0
References
ISBN 978-86-900284-2-9
10 11 12 13 14 15 16 17 18 19 20 21
[...]
9
9 Measurement uncertainty
8
8 Tests and functions fitting
7
7 Variances
In front of you is the Companion with
concisely described subjects that are
indispensable in all measurements.
The Companion consists of about
minimal and sufficient set of concepts
and methods required to compute a
measurement result and its
measurement uncertainty. The text is
in the form that enables direct
applications in computations, manual
or spreadsheet, and in writing
software.
6
6 Outlines of the statistics
5
5 Measurement standards and
calibrations
The essence of measurement is the
same in all areas. The purpose of
this publication is to be a companion
to people dealing with
measurements, that is experiments or
observations, from production to
research, from physics and chemistry
to biology and medicine. The
publication is also intended for
students.
4
4 Measurement results, accuracy
and errors
Goran Kostić
3
3 Devices for measurement
2
2 Measurements
From a shop floor to the fundamental
sciences, statistical analysis of
results of repeated measurements is
required. The purpose of the
analysis is to determine the value of
a measured quantity, as well as the
accuracy of the determined value. It
is requested that accuracy is
described by the standard
measurement uncertainty to enable
traceability and ensure computing of
the confidence level and the
confidence interval.
1
1 Quantities and units of
measurement
From the Preface
Goran Kostić
Chapters
THE NEW SI E-BOOK ON CONCEPTS AND
COMPUTATIONS IN MEASUREMENTS
Goran Kostić
PAGES FROM:
Metrology
Companion
To buy this book, click www.symmetry.rs.
Leskovac, 2019
CONTENTS
CONTENTS 5
Abbreviations 12
Letter symbols 13
Greek alphabet 14
PREFACE 15
INTRODUCTION 19
PART I FUNDAMENTALS OF MEASUREMENTS 21
1 QUANTITIES AND UNITS OF MEASUREMENT 23
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
1.21
Measurable quantity 23
Quantities of the same kind 23
System of quantities 24
Base quantity 25
Derived quantity 26
Ordinal quantity 26
Dimension of a quantity 27
Unit of measurement 28
Base unit of measurement 29
Derived unit of measurement 29
Coherent unit of measurement 29
Value of a quantity 30
Numerical value of a quantity 31
Conventional value of a quantity 31
Reference value 31
Constant 32
Quantity equation 34
Numerical value equation 34
Unit equation 35
Conversion factor between units 35
System of units 36
1.21.1 SI base units 37
1.22 Off-system unit of measurement 41
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5
1.23 Decimal units of measurement 41
1.24 Binary unit of measurement 42
1.25 Expressing the value of a quantity in the SI 43
2 MEASUREMENTS 51
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
Measurement 51
Metrology 51
Measurand 52
Principle of measurement 52
Method of measurement 52
Measurement procedure 53
Robustness of procedure 53
Standard operating procedure 53
Standard measurement procedure 54
Qualification and validation 55
Validation of measurement procedure 55
Accreditation and certificate 57
About design of experiment 58
Mathematical model 59
2.14.1 Examples 59
2.15 Simulation 60
3 DEVICES FOR MEASUREMENT 61
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
6
Device for measurement 61
Indicating measuring instrument 62
Material measure 62
Measuring chain 63
Measuring system 63
Nominal value 64
Indication of a device for measurement 64
Indication interval 65
Measuring interval 65
Response function 65
Measuring transducer 66
Sensor 66
Detector 66
Measurement signal 66
Resolution 67
Sensitivity 67
Response time 67
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3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
3.30
3.31
3.32
3.33
3.34
3.35
3.36
3.37
Limit of detection 68
Limit of quantification 68
Dynamic interval 68
Dead interval 69
Hysteresis 69
Influence quantity 69
Blank indication 69
Stability 70
Selectivity 70
Transparency 70
Operating conditions 71
Reference operating conditions 71
Limiting conditions 71
Gauging of a device for measurement 72
Error of indication of a device for measurement 72
Datum error of a device for measurement 72
Zero error of a device for measurement 72
Bias of a device for measurement 73
Intrinsic error of a device for measurement 73
Accuracy class 73
3.37.1 Examples 73
3.38 Type approval of a device for measurement 74
3.39 Verification of a device for measurement 74
4 MEASUREMENT RESULTS, ACCURACY AND ERRORS 75
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
Measurement result 75
Accuracy 75
Absolute error 76
Relative error 77
Random error 78
Systematic error 78
Combined error 79
4.7.1 Example 79
Limits of error 80
Outlier 80
Correction 81
Precision of measurement results 82
Repeatability of measurement results 82
Reproducibility of measurement results 82
Rounding of number 83
Truncation of digits 84
Significant digits 84
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5 MEASUREMENT STANDARDS AND CALIBRATIONS 85
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
Measurement standard 85
Reference material 86
Primary measurement standard 88
Secondary measurement standard 88
International measurement standard 88
National measurement standard 89
Transfer measurement standard 89
Reference measurement standard 89
Working measurement standard 89
Relativistic effects on the values of standards 90
Calibration 91
Calibration report 92
PART II STATISTICAL ANALYSIS OF MEASUREMENT RESULTS 93
6 OUTLINES OF THE STATISTICS 95
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
6.23
8
About statistical analysis 95
Statistical analysis of results of repeated measurements 97
Probability 98
Random variable 99
Dependent variable and independent variable 100
Correlated variables 100
Sample 101
Maximum likelihood value 102
Best estimate 102
Biased evaluation 103
Distribution of a variable 104
Normal distribution 106
Means 108
Mode 110
Median 111
Arithmetic mean 112
Experimental arithmetic mean 114
Indicators of dispersion of values 115
Natural Deviation 116
Experimental standard deviation 118
Deviation of the arithmetic mean 121
Deviation of the standard deviation 122
Standard deviation from a median, range and sample size 124
Kostić • 2019 • Metrology Companion
6.24 Degrees of freedom 125
6.24.1 Examples 126
6.25 Effective degrees of freedom 127
6.25.1 Example 127
6.26 Degrees of freedom from the deviation of the standard
deviation of the arithmetic mean 128
6.26.1 Example 128
6.27 Confidence level and confidence interval 129
6.27.1 Example 131
6.28 Standard deviation and its degrees of freedom from the level
and interval of confidence and sample size 131
6.29 T- distribution 133
6.30 Uniform distribution 136
6.31 Triangular distribution 138
6.32 U-distribution 140
6.33 Lognormal distribution 142
6.34 Chi-square distribution 144
6.35 F-distribution 145
6.36 Indicator of asymmetry of a distribution 146
6.37 Indicator of peakedness of a distribution 147
6.38 Histogram 148
6.38.1 Example 150
6.38.2 Example 151
6.39 Convolution 152
6.40 Filtering 154
6.41 Operations on distributions and central limit theorem 155
6.42 Monte Carlo simulation 158
6.42.1 Example of estimate of a value and its statistical
parameters 162
6.42.2 Example of estimate of a length of gauge block and
its statistical parameters 166
The contents continues on the next page.
