confidence level
v Degrees of freedom
Table 1 (continued)
σ Standard deviation of the normal distribution
χν()
Chi-squared distribution with the degree of freedom, v, and a confidence level of (1 − α) %
1−α
4 Evaluating uncertainty of measurement
4.1 General
The general concept is documented in ISO/IEC Guide 98-3, which represents the international view of
how to express uncertainty in measurement. It is just a rigorous application of the variance-covariance
law, which is very common in geodetic and surveying data analysis. However, the philosophy behind it
has been extended in order to consider not only random effects in measurements, but also systematic
errors in the quantification of an overall measurement uncertainty.
In principle, the result of a measurement is only an approximation or estimate of the value of the specific
quantity subject to a measurement; that is the measurand. Thus, the result is complete only when
accompanied by a quantitative statement of its quality, the uncertainty.
The uncertainty of the measurement result generally consists of several components, which may be
grouped into two categories according to the method used to estimate their numerical values:
a) those which are evaluated by statistical methods;
b) those which are evaluated by other means.
Basic to this approach is that each uncertainty component, which contributes to the uncertainty
of a measuring result by an estimated standard deviation, is termed standard uncertainty with the
suggested symbol u.
The uncertainty component in category A is represented by a statistically estimated experimental
standard deviation, s , and the associated number of degrees of freedom, v . For such a component, the
i i
standard uncertainty u = s . The evaluation of uncertainty components by the statistical analysis of
i i
observations is termed a Type A evaluation of measurement uncertainty (see 4.2).
In a similar manner, an uncertainty component in category B is represented by a quantity, u , which may
j
be considered an approximation of the corresponding standard deviation and which may be attributed
an assumed probability distribution based on all available information. Since the quantity u is treated as
j
a standard deviation, the standard uncertainty of category B is simply u . The evaluation of uncertainty
j
by means other than statistical analysis of series of observations is termed a Type B evaluation of
measurement uncertainty (see 4.3).
Correlation between components of either category are characterized by estimated covariances or
estimated correlation coefficients.
8 © ISO 2014 – All rights reserved
Input: Output:
vector x , U
vector y and u
x y
input quantity x and its output quantity y and its
j
k
uncertainty u(x )
standard uncertainty u (y )
j
k
Type A:
expanded uncertainty
observations, measurement
U(y )
k
data analysed by statistical
Model
methods
x, U
x
of evaluation: y, u
y
Final result:
x , U
A x(A)
T
y = f (x ) y ± U(y )
k k
Type B:
previous, external
Can be used as input
measurement data analysed
quantity in further
by other means
applications
x , U
B x(B)
Figure 1 — Universal mathematical model and uncertainty evaluation
4.2 Type A evaluation of standard uncertainty
4.2.1 General mathematical model
In most cases, a measurand, Y, is not measured directly, but is determined by N other quantities
X , X , ., X through the functional relationship given as Formula (1):
1 2 N
Y = f (X , X , ., X )
1 2 N
(1)
An estimate of the measurand, Y, the output estimate, y, is obtained from Formula (1) by using the input
estimates, x , x , ., x , thus the output estimate, y, which is the result of measurements, is given by
1 2 N
Formula (2):
y = f (x , x , ., x )
1 2 N
(2)
In most cases, the measurement result (output estimate, y) is obtained by this functional relationship.
But in some cases, especially in geodetic and surveying applications, the measurement result is composed
of several output estimates, y , y , ., y which are obtained by multiple, e.g. N, measurements (input
1 2 n
estimates).
From this follows the general model function (see Figure 1) given as Formula (3):
T
y = f(x )
(3)
Assuming that
x is a vector (N × 1) of input quantities x ( j = 1, 2, ., N);
j
y is a vector (n × 1) of output quantities y (k = 1, 2, ., n);
k
T
f is a vector (n × 1) with the elements f (x ) (k = 1, 2, ., n);
k
f can be understood as a suitable algorithm to determine the output quantities y (see Annex C).
4.2.2 General law of Type A uncertainty propagation
Often in geodetic measuring processes, the input quantity, x , is a function of several observables, the
j
random variables:
T
l = (l , l , l , ., l )
1 2 3 m
(4)
The reason for this can be, for example, internal measuring processes of the instrument, correction
parameters obtained by calibration or even multiple measurements of the same observable.