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7 VARIANCES 171
7.1
7.2
7.3
7.4
7.5
7.6
Natural variance 171
Experimental standard variance 173
Evaluations of grouped results 175
Experimental standard covariance 176
7.4.1 Example of compution of the standard covariance 179
7.4.2 Example of compution of the standard covariance of
the means 180
Standard correlation coefficient 181
Combined variance 183
7.6.1 Example of the combined variance of an evaluation of
resistance of electrical circuit element and the
maximum possible variance of this variance 191
7.6.2 Example of the combined variance of an evaluation of
length of a gauge block and the maximum possible
variance of this variance 193
7.6.3 Example of the combined variance of pH value 198
7.6.4 Example of the combined variance of resistance of
resistors in a series 199
8 TESTS AND FUNCTIONS FITTING 203
8.1
Detection of outliers 203
8.1.1 Example 205
8.2 Fitting 206
8.3 Method of the least sum of squares 206
8.3.1 Example 209
8.4 P-value 212
8.5 Goodness-of-fit test 213
8.6 Estimating the type and parameters of a distribution 215
8.7 Pearson chi-square test for the goodness of fit of
distributions 217
8.7.1 Example 221
8.8 Shapiro-Wilk test for normality of a population 223
8.8.1 Example 225
8.9 White noise test 230
8.10 Analysis of variance (ANOVA) 231
8.10.1 Example 233
8.11 Transformation of a variable 236
8.11.1 Example 237
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9 MEASUREMENT UNCERTAINTY 239
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
9.12
9.13
9.14
Measurement result and its uncertainty 239
Type A standard measurement uncertainty 241
Type B standard measurement uncertainty 242
9.3.1 Example 243
9.3.2 Example 245
Combined standard measurement uncertainty 247
Expanded standard measurement uncertainty 248
Relative standard measurement uncertainties 249
Evaluations from non-normally distributed results 249
Expressing the standard measurement uncertainty 251
Expressing the standard measurement uncertainty when
corrections are not applied 252
General guidance on reporting a measurement result 253
Procedure for estimating the measurement result and
associated data 254
Example of calibration of a thermometer 256
9.12.1 The measurement problem 256
9.12.2 Least squares fitting 256
9.12.3 Standard uncertainty of the predicted correction
addend 259
9.12.4 Elimination of the correlation between the slope and
the intercept point 260
Resistance, reactance and impedance, computed from the
same component measurement results 261
9.13.1 The measurement problem 262
9.13.2 Mathematical model of the measurement 262
9.13.3 Final results and their standard uncertainties,
standard covariances and standard correlation
coefficients - approach a 263
9.13.4 Final results and their standard uncertainties,
standard covariances and standard correlation
coefficients - approach b 263
Metrological traceability 268
9.14.1 Example 269
REFERENCES 271
Kostić • 2019 • Metrology Companion
11
Preface
From a shop floor to the fundamental sciences, statistical analysis of
results of repeated measurements is required. The purpose of the
analysis is to determine the value of a measured quantity, as well as the
accuracy of the determined value. It is requested that accuracy is
described by the standard measurement uncertainty to enable traceability
and ensure computing of the confidence level and the confidence interval.
The essence of measurement is the same in all areas. The purpose of
this publication is to be a companion to people dealing with
measurements, that is experiments or observations, from production to
research, from physics and chemistry to biology and medicine. The
publication is also intended for students.
In front of you is the Companion with concisely described subjects that
are indispensable in all measurements. The Companion consists of about
minimal and sufficient set of concepts and methods required to compute
a measurement result and its measurement uncertainty. The text is in the
form that enables direct applications in computations, manual or
spreadsheet, and in writing software.
The manner of presentation is encyclopedic. However, the order of
presentation allows reading the text as a book or textbook. It seems that
the form of encyclopedias is suitable for expert writings because it
facilitates understanding of relations among concepts that make the
essence of every exact knowledge.
Most of this text is a terminologically consistent and compliant
synthesis of international recommendations and regulations, articles and
encyclopedic texts. References are given in brackets.
Special attention was paid to the definitions of terms and their
consistent use. Most of the terms are according to the second and third
editions of the International vocabulary of metrology (see [VIM 2] and
[VIM 3]) which was issued by the Joint Committee for Guides in Metrology
(JCGM). These glossaries were not applied when it was thought that they
do not correspond to what they should express, or that there are far better
terms. Any such exceptions from the glossaries are noted with the
references.
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15
The second part of the Companion, regarding statistical analysis of
measurement results, is the basis for computation of the standard
measurement uncertainty. It is intended for anyone involved in the
analysis of the measurement results.
The chapter on the standard measurement uncertainty is according to
the instructions published by the JCGM in the Guide to the expression of
uncertainty in measurement (see [GUM]).
The level and interval of confidence of the measurement result can be
considered as a purpose of determination of all measurement errors. An
accurate evaluation of the level and interval of confidence is enabled by
the standard measurement uncertainty of a traceable value. In this
Companion, the traceability is described in the short, but comprehensive
text given at the end of the section concerning measurement uncertainty.
At the end of the Preface, attention is drawn to the following viewpoint
expressed in [GUM]. “The evaluation of uncertainty is neither a routine
task nor a purely mathematical one; it depends on detailed knowledge of
the nature of the measurand and of the measurement. The quality and
utility of the uncertainty quoted for the measurement result therefore
ultimately depend on the understanding, critical analysis, and integrity of
those who contribute to the assignment of its value.”
All equations in the Companion are for the coherent units.
The following marking is used in the Companion.
● Bold fonts are used for terms when defined.
● The abbreviated terms that may be used are provided next to the
terms, in parentheses, when this will not lead to confusion.
● Underlined text is linked to detailed explanations of a concept which
underlined text indicates. A “see...” is also linked to a reference.
● Equations, tables and figures are marked with the number of the
chapter, then by a point, and subsequently by their order number in the
chapter.
● A blue text needs special attention.
For the help in the creation of the Companion, I wish to thank to
engineer Siniša Hristov, professor Dr Gligorije Perović, Dr Walter Bich,
professor Dr Dragić Banković, professor Dr Vesna Jevremović and
Dr Emina Krčmar. To my longtime colleague, engineer Zlatko Sudar, I am
grateful for comments which have led to an improved text.
16
Kostić • 2019 • Metrology Companion
I am grateful to professor Ljiljana Sudar and professor Dr Živomir
Petronijević for their efforts and valuable remarks of the trial readers.
Thanks to Ljiljana D. Kostić for her trust and material support.
This Companion was created as a part of the development project of
measurement and control equipment, which is in progress in the
laboratory for development of electronic products, Symmetry, based in
Leskovac, Serbia.
This is an updated e-edition that comes after the fundamental revision
of the International System of Units (SI) as well as one English and two
Serbian editions.
Please send comments to the symmetry@ptt.rs.
Leskovac, 15th August 2019
Goran Kostić
Kostić • 2019 • Metrology Companion
17
Introduction
The measurements are essential for human activity - from the trade to
the fundamental sciences.
International trade has been one of the important reasons for the
establishment of the International Bureau of Weights and Measures
(BIPM). The Bureau contributes to the reduction of technical barriers to
trade, ensuring that measurements and tests carried out in different
countries are considered equivalent.
Scientists create their theories on a basis of measurement results
which are for them facts about the material world, true within limits of
measurement errors.
The International System of Units was conceived after ancient
discussions of European scientists about the need for a new system of
measurement. The first proposal of a unified system of units, based on
the metre (metric) and decimal, was made as early as 1670 by Gabriel
Mouton, a priest from Lyon, France. That system replaced national and
regional variants that made scientific and commercial communication
difficult. The first decimal metric system was established in France in
1799 as an important fruit of the French Revolution. The world became
metric in early 1993 when the International System of Units (SI) became
primary in the USA government institutions.
The current SI has emerged as perhaps the most significant revision
of the SI. This revision was voted at the 26th General Conference on
Weights and Measures (CGPM) in November 2018. The current
definitions of SI base units are fundamentally different from those
previously used. Now, the seven SI base units are defined using only
seven “defining constants” whose values are taken as exact. The
definitions are published in the 9th edition of the SI Brochure and are
applied from 20th May 2019.
The International Bureau of Weights and Measures was created by
the Metre Convention signed in Paris on 20th May 1875. The Convention
was amended in 1921 and has remained the basis of the international
agreement on measuring units until now. The original signatories of the
Convention were representatives of 17 States, and on 15th August 2019,
there were 60 Member States of the Convention and 42 Associates of the
Kostić • 2019 • Metrology Companion
19
CGPM. Since its establishment, the headquarters of Bureau is in Sevres
near Paris.
The purpose of Bureau is to ensure worldwide unification of
measurements in order to enable traceability of measurement results to
the International System of Units. Therefore the task of Bureau is to:
● “represent the worldwide measurement community, aiming to
maximize its uptake and impact
● be a centre for scientific and technical collaboration between the
Member States, providing capabilities for international measurement
comparisons
● be the coordinator of the worldwide measurement system, ensuring
it gives comparable and internationally accepted measurement results.”
The Bureau operates under the exclusive supervision of the
International Committee for Weights and Measures (CIPM) which itself
comes under the authority of the General Conference on Weights and
Measures. The International Committee and the General Conference was
established by the Metre Convention, at the same time as the Bureau.
By harmonizing and providing complete coverage of all areas, the
International System of Units is an outstanding tool of the modern
sciences and practices that created it.
[SI] Appendix 4. Part 2, 1.1, PP 117 [Britannica] [NIST]
20
Kostić • 2019 • Metrology Companion
1 Quantities and units of measurement
1.1 Measurable quantity
Measurable quantity (quantity) is a property of a body, a
substance, or a phenomenon, that may be distinguished qualitatively
and determined quantitatively.
Quantity is not a body, a substance or a phenomenon.
Quantity is independent of measurement and system of quantities.
The term quantity may refer to a quantity in a general sense (length,
time, mass, temperature...) or to a value of a particular quantity (the
length of a given rod, the electric resistance of a given wire, a number of
entities...).
[VIM 2] 1.1 [VIM 3] 1.1 [Sonin] 2.3
1.2 Quantities of the same kind
Quantities of the same kind are quantities whose values may be
added or subtracted, but that it has a physical representation.
Examples. The addition of values of all forms of energies, such as the
amount of heat, kinetic energy and potential energy, has a physical
representation. Thus, all forms of energies, are quantities of the same
kind. A value of energy and a value of moment of force have the same
dimension but there is no physical representation of addition or
subtraction of their values, so those quantities are not of the same kind.
In a given system of quantities:
- quantities of the same kind have the same dimension
- quantities of the same dimension are not necessarily of the same kind
- quantities of the different dimension are always of a different kind.
Quantities of the same kind are grouped into categories of
quantities. Examples of the categories of quantities are:
- thickness, circumference, wavelength
- heat, work, energy.
[Quinn] [VIM 2] 1.1 Notes [VIM 3] 1.2, 1.7
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1.4 Base quantity
Base quantity, together with other base quantities of a system of
quantities, enables obtaining derived quantities.
The selected base quantity always has a physical representation so
that it may be realized using its definition.
A base quantity is a property for which the following mathematical
operations are defined in physical terms: comparison, addition,
subtraction, multiplication by a quantity of dimension one and division by
a quantity of dimension one. Each of these operations is performed on
physical properties of the same kind and yields a physical property of that
kind. Each of these physical operation obeys the same rules as the
corresponding mathematical operation for pure numbers, that is,
quantities of dimension one. This makes possible to describe the physical
properties by the language of mathematics.
The choice of a set of base quantities is somewhat arbitrary. However,
for the selected base quantities it must be possible to define the
operations previously specified. It is preferred that the base quantities of a
set are mutually independent. The set may be minimal, or somewhat
larger, than one out of which it is possible to obtain all necessary derived
quantities.
Example. The base quantities may be: the length, mass, area, speed
and force. A shape and color cannot be base quantities because they
lack acceptable addition.
Things from nature can be described only by comparison with the
several well-known things among which are the base quantities.
[Sonin] 2.2, 2.1, 2.8 [SI] 2
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1.21 System of units
System of units is formed by the base units, the derived units and
equations that define relations between these units.
International System of Units, SI, is based on the International
System of Quantities. The General Conference on Weights and Measures
recommends the SI.
SI coherent units are formed only by seven SI base units (base
units of the SI) and twenty-two SI coherent derived units.
The complete set of SI units (SI units, or units of the SI) is formed
by SI coherent units and SI decimal units.
The complete set of SI units is not coherent.
The complete set of SI units is divided into the class of SI base units
and the class of SI derived units.
The definitions of seven SI base units are provided in Table 1.1 and
described in the following section 1.21.1.
Out of seven SI base units, in almost all areas are used: metre,
kilogram and second. The other four units are used to define derived
units that are specific to particular areas:
● ampere, for electrical and magnetic units
● kelvin, for heat units
● mole, for units of chemistry and molecular physics
● candela, for light units.
The SI coherent derived units with special names and symbols are
listed in Table 1.2. These special names and symbols may be used in
combination with the names and symbols for base units and for other
derived units to express the units of other derived quantities.
The “SI Brochure” [SI] recommends using the complete set of SI units
and only in particular circumstances (to satisfy the needs of commercial,
legal, or specialized scientific interests) non-SI units listed in Table 1.3.
When the non-SI unit is used, it is recommended to define the non-SI
unit in terms of the SI units.
By harmonizing and providing complete coverage of all areas, the SI is
an outstanding tool of the modern sciences and practices that created it.
[Sonin] 2.7 [SI] 2, 3, 4, in which “decimal multiples and sub-multiples of SI units” is
used instead of “SI decimal units”
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1.21.1 SI base units
Seven SI base units are defined using only the seven defining
constants. The values of defining constants were fixed after being
identified as the best estimates at the time just before the decision on the
revision of the SI. In this way the SI is scaled to the seven physical
constants of high reproducibility, and these constants become exact. See
Figure 1.1 and Table 1.1. The reference definitions of base units are
published in the SI Brochure, see [SI].
The current definitions of SI base units have brought important benefits.
The most important are: 1) a large number of physical constants become
either exactly known or known with higher accuracy; 2) values of quantities
that are much smaller or much larger than the base units can be measured
with unreduced accuracy; 3) liberation of mass metrology from the unit
based on the mass standard which is undoubtedly changing with time; 4)
the Josephson effect and Hall effect can be used to directly realize the SI
definitions of most electrical units; 5) the definition of the kelvin does not
make reference to a water having the defined isotopic composition.
Some of the consequences of the current definitions are that the following
constants are not exactly known, and must be measured: 1) the magnetic
constant, the electric constant and the characteristic impedance of a vacuum;
2) the temperature of triple point of water; 3) the molar mass of carbon 12.
The current definitions of SI base units are fundamentally different from
those previously used. However, as it was the case in the past revisions
of the SI, the transition to the current definitions was realized without
noticable impact on daily life, while the measurements based on previous
definitions of the units remain valid within their measurement uncertainties.
Figure 1.1. The SI
base units are defined
using the set of only
seven defining
constants whose values
are taken as exact.
Each base unit is
defined, either, through
the corresponding
constant from the set, or
through the constant
from the set and other
base units that are also
defined through the
constants from that set.
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1.25 Expressing the value of a quantity in the SI
A numerical value of a quantity and unit symbol are printed in upright
typeface. The numerical value always precedes the unit symbol, and a
space is always used to separate the number from the unit symbol. The
only exceptions, in which no space is left between the number and the unit
symbol, are the unit symbols for plane angle: degree, °, minute, ’, and second, ”.
A prefix symbol precedes the unit symbol without a space, and it is
considered as a new inseparable unit symbol, as well as mathematical entity.
A numerical value of a quantity with a unit symbol is regarded as the
product, and they are treated using the ordinary rules of algebra.
One must neither use the plural of a unit symbol, nor the period after a
symbol except at the end of a sentence.
A prefix symbol, formed by placing together two or more prefix
symbols, are not permitted. This also applies to prefix names.
A prefix symbol and a prefix name will not be used with the following
symbol and name of the units of time: min, minute; h, hour; d, day.
No space or hyphen is used between the prefix name and the unit
name. The combination of prefix name plus unit name is a single word.
In an expression, only one unit is used. An exception is in expressing
the values of time, and of plane angles, using non-SI units. However, for
plane angles it is generally preferable to divide the degree decimally, except
in fields such as navigation, cartography, astronomy, and in expressing
very small angles. Example, one would write 22.20° rather than 22° 12’.
Unit symbol is never qualified by further information about the property
of the quantity. Extra information on the property of the quantity should be
attached to the quantity symbol, and not to the unit symbol. Example,
U max = 50 V, but not U = 50 Vmax.
In forming products and quotients of unit symbols the normal rules of
algebraic multiplication or division apply. Multiplication must be indicated
by a space or a half-high dot (⋅). Division is indicated by a horizontal line
(――), by a solidus (/), or by negative exponents. When several unit symbols
are combined, care should be taken to avoid ambiguities, for example by
using brackets or negative exponents. To remove ambiguities, a solidus
without brackets, must not be used more than once in a given expression.