The associated uncertainty matrix may be given by Formula (5):
u 0
U =
l
0 u
m
(5)
Assuming the general function
x = g (l) ( j = 1, 2, ., N)
j j
(6)
the linearized model
T
g + gl
0 j
x = (7)
j
with
∂g ∂g ∂g
jj j
T
g =…(,gg ,,g )(= ,, …,)
j jj12 jm
∂l ∂l ∂l
12 m
(8)
yields the standard uncertainty of the input quantity, x , as given by Formula (9):
j
T
ux()= gU g
jj lj
(9)
Under the assumption that the observables are random,
u(x ) = s(x )
j j
(10)
which is called the experimental standard deviation of x .
j
Of course, u can also be introduced in Formula (5) covariances such that U becomes a fully occupied
jk l
matrix.
The numerical example in C.1 illustrates this approach of a Type A evaluation for calculating the standard
uncertainty.
10 © ISO 2014 – All rights reserved
If there are N functions of X, all dependent on the observables l, they are treated according to Formula (7):
xg=+Gl
(11)
With the Jacobian matrix:
gg
11 lm
G =
gg
NN1 m
(12)
Finally, Formula (9) can be written in the general form of the known law of error propagation:
ux() ux(,xu)( xx,)
11 21 N
T ux(,xu)(xu)( xx,)
21 22 N
UG==UG
xl
ux(,xu)(xx,) ux()
MM12 N
(13)
From the diagonal elements, the standard uncertainties can be derived as given by Formula (14):
T
u = ux(),(ux ),.,ux()
[]
xN12
(14)
Respectively, the empirical standard deviations are
T
s = sx(),(sx ),.,ss()
xN 12
(15)
Following the flowchart of Figure 1 in which the output quantities are obtained from the input estimates
x by a linear transformation, then
T
y = f(x ) = h +H(x)
(16)
Taking Formula (11) into account,
yh=+Hg(+Gl)=h +HGl
00 0
(17)
and, according to Formula (13), the uncertainty matrix becomes:
T T T
U = HU H = HGU G H (18)
y x l
The diagonal elements of the matrix U incorporate the standard uncertainty vector given as Formula (19):
y
T
u = uy(),(uy ), ., uy()
yN12
(19)
of the output estimates y , y , ., y .
1 2 N
Again, if the input quantities vary randomly, the standard uncertainties in Formula (19) match the
empirical standard deviations of the output estimate y.
u = s or u(y ) = s(y ) (k = 1, 2, ., n)
y y k k
(20)
The nesting in Formula (18) can be arbitrarily enhanced for further applications (see Figure 1), e.g.
z = M(y).
The numerical example in C.2 illustrates this approach of a Type A evaluation for calculating the standard
uncertainty.
4.2.3 Least squares approach
Often, more model equations according to Formula (3) are given than output quantities, y , have to be
k
determined. In such a case (N > n), it is suitable to solve the equation system by the known method of
a least-squares adjustment. For this, it is necessary to restate the model function of Formula (3) in a
system of (nonlinear) observation equations:
x + r = F(y)
(21)
or in a linearized notation (neglecting higher-order terms):
∂F
xr+=Fy()+ ()yy−
∂y
(22)
where
x is the vector (N × 1) of the observations or measurable input quantities;
r is the vector (N × 1) of the residuals;
y is the vector (n × 1) of unknowns, output estimates;
y is the vector (n × 1) of the approximate values of y.
Substituting in Formula (22):
yy−= y
,
12 © ISO 2014 – All rights reserved
xF−=()yl
(23)
and
∂F ∂F
1 1
∂y ∂y
1 n
∂F
=
=A
∂y
∂F ∂F
NN
∂y ∂y
1 n
(24)
yields Formula (25):
rA=−yl
(25)
Often, it is necessary to introduce a stochastic model by the weight matrix of the measurable input
quantities:
p 0
s
0
P =
p =
j
2
0 p s
N j
with (26)
The weights, p , can be determined under consideration of Formula (13), respectively Formula (15).
j
Following the Gauß-Markov model, the solution vector is:
TT−−11
y ==()APAA Pl Nn
(27)
With the results of Formula (27), the residuals can be calculated from Formula (25). Thus, the a posteriori
variance factor can be derived from Formula (28):
T
rPr
s =
v
(28)
where
v = N−n (degree of freedom).