Among the base units of the SI, the kilogram is the only whose name
and symbol, for historical reasons, includes a prefix (“k”). Symbol and
name for the decimal unit of mass are formed by attaching prefix symbol
to the unit symbol “g”, and prefix name to the unit name “gram”. Example,
−6
10 kg is written as 1 mg, not as 1 µkg.
[SI] 3, 5
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Table 1.2 SI coherent derived units with special names and symbols
[SI] 2.3.4
Expressed
in terms
of other
SI units
SI derived quantity
Special
name of
SI unit
Special
symbol
for SI
unit
plane angle
radian
rad
m · m −1 = 1
solid angle
steradian
sr
m2 · m − 2 = 1
frequency
hertz
Hz
s−1
force
newton
N
kg · m · s − 2
pressure, stress
pascal
Pa
N / m2
kg · m−1 · s−2
energy, work, amount of heat
joule
J
N·m=
W·s
kg · m2 · s − 2
power, radiant flux
watt
W
J/s
kg · m2 · s − 3
coulomb
C
F·V
A·s
volt
V
W/A
kg · m2 · s−3 · A−1
ohm
Ω
V/A
kg · m2 · s−3 · A−2
electric conductance
siemens
S
A/ V
kg · m · s · A
electric capacitance
farad
F
C/V
kg · m · s · A
electric charge, amount of
electricity
(electric) potential difference,
electromotive force, voltage,
(electric) tension
electric resistance
electric inductance
henry
Expressed in
terms of SI
base units
−1
−2
−1
3
−2
2
2
4
−2
2
−2
H
Wb / A
kg · m · s · A
kg · m2 · s−2 · A−1
magnetic flux
weber
Wb
V·s=
T · m2
magnetic flux density,
magnetic induction
tesla
T
Wb / m2 =
N / (A · m)
kg · s − 2 · A−1
Celsius temperature
degree
Celsius
°C 4)
K 4)
K 4)
luminous flux
lumen
lm
cd · sr
m · m · cd = cd
illuminance
lux
lx
lm / m2
m − 2 · cd
activity referred to a radionuclide
(ionizing radiation) absorbed
dose, specific (imparted)
energy, kerma
(radiation) dose equivalent
becquerel
Bq
gray
Gy
J / kg
m2 · s − 2
sievert
Sv
J / kg
m2 · s − 2
catalytic activity
katal
kat
2
−2
s −1
mol · s −1
4)
The units °C and K are equal only when expressing temperature difference or
width of temperature interval. The numerical value of a Celsius temperature expressed
in degrees Celsius is related to the numerical value of the thermodynamic temperature
expressed in kelvins by the relation: t [°C] = T [ K] − 273.15. [SI] 2.3.1 The kelvin
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2 Measurements
2.1 Measurement
Measurement is a determination of a value of a quantity by its
comparison with the known value of the quantity.
The value of a quantity may be measured with very high accuracy
except perfect.
The following are necessary for measurement: a specification of
measurand, a description of measurement procedure and a device for
measurement.
Measurement does not apply to nominal properties. Examples: sex of
a human being; ISO two-letter country code; the sequence of amino acids
in a polypeptide...
[Sonin] 2.3 [GUM] 3.1.1 [VIM 2] 2.1 [VIM 3] 2.1, 1.30 Examples
2.2 Metrology
Metrology is the science of measurements.
Metrology includes all theoretical and practical aspects of
measurements, whatever the measurement accuracy and field of
application.
Legal metrology is the part of metrology relating to measurements
specified in the law.
The scope of legal metrology commonly includes:
● protection of the interests of individuals and enterprises
● protection of national interests
● protection of public health and safety, especially those related to the
environment and medical services
● meeting the requirements for commerce and trade.
[VIM 2] 2.2 [VIM 3] 2.2 [VIML] 1.03
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2.3 Measurand
Measurand is a particular quantity subject to measurement.
The requested accuracy determine how many details should be given
in a specification of a measurand. For example, if the length of a
nominally one-metre long steel bar is to be determined to micrometre
accuracy with relative error of about 5 · 10 − 6, its specification should
include the temperature and pressure of the environment in which is the
bar.
[VIM 2] 2.6 [VIM 3] 2.3 [GUM] 3.1.3
2.4 Principle of measurement
Principle of measurement (or measurement principle) is a
phenomenon serving as the basis of a measurement.
Example: the thermoelectric effect applied to a measurement of
temperature.
[VIM 2] 2.3 [VIM 3] 2.4
2.5 Method of measurement
Method of measurement (or measurement method) is a concept of
applying the principle of measurement.
Primary method of measurement (or primary measurement
method, primary method) is a method having the highest metrological
qualities, whose procedure is completely explained and described, and
for which the combined standard uncertainty traceable to the SI units is
determined.
Primary ratio method is a method used to determine a quotient of
value of measurand and value of measurement standard (where both
values are of the same kind).
Primary direct method (primary method) is a method in which a
value of measurand and values of used measurement standards, are of
different kind.
[BIPM] [NIST]
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3 Devices for measurement
3.1 Device for measurement
Device for measurement is a device, or combination of devices,
designed for measurement.
The device for measurement may be a measuring instrument or a
measuring transducer. A measuring instrument is an indicating
measuring instrument or a material measure. A material measure is: a
reference material, weight, volume measure, gauge block, standard
electric resistor, standard signal generator...
Devices for measurement arranged in approximate order of increasing
complexity are: a part, material measure, reference material, measuring
transducer, measuring instrument, measuring chain, measuring system,
measuring installation.
Device for measurement is periodically checked, or qualified, by a
user or a recognized laboratory. The checks determine whether an error
of a device for measurement, in operating conditions, is within a specified
interval and whether it will be within this interval in future.
[VIM 2] 4 introduction, 4.1, 4.3, 4.6, 4.2 [VIM 3] 3, 3.1, 3.7, 3.3, 3.6
Figure 3.1. A measuring transducer and a measuring installation. Left:
an ion-selective probe for concentration of sodium measurement
(Thermo Scientific). Right: a highly automated measuring installation, 18 m
long, for biochemical analysis in a hospital laboratory (Roche Diagnostics).
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3.2 Indicating measuring instrument
Indicating measuring instrument is a measuring instrument
providing an output signal that contains information about the value of a
measurand.
Examples: a thermometer, micrometer, electronic balance, barograph.
Displaying measuring instrument (or meter) providing an output
signal in visual form.
Recording measuring instrument provides a record of its indication
as an output signal.
[VIM 2] 4.6 [VIM 3] 3.3
3.3 Material measure
Material measure (measure) is a device for measurement intended
to supply, or reproduce, one or more known values of a quantity.
Examples: a weight, volume measure, gauge block, standard electric
resistor, standard signal generator, reference material.
An indication of a material measure is its assigned value.
An error of a material measure is the error of the indication of the
material measure.
End gauge is a material measure which supplies one known value.
Example. A length end gauge has the form of: parallelepipeds
(gauge block), pins, blades, plugs, rings, snaps.
Go - no go gauge consists of two end gauges which determine
whether a value of a measurand is within an interval given by the end
gauges.
[VIM 2] 4.2, 3.2 Note 3, 5.20 Note 3 [VIM 3] 3.6, 4.1
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4 Measurement results, accuracy and
errors
4.1 Measurement result
Measurement result (or result of measurement, or estimated
value) is a value assigned to a measurand on the basis of one or more
results of measurements.
A measurement result is obtained in the manner specified in the
description of the measurement procedure: ● taking a result of a single
measurement (or particular result), ● computing the arithmetic mean of
the results of repeated measurements, ● taking the value that appears
most often in the results of repeated measurements (that is, taking the
mode), ● computing on the basis of a functional relationship between the
value of a measurand and component values (or component results,
or component variables, or component quantities, or input
quantities, or components) on which the value of the measurand
depends...
Results of repeated measurements of the same quantity are represented
by the random variable which is characterized by statistical parameters.
A complete measurement result includes information about its accuracy.
[VIM 2] 3.1 [GUM] [VIM 3] 2.9
4.2 Accuracy
Accuracy of a value is a qualitative concept which indicates a
closeness of this value to a reference value.
Accuracy of a measurement result (accuracy) is a qualitative
concept which indicates a closeness of this measurement result to a
value of the measurand.
Accuracy of a measurement result is determined in the manner
specified in the description of the measurement procedure.
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Accuracy of a measurement result depends on errors which are
classified into the two groups. The first group is random errors, whose
influence on the final measurement result can be reduced by increasing
the number of repeated measurements and then computing the arithmetic
mean of the measurement results. The second group is systematic errors,
whose influence on the final measurement result can be reduced by
applying certain correction.
A complete measurement result includes information about its
accuracy. However, if inaccuracy is considered to be negligible for the
intended purpose, the information about accuracy may be omitted. In
many fields, the ordinary way of expressing a measurement result does
not include the information about accuracy.
In order to enable the use of one measurement result and its accuracy
as component values of another measurement result and its accuracy,
the information specified in 9.10 General guidance on reporting a
measurement result should be provided.
Results of repeated measurements of the same quantity are
represented by the random variable which is characterized by statistical
parameters. The description of the accuracy of the result of repeated
measurements is its standard measurement uncertainty.
The term precision should not be confused with accuracy.
[VIM 2] 3.5 [VIM 3] 2.13, 2.15, 2.9 Note 2 [GUM] 3.2
4.3 Absolute error
Absolute error (error, or difference, or divergence) of a value is
given by formula:
ABSOLUTE_ERROR = VALUE – REFERENCE_VALUE.
(4.1)
VALUE = REFERENCE_VALUE + ABSOLUTE_ERROR
(4.2)
REFERENCE_VALUE = VALUE – ABSOLUTE_ERROR
(4.3)
ABSOLUTE_ERROR =
= SYSTEMATIC_ERROR + RANDOM_ERROR
(4.4)
In practice, the conventional value is used as the reference value.
The value and its absolute error have the same dimension.
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5 Measurement standards and calibrations
5.1 Measurement standard
Measurement standard (standard, or etalon) is a device for
measurement intended to realize, reproduce or define the value of a
quantity to serve as a reference in measurements.
Examples: a Kibble balance (or a watt balance); 1 kg mass standard;
100 Ω standard resistor; standard ammeter; caesium frequency standard;
hydrogen reference electrode; reference solutions of cortisol, in human
serum, with specified concentration.
Figure 5.1. Left: a Kibble balance in the International Bureau of
Weights and Measures. Among other things, the balance is used for
measuring a 1 kg mass standard with the combined standard
measurement uncertainty of 2 · 10 − 8. The measurements are performed
through the Planck constant, gravitational acceleration, velocity, plus
electromagnetic quantities, and represent the realization of the definition
of the kilogram. Right: a Josephson voltage standard used for driving a
stable current through the coil of the above-mentioned Kibble balance
using a standard resistor. (Courtesy of the BIPM.) [BIPM]
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Intrinsic measurement standard (intrinsic standard) is based on an
inherent and reproducible phenomenon in nature. Examples: the speed of
light in vacuum; the elementary charge; a triple-point-of-water cell as a
standard of temperature; a Josephson array as a voltage standard; a
quantum Hall effect as a standard electric resistor.
Collective standard (or ensemble of standards) is a set of
standards that, through their combined use, constitute a standard.
Group standard is a set of standards that, individually, or in
combination, provide a sequence of values of quantities of the same kind.
Table 5.1 presents the most accurate standards, or measurements, of
the base quantities and their approximate combined standard
measurement uncertainties.
[VIM 2] 6.1 [VIM 3] 5.1 in which the associated measurement uncertainty of a
measurement standard is required, 5.10
5.2 Reference material
Reference material, RM, is material with homogeneous, stable and
accurately defined properties used as a reference in measurements.
Reference material may be a pure or mixed: gas, liquid or solid.
Examples: a solution used in calibration of measuring instruments for
chemical analyses; a color chart; a fish tissue containing a specified mass
concentration of dioxin.
Certified reference material, CRM, is the reference material
accompanied by a certificate specifying the values of the material
properties and traceable standard measurement uncertainty of these
values.
Example. Human serum accompanied by a certificate specifying the
value of the concentration of cholesterol and the associated traceable
standard uncertainty.
[NIST] [VIM 2] 6.13, 6.14 [VIM 3] 5.13, 5.14
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Table 5.1 The most accurate standards, or measurements, of the
base quantities, state in 2019 [BIPM] [Newell 2014] [NIST]
[SI realization]
Quantity
Conditions
Absolute
standard
uncertainty
Relative
standard
uncertainty
time
The best of primary standards which
realizes the SI second in the BIPM.
1 · 10−16 s
1 · 10 − 16
The length of the cubic unit cell edge
of pure 28Si (5.431 · 10 −10 m)
measured using interferometry.
3 · 10 −18 m
5 · 10 − 9
A steel gauge block of the length of
100 mm measured using optical
interferometry.
10 nm
1 · 10 − 7
The Earth-Moon separation
(376 203 km) measured by a laser
and a retroreflector on the Moon.
1 mm
3 · 10 −12
A mass of single atom (an order of
magnitude of 10 − 25 kg) through
frequency measurements of photons
during absorption or emission of that
atom.
ca 10−33 kg
2 · 10 − 9
A 1 kg weight measured on a Kibble
balance.
20 µg
2 · 10 − 8
electric
current
Generating a current up to 10 mA
using a Josephson voltage standard
and a Hall standard resistor.
< 6 · 10
thermodynamic
temperature
−155 °C to 30 °C measured using the
relative primary acoustic gas
thermometry in the NPL.
< 1 mK
amount of
substance
The number of atoms in a single
crystal of 1 kg of the pure 28Si
(2.153 · 10 25 atoms) using volumetric
and interferometric measurements.
4 · 10
atoms
2 · 10 − 8
luminous
intensity
The light source, from 10 − 3 cd to
10 4 cd, directly measured with a
standard photometer.
2 · 10 − 6 cd
to 20 cd
2 · 10 − 3
length
mass
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−12
A
17
6 · 10 −10
for kelvin
scale
< 7 · 10 − 3
87
6 Outlines of the statistics
6.1 About statistical analysis
Statistical methods are used to determine the representative value of
an investigated property for a group of elements. The elements within the
group can have significantly different values of the property.
Evaluation (or estimation, or “statistic”) is a value estimated by the
statistical method.
Population is a group of all elements for which we determine the
representative value of a property by the statistical method. The
population has a finite or infinite number of elements.
Figure 6.1. The example of four samples of values which yield: the
equal arithmetic means of variables x and y, the equal deviations of x, the
same linear regression function obtained by the method of least
squares... [Anscombe]
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The investigated property of a population is determined by the
statistical method usually on a basis of a sample of the population that
truly represents the population. The sample truly represents the
population when a number of its elements is sufficiently large and when
the elements are randomly selected from the population.
In order to prove that the conclusions drawn from the sample are with
a certain probability valid for the population can be proved by
goodness-of-fit tests.
In order to reduce the possibility of errors in statistical analysis, data
graph should be necessarily considered. The graph provides an overview
of qualitative features of data. The diagrams in Figure 6.1 demonstrate
the importance of graphical presentation and visual analysis of data.
A property of a population determined by a statistical method is the
property of a large number of elements of that population. On a basis of
statistical evaluation, we can determine only the probability that a
particular element of a population has a certain property.
Statistics is subjective; statisticians try to explain or predict the
material world through arbitrary, but reasonable way, using probability
theory, mathematics and common sense.
Unlike the statistics, probability theory for a fully defined problem gives
a unique and a reproducible solution. Probability theory is the algebra of
inductive reasoning. (Boolean algebra is the algebra of deductive
reasoning.)
Statistical methods are among the most widespread quantitative
methods, with important applications in almost all human activities.
[Njegić] 1.1., 1.2. [Ivković] II.1 [Britannica] Probability theory [Drake] 7-1, 7-6
[Lazić] I, 1.2 [Anscombe]
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6.2 Statistical analysis of results of repeated
measurements
The purpose of statistical analysis of results of repeated
measurements of a particular quantity is to determine the best estimated
value of the quantity. And also, to determine an interval about the best
estimated value which, with requested probability, encompass a true
value of the quantity.