From this, the experimental standard deviation of the output estimates, y, can be calculated by the
known relationships
sQ()ys=
ky0 y
kk
k = 1, 2, ., n (29)
with
−1
Q = diagQ and Q = N (30)
ykyk y y
Finally, the standard uncertainties, Type A evaluation, of all output estimates y can be stated as
k
Formula (31):
u = s or u(y ) = s(y ) k = 1, 2, ., n (31)
y y k k
But, the adjusted input values can also be quoted by Formula (32):
xl =+r
(32)
x
and the estimated variance covariance matrix of by Formula (33):
21− T
SA=s NA
x 0
(33)
Finally, from its diagonal elements, the experimental standard deviation is given by Formula (34):
sS(,ss ,.,sd) iag
xx xx x
12 N
(34)
Thus, the standard uncertainty of the adjusted input estimates, , yields Formula (35):
ux()=sx()
us=
jj
xx
or ( j = 1, 2, ., N)
(35)
The numerical example in C.3 illustrates this approach of a Type A evaluation for calculating the standard
uncertainty.
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==
4.2.4 Special cases
ux(), x
i i
4.2.4.1 Calculation of the standard uncertainty, of the arithmetic mean or average for
the ith series of measurements
Often, the input quantity X is estimated from j = 1, 2, ., n independent repeated observations x .
i i,j
Following Formula (27), the best available estimate is Formula (36):
TT−1
x =()ePee Px
i i
(36)
With its experimental standard deviation, given as Formula (37):
ss
sx()==
i
T
p
ePe
∑ ij
(37)
For uncorrelated equal accurate input estimates, x , the average yields Formula (38):
i,j
n
x = x
ii∑ j
n
j=1
(38)
and the experimental standard deviation yields Formula (39):
T
s rr
sx()==
i
re=−xx
nn()−1
n
ii
, with (39)
Then, the standard uncertainty is given by Formula (40):
ux()=sx()
ii
(40)
uy(), y
i i
4.2.4.2 Calculation of the standard uncertainty, of the arithmetic mean or average for
the ith series of double measurements
yi(.= 1, 2, ., n)
i
Often the output quantities, Y , are estimated by the mean of pairs of measurements
i
(two measurements with the same measurand):
T
l =(,ll ,.,l )
jj12jjn
(l , l ) with and j = 1, 2.
1 2
(41)
The vector of the output estimates reads as Formula (42):
yl=+()l
(42)
The following evaluation implies that the measurement procedure eliminates systematic errors; this
means that, for the expectation of the difference vector, it follows that:
Ed()=−E0()ll =
(43)
Furthermore, it is assumed that the same standard uncertainty u , with j = 1, 2, can be attributed to all
l,j
pairs of measurements. Therefore
PPP
ll
(44)
and
T
dPd
s =
2n
where
d = (l − l )
2 1
(45)
If the same weight can be allocated to all observations, the experimental standard deviation reads as
given in Formulae (46), (47) and (48):
for the measurements l :
j,i
T
dd
s =
l
2n
(46)
for the differences d :
i
T
dd
s =
d
n
(47)
and
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==
y
i
for the output estimates :
T
dd
sy()=
i
4n
(48)
To check if the assumption in Formula (43) is fulfilled, the following rule should be applied.