Normally, for the best estimated value of the quantity is taken the
maximum likelihood value of this quantity.
The results of repeated measurements most often have normal
distribution because the central limit theorem is most often valid. In the
case of normal distribution, and after correction for all significant
systematic errors, the maximum likelihood value of the quantity is an
arithmetic mean of the results of measurements.
For the normal distribution, the interval:
ARITHMETIC_MEAN ± DEVIATION_OF_THE_ARITHMETIC_MEAN
is the interval in which a true value of the quantity is with the probability of
0.68.
[GUM] 3, 4
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6.24 Degrees of freedom
Degrees of freedom, ν, of an evaluation estimated from a sample of
size n is given by formula (6.64). The R is a number of evaluations
estimated from this sample and used to estimate this evaluation. See
also 6.24.1 Examples.
ν=n–R
(6.64)
The least possible sample size is 1, and from this sample it is possible
to estimate an evaluation of the degree of freedom equal to 1.
Evaluations with the degrees of freedom equal to 1 have the greatest
possible dispersion. The dispersion is reduced by increasing the sample
size, that is, the degrees of freedom, so the degrees of freedom is the
indicator of reliability of an evaluation. The degrees of freedom of an
evaluation is equal to the number of elements of the sample above the
least number of elements necessary to compute the evaluation, plus, 1.
The degrees of freedom defined by (6.64) gives a value which is a
parameter of different distributions. For example, a sample from population
with the normal distribution, for which (6.64) gives a degrees of freedom ν,
has the T-distribution with parameter “degrees of freedom” equal to ν. Also,
the standard uncertainty, for which (6.64) gives a degrees of freedom ν,
has the T-distribution with parameter “degrees of freedom” equal to ν.
If for a sample from a population with the normal distribution, is known
its standard deviation s, and a degrees of freedom of this deviation, ν,
formula (6.65) gives half-width of confidence interval, Δ, for a confidence
level of an interval, P. The coefficient k is the coverage factor which can
be determined from the relation among P, ν and k listed in Table 6.3 for
the T-distribution. Formula (6.65) is derived from (6.77).
Δ = k·s
(6.65)
The degrees of freedom enables accurate computation of a level and
interval of confidence, even in the case of small sample size.
The quantities Δ and k from (6.65) can be used as an expanded
standard measurement uncertainty and its coverage factor.
Without giving a general definition, the concept of the degrees of
freedom first started to use Sealy Gosset (“Student”) in 1908 and Ronald
Fisher in 1915. Since then, the concept is based on an incomplete theory.
A particular problem is the computation of the degrees of freedom.
[Njegić] 5.6. [GUM] 4.2.6, G.3.2, G.3.3, G.6.4 [Eisenhauer]
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6.27 Confidence level and confidence interval
Confidence level (or level of confidence, or coverage probability)
is a probability that a value of variable is from a specified interval.
Confidence interval (or interval of confidence, or two-sided
confidence interval, or coverage interval) is the interval which has the
specified probability that the value of variable is from it. Confidence
limits are the limits of confidence interval.
The number of measurement results is from a particular interval. The
probability that the result is from mentioned interval is equal to the
quotient of the number of results from mentioned interval, and the total
number of results. The mentioned interval is the confidence interval. The
mentioned probability is the confidence level.
Generally, there is more than one coverage interval for a specified
probability. Shortest confidence interval of a variable is a confidence
interval with the narrowest width among all coverage intervals having the
same coverage probability. In the case of unimodal distribution, the
shortest confidence interval is to the left of the mode and to the right of
the mode. Unless otherwise indicated, the confidence interval means the
shortest confidence interval.
A confidence level, P, of a variable x with a probability density function
f(x), for confidence interval [a, b], is given by (6.76). With (6.76) there is a
graphical example for the normal distribution.
P = P(a ≤ x ≤ b ) =
b

f( x ) ⋅ d x
(6.76)
a
For a variable with the normal distribution, it can be computed an
arithmetic mean x , and standard deviation of the mean, s. For a requested
confidence level, P, a confidence interval can be given in the form of:
[ x − k ⋅ s , x + k ⋅ s ] or simplified x ± k ⋅ s .
(6.77)
In the case of the normal distribution, the coefficient k can be determined
from function (6.78) which gives the relation between P and k. The
values in Table 6.2 are obtained from (6.78).
P(k ) =
2
⋅
π
k

0
1
k2
2
⋅ dk
(6.78)
e
The coefficient k is called the coverage factor. Product k · s is a
half-width of confidence interval (or one-sided confidence interval).
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A standard deviation of a variable with the normal distribution is a
half-width of a confidence interval of the variable at the confidence level
(of the interval) equal to 0.6827.
One or more measurement results, from a population with the normal
distribution, have the T-distribution. For details and about levels and
intervals of confidence, see 6.29 T-distribution.
The level and interval of confidence of the measurement result can be
considered as a purpose of all determinations of measurement errors.
[Ivković] PP 79, PP 81, PP 17 [GUM] 7.2.4, G.3.2, G.5.3, G.1.4
Table 6.2 For the normal distribution, N(x): relations between the
confidence level, P, and coverage factor k
P
0.079 7
0.100 0
0.158 5
0.200 0
0.300 0
0.382 9
0.400 0
0.500 0
0.600 0
0.682 7
0.700 0
0.800 0
0.850 0
0.900 0
0.950 0
0.954 5
0.970 0
0.980 0
0.990 0
0.995 0
0.997 3
0.998 0
0.999 0
0.999 9
0.999 99
0.999 999
130
k
0.100
0.126
0.200
0.253
0.385
0.500
0.524
0.675
0.842
1.000
1.036
1.282
1.440
1.645
1.960
2.000
2.170
2.326
2.576
2.807
3.000
3.090
3.291
3.891
4.417
4.892
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6.38 Histogram
The sample of values can be splitted into disjoint subintervals of equal
widths (or cells, or bins). The histogram is the graphic representation of
a number of the values that fall into each of the subintervals.
The histogram is made in the following way. First, split a part of the
horizontal axis which contains the values from the sample into disjoint
subintervals of equal width, see Figure 6.14 b). Then draw rectangles
whose bases overlap subintervals on the horizontal axis and whose
heights are proportional to the number of values that fall into the
subinterval of their base.
The histogram appearance is significantly affected by the subintervals
width and their position along the axis. The limits of subintervals can be
determined according to the following two recommendations.
a) If we make the histogram for a sample with a small number of
possible discrete values, the subintervals should be chosen so that each
discrete value corresponds to one subinterval. This is the case in
Example 6.38.1, in which 8033 discrete measurement results have one of
16 possible discrete values, out of which only 5 values are distinguished
by the number of appearances. Histograms, such as in this case, are
bases for reliable estimating.
b) If we make the histogram for a sample with a large number of
possible values, subintervals width and position in the horizontal axis can
be determined as follows.
If the n values in the sample have the normal distribution and n is
larger than about 25, the optimal subinterval width w, is given by formula
(6.116) with the parameter C = 3.5. The standard deviation of values in
the sample is denoted by s. This formula is derived from the criterion of
the minimum of integrated mean squared error (IMSE).
w = C⋅
s
3
n
(6.116)
If the values in the sample do not have the normal distribution, and n
is larger than about 25, based on the experience we can also recommend
using (6.116) with C = 2.
In the case of discrete values, it is necessary to round the subintervals
width so that each subinterval contains the same number of possible values
(that is, round subintervals width to an integer multiple of a resolution).
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The limits of subintervals should be chosen so that more of the values
are around the middle of subintervals.
In the histograms described in b), it is classified the case in Example
6.38.2, in which 78 discrete measurement results have one of 30 possible
values. This example shows how the subintervals width affects the
histogram appearance.
In most statistical software a default subinterval width and position are
not optimal.
The height of a rectangle of a histogram is proportional to the
probability that values fall into the subinterval of the rectangle base, so
the histogram is the variation of the probability density function.
By visual inspection of the well-made histogram, we can estimate the
following properties of the distribution: a type; a mode; an existence of
outliers; a dispersion of values; an existence of several maximums; a
symmetry; and a peakedness. The lack of an evaluation based on the
histogram is subjectivity.
The approximate arithmetic mean of n values represented by a
 is given by formula (6.117) derived from (6.19). The bi is a
histogram, x,
number of values in a subinterval with the middle xi .
x ≈
1
1 n
⋅ (b1 ⋅ x1 + b2 ⋅ x2 + ... + bn ⋅ xn ) =
⋅  bi ⋅ xi
n
n i =1
(6.117)
[Britannica] Statistics [Perović, communication] [Scott] [Hristov, communication]
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6.40 Filtering
Filtering is a procedure to reduce random differences of values in a
sequence.
Filtering is most often performed by averagings. Moving average filtering
uses the convolution, it is simple and optimal to reduce random differences
of values in a sequence. This filtering retains sharp changes but gives the
wrong idea about values and widths of peaks in the sequence. These
disadvantages are worsened by decreasing a bandwidth of the filtering.
In Figure 6.17 there is an
example of filtering by using the
convolution. The weight
function (that is, window) is the
Chebyshev type sequence with
seven coefficients (that is,
points) scaled so that their sum
equals to one. (The moving
average filtering using the
weight function with equal
coefficients.) Filtering is applied
to the fluctuating discrete
measurement results. These
results with the corresponding
histogram are also shown in
Figure 6.15.
In addition to averaging,
filtering is also implemented by
fitting. Savitzki - Golay
filtering uses local polynomial
fitting across a moving window
within the values in the
sequence. The polynomial is
often of the fourth degree, and
it is fitted by the method of least
squares. Such filtering tends to
preserve the values and widths
of peaks in the sequence. See
[Savitzky].
[Savitzky]
[Hristov, communication] [Guiñón]
154
Figure 6.17. Filtering of the
sequence of discrete measurement
results by using the convolution: a) the
Chebyshev type weight function, b)
the sequence of discrete
measurement results and c) the
convolution of these two functions.
[Hristov, communication]
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6.41 Operations on distributions and central limit theorem
When a die is thrown, each number from 1 to 6 comes up with equal
probability. The probability function of these numbers, P(x1), is uniform
and discrete, and it is represented pictorially in the following figure.
Think about the probabilities to happen values which are the sum of
two numbers obtained from two tosses of a die. The possible values of
the sum are 2 to 12. There is only one way of making 2, two ways of
making 3, etc. See the following picture. The sum 7 is the most probable
because it can be achieved in the greatest number of ways.
The random variable x1 , which assume values obtained from tosses of
the die, has the uniform and discrete probability function. The random
variable x2 , whose values are the sum of the two numbers obtained from
two tosses of the die, has the triangular and discrete probability function.
As an example, using the properties of probability, we can compute the
probability that the sum of x2 is 5:
P(x2 = 5) = P[(4  1)  (3  2)  (2  3)  (1  4)] =
= P(4  1) + P(3  2) + P(2  3) + P(1  4) =
= P(4) · P(1) + P(3) · P(2) + P(2) · P(3) + P(1) · P(4) =
2
= 4 · (1/6) = 4/36 = 1/9.
We see that P(x2 = 5) can be written as P( x2 = 5) =
4
 P(5 − i ) ⋅ P(i ),
i =1
which is the convolution of discrete function P with itself. Based on previous
example, we can determine the probability function of the x2 , P2 (x2 ):
 1/ 6, x1 = 1, 2, ..., 6
P1 ( x1 ) = 
0 ≥ x1 ≥ 7
 0,
P 2 ( x2 ) =
∞
 P1 ( x2 − i ) ⋅ P1 (i ) = P1 ( x2 ) ∗ P1 ( x2 ).
i = −∞
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Analogous to the previous discrete probability functions, we can
determine the probability function of the sum of variables with continuous
probability functions. See [Osgood] 3.6.6.
For discrete and for continuous probability density functions the
following applies. The probability density function of a sum of two
independent random variables is the convolution of the probability density
functions of those two variables. That is, if f is the probability density
function, and if x1 and x2 are independent variables, then:
f (x1 + x2 ) = f (x1 ) * f (x2 ).
(6.127)
From the example in Figure 6.18, we can see the following. The sum
of two independent variables, with the uniform distributions of the same
width, has a triangular distribution. Approximately normal distribution has
the sum of the variable with the triangular distribution and variable with
the uniform distribution, as well as the sum of variables with normal
distribution and variable with the uniform distribution. The example in
Figure 6.18 shows how from chaotic differences of variables the normal
distribution emerges necessarily which suggests the central limit theorem.
Central limit theorem states that a sum of independent variables,
with any distributions, approaches the normal distribution as a number of
the variables increases, but if there is no variable with multiple times
greater differences than other differences. See [Osgood] 3.7 and 3.10.
[Osgood] 3.6, 3.7, 3.10 [Box 2005] PP 28
156
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Figure 6.18. The variable x1 in
time, its distribution and the
distributions of the sums of
independent variables xi with the
distributions as x1.
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157
6.42 Monte Carlo simulation
Monte Carlo simulation, MCS (or Monte Carlo method, MCM) is
used to estimate a value by performing mathematical model computations
for a large number of combinations of input values.
The combinations of input values are chosen so that intervals of
interest are simulated as uniformly as possible and at enough points. The
input values can be a sequence of numbers represented in the diagrams
below, each with one hundred pairs of the: a) equidistant numbers x and y,
b) numbers with the uniform distributions, c) numbers with quasi-random
distributions, d) numbers with the normal distributions and e) linearly
correlated numbers with the normal distributions.
Input numbers with quasi-random distributions are suitable when a
model has many variables and depends strongly on some of the
variables. If with such model we use numbers with other distributions, and
with the same total number of points, we obtain significantly poorer
resolution per variable. See [Sobol].
The Monte Carlo simulation is for computer implementation.
In the metrology, in order to simulate values of component quantities,
input values produce number generators for given best estimated
component quantities and distributions of these evaluations. The
simulated input values must have all parameters of a distribution equal to
parameters of a distribution of simulated evaluation, including possible
correlation. This simulation truly represents a measurement procedure.
The simulation can be used to:
● determine a distribution of evaluations of measurand
● determine a confidence interval of evaluation
● determine the best estimated value of measurand
● determine a standard measurement uncertainty of this evaluation
● determine the degrees of freedom of this uncertainty
● determine sensitivity coefficients
● to validate procedures according to the GUM.
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The Monte Carlo simulation applies if the conditions to use the primary
procedures according to the GUM are not fulfilled, or if it is unclear whether
they are fulfilled. The simulation can also be applied if the condition to use
the convolution is not fulfilled (the condition is uncorrelated input values).
The simulation is commonly used in the following cases:
● linearization of a model of measurement is not satisfactory
● it is inconvenient to provide partial derivatives for a model of
measurement
● a model of measurement is too complex
● distribution functions of input values are highly skewed
● standard measurement uncertainties of input values are not roughly
equal
● a distribution of output values departs appreciably from the normal
distribution
● an output value and its standard measurement uncertainty are
roughly equal.
Conditions for application of the Monte Carlo simulation:
● the possibility of computing a sufficiently large number of Monte
Carlo trials
● in the neighborhood of the best estimated input quantities the model
of measurement is continuous (not necessarily the derivatives of the model)
● if for output values an arithmetic mean and deviation exist
● if the confidence interval is to be determined:
- the cumulative distribution function of output values is continuous
and strictly increasing
- the probability density function of output values is continuous over
an interval for which it is larger than zero, it is unimodal and it is strictly
increasing, or zero, to the left of the mode and strictly decreasing, or zero,
to the right of the mode.
The primary advantages of the Monte Carlo simulation:
● it is not necessary to determine partial derivatives of a model of
measurement
● the simulation gives the distribution of output values
● the simulation has broader applicability than the procedures
according to the GUM
● the same kind of simulation results and results of procedures
according to the GUM.
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Estimate of a value of a quantity and its statistical parameters can be
performed by the Monte Carlo simulation in the following steps.
1) Specify the input data: a model of measurement and component
quantities with their parameters.
2) Determine the required number of Monte Carlo trials, that is, sets of
input values, M. This number depends on the requested accuracy of the
4
simulation results and on distributions of input values. Often satisfies 10
6
to 10 trials.