If Formula (49)
TT2
()ed
(49)
E0()d =
is true, it can be expected that . In this case, the standard uncertainty is given as Formula (50):
uy()=sy()
ii
(50)
4.2.4.3 Calculation of the overall standard uncertainty, u, for m series of measurements
The experimental standard deviation obtained for each of the m series of measurements is considered
to be a separate estimate of the overall experimental standard deviation of the measurements. It is
assumed that each of these estimates is of the same order of reliability, v = v = v = . = v . Formulae (51)
i 1 2 m
and (52) indicate how the individual experimental standard deviations are combined to give one overall
experimental standard deviation which takes equal account of the experimental standard deviations
calculated for each series of measurements.
m
22 2 22
ss== ss++.+ s
∑ ∑ im1 2
il=
(51)
where
m is the number of series of measurements;
s is the experimental standard deviation of a single measured value within the ith series of
i
measurements;
is the sum of squares of all standard deviations, s , of the m series of measurements.
s i
å
The overall experimental standard deviation, s, of m series of measurements yields Formula (52):
s
∑
s=
m
(52)
The number of degrees of freedom of all m series of measurements is obtained by Formula (53):
m
vv==mv×
ii
∑
i=1
(53)
Finally, the overall standard uncertainty can be written as Formula (54):
u = s (54)
Numerical examples in C.4 and C.5 illustrate these approaches of a Type A evaluation for calculating
standard uncertainties.
4.3 Type B evaluation of standard uncertainty
4.3.1 General
Often, not all uncertainties of the N input quantities can be estimated by a Type A evaluation; this number
of uncertainties, obtained by the Type A evaluation, is therefore assumed, M, so that the uncertainties of
N − M input quantities have to be determined by other means, namely by a Type B evaluation.
For an estimate x , M < j ⩽ N of an input quantity, which has not been obtained from repeated observations
j
or was derived from small samples, the evaluation of the standard uncertainty u(x ) is usually based on
j
scientific judgment using all available information, which may include
— previous measurement data,
— experience with, or general knowledge of, the behaviour and properties of relevant materials and
instruments,
— manufacturer’s specifications,
— data provided in calibration reports,
— uncertainties assigned to reference data taken from handbooks.
Examples of such a Type B evaluation, which can be very helpful for practical use, are given in the
following subclauses.
4.3.2 Quantity in question modelled by a normal distribution (see Annex A)
— Lower and upper limits are estimated by a and a .
− +
— Estimated value of the quantity: (a + a )/2.
+ −
— 50 % probability that the value lies in the interval a to a .
− +
Then, the standard uncertainty yields Formula (55):
ua»14, 8
j
(55)
where a = (a − a )/2
+ −
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4.3.3 Quantity in question modelled by a normal distribution (see Annex A)
— Lower and upper limits are estimated by a and a .
− +
— Estimated value of the quantity: (a + a )/2.
+ −
— 67 % probability that the value lies in the interval a to a .
− +
Then, the standard uncertainty yields Formula (56):
ua»
j
(56)
where a = (a − a )/2
+ −
4.3.4 Quantity in question modelled by a uniform or rectangular probability distribution (see
Annex A)
— Lower and upper limits are estimated by a and a .
− +
— Estimated value of the quantity: (a + a )/2.
+ −
— 100 % probability that the values lies in the interval a to a .
− +
Then, the standard uncertainty yields Formula (57):
a
u =≈05, 8a
j
(57)
where a = (a − a )/2
+ −
4.3.5 Quantity in question modelled by a triangular probability distribution (see Annex A)
— Lower and upper limits are estimated by a and a .
− +
— Estimated value of the quantity: (a + a )/2.
+ −
— 100 % probability that the values lies in the interval a to a .
− +
Then, the standard uncertainty yields Formula (58):
a
u =≈04, 1a
j
(58)
where a = (a − a )/2
+ −
The numerical Examples in C.6 illustrate these approaches of a Type B evaluation for calculating
standard uncertainties.
4.4 Law of propagation of uncertainty and combined standard uncertainty
The combined standard uncertainty, u (y ), of a measurement result y is taken to represent the estimated
c k k
standard deviation of the final result. It is obtained by combining the individual standard uncertainties,
u(x ), and, if available, the covariances u(x , x ) of the input estimates x , x , ., x , x , x , ., x , whether
i i j 1 2 M M+1 M+2 N
arising from a Type A evaluation or a Type B evaluation. This method is called the law of propagation
of uncertainty or in the parlance of geodetic metrology the root-sum-squares method of combining
standard deviations.
It is assumed that for the input estimates
T
(,xx ,.,x )= x
12 MA
(59)
the standard uncertainties are from a Type A evaluation and given by Formula (60):
ux() 00
0 ux()
U =
xA()
...
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