3) Produce by number generators M sets of randomly ordered input
values with appropriate distributions and possible correlations. See
Example 6.42.1 and [GUM-S1].
4) For each set of input values from 3), by the use of a model of
measurement, compute the model output value, ym (m = 1, 2, ..., M ).
5) From the M model output values, the histogram is obtained in the
usual way according to 6.38 Histogram.
6) The best estimated value of a quantity is obtained by:
● using the model of measurement and other input data - if the input
values have symmetric distributions with approximately equal
measurement uncertainties which are relatively small in relation to the
means of input values, or
● if the input values do not have distributions as in the preceding
clause - based on the distribution of the model output values, and
according to 6.9 Best estimate.
7) When n results of repeated measurements of component quantities
are simulated, the M output values are divided into J = M / n groups with
the n values.
8) The maximum likelihood value of the standard deviation of the
value obtained from the n results of repeated measurements is computed
in the usual way from (6.56) so that the standard deviation of all M values
is divided by n . This deviation is the combined standard measurement
uncertainty of the mean of n results of repeated measurements. (The
standard deviation of all M values is approximately equal to the average
(that is, arithmetic mean) of the deviations of the n values within the groups.)
9) For each group with the n values, the standard deviation of the mean
of the group is computed in the usual way from (6.56). See Table 6.4.
10) For all J groups, the deviation of the standard deviations of the
mean of the group, from 9), is computed in the usual way from (6.49).
See Table 6.4.
11) For all J groups, the standard deviation of the means of the group is
computed in the usual way from (6.49). See Table 6.4.
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12) The degrees of freedom of the standard deviation from 8) is
approximately computed from (6.75), using the deviation of the standard
deviations of the mean of the group from 10) and the standard deviation
of the means of the group from 11). See Table 6.4.
13) In order to determine the needed number of the Monte Carlo trials,
repeat, for example, 10 simulations with a different number of trials.
Compute the standard deviation for the results of the repeated simulations.
Based on these deviations estimate the required number of trials.
The simulation on a personal computer often takes only a few seconds.
Example
In Figure 6.19 the instructive
example of Monte Carlo simulation
is represented. The simulation
gives probability density function
of evaluations of a measurand,
f( y), for uncorrelated component
quantities with the probability
density functions f(x1) and f(x2)
and for the model of measurement
y = x1 + x2 . Based on the
distribution f( y), it is determined
the mode ymode = 0.96, which is
the best estimated measurand,
y, as well as the standard
measurement uncertainty of this
evaluation of y, u( y) = 2.149.
The procedures according to
the GUM gives the best
estimated measurand
y = x1 + x2 mode = 0.500 + 0.374 =
= 0.874 and its standard
measurement uncertainty
u(y=
)
u2 (x1 ) + u2 (x2 =
)
= =
0.16672 + 2.1592
= 2.165.
[Hristov, communication] [Sobol]
[GUM-S1] Introduction, 1, 8.1.1, 5.11,
5.10, 5.9.6, 7
Kostić • 2019 • Metrology Companion
Figure 6.19. The Monte
Carlo simulation gives probability
density function of estimates,
f( y), for uncorrelated component
quantities with the density
functions f(x1) and f(x2).
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7 Variances
7.1 Natural variance
Natural variance (variance) V , of a group of m values xi
(i = 1, 2, ..., m), is an arithmetic mean of squared differences of these
values from an accurate arithmetic mean of these values, x :
m
V =
 ( xi − x )2
i =1
(7.1)
.
m
The natural variance V , of a group of m values xi , can be computed
also from the squared differences of these values from pairs obtained by
combinations (without repetition) of second order from m elements:
m −1 m
V =
  ( xi − x j )2
i =1 j = i +1
(7.2)
.
m2
Population variance (variance of a population, or expected
2
variance) σ , of all m values in a population, xi (i = 1, 2, ..., m), is
computed from (7.1) by taking for the accurate arithmetic mean of these
values, x , the accurate arithmetic mean of a population, μ:
m
σ2 =
 ( xi − μ)2
i =1
(7.3)
.
m
The variance Vm , of values xi which arrive continuously, is computed
by the recursive method using (7.4) and (7.5), derived from (6.19) and
(7.1). This computation does not require memorizing all m received
values, but only the last xm , and the previously computed arithmetic
mean x m − 1 and the variance Vm − 1 .
x m =
(m − 1) ⋅ x m − 1 + xm
m
⋅
m −1 
m −1
Vm =
⋅ Vm − 1 +
⋅ ( xm − x m − 1 )2
m
m2
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(7.4)
(7.5)
171
Computation using (7.5) in practice gives more accurate results than (7.1).
Using (7.5) is especially appropriate in the following cases: ● when the
values for which the variance is computed arrive continuously; ● when the
values have a small relative deviation; ● if it is required to determine the
variance during a process (for example during Monte Carlo simulation, in
which trials are stopped when the variance of the simulation results
becomes small enough).
The degrees of freedom of the variance is equal to the number of
values: ν = m.
The variance is an indicator of dispersion of values in a group. It is
expressed in units which are the squared unit of these values.
The variance is valid for values with any distribution. See Example
6.42.2.
The variance of a group of values is equal to the squared deviation of
this group.
The properties of the variance V are listed in the following text.
Variables are denoted by x and y, constants by C and D, function (7.1) by
 a ), and the standard covariance of x and y, given by (7.21), by V(x, y).
V(
● V ≥ 0.
(7.6)

= 0.
● V(C)
(7.7)
 + x ) = V(
 x ).
● V(C
(7.8)
 ⋅ x ) = C2 ⋅ V(
 x ).
● V(C
(7.9)
● If x and y are uncorrelated:
 ⋅ x + D ⋅ y ) = C2 ⋅ V(
 x ) + D2 ⋅ V(
 y ).
V(C
(7.10)
● If x and y are correlated:
 x ) + D2 ⋅ V(
 y ) + 2 ⋅ C ⋅ D ⋅ V( x, y ).
V (C ⋅ x + D ⋅ y ) = C2 ⋅ V(
(7.11)
● If x and y are uncorrelated:
 x ⋅ y ) ≈ y 2 ⋅ V(
 x ) + x 2 ⋅ V(
 y ).
V(
(7.12)
[GUM] C.2.20, C.3.2, C.2.12 [Box 2005] PP 27 [Hristov, communication] [Welford]
[Ivković] PP 45
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7.6 Combined variance
Combined variance is a variance of estimated value arising from
variances of component variables.
The computed combined variance of the evaluation is valid for any
distribution of the component variables and the evaluation. Moreover,
evaluations approach the normal distribution as the number of component
variables increases, but if the variables are independent and if there is no
a variable with multiple times greater variance than other variances
(central limit theorem). See Example 6.42.2.
Let the relation between the evaluation y and J component variables xj
be given by the function
y = y (x1 , x2 , ..., xJ ),
let the component variables be uncorrelated and let the nonlinearity of the
function y be insignificant. Then the combined variance of the evaluation y,
VC ( y), is computed from function (7.40). The partial derivative ∂ y / ∂ x j is
called the sensitivity coefficient to x j and is usually denoted by c (x j ) or cj .
The variance of the component variable xj is denoted by V(x j).
2
2
2
 ∂y 
 ∂y 
 ∂y 
VC ( y ) = 
 ⋅ V( xJ ) =
 ⋅ V( x1 ) + 
 ⋅ V( x2 ) + ... + 
 ∂x1 
 ∂x2 
 ∂xJ 

∂y
=  


j =1  ∂x j

J
=
2


 ⋅ V( x j ) =




J
 ( c2 ( x j ) ⋅ V( x j ))
(7.40)
j =1
Function (7.41) gives the variance Vmax (VCy ), which is the maximum
possible variance of variance VC ( y ) from (7.40). If the maximum variance
of variance, Vmax (VCy ), is several times smaller than the variance VC ( y ),
the nonlinearity of the function y is insignificant considering the variances
of its component variables, so (7.40) gives a valid value.
Partial derivatives of (7.41) are at the changing point
( x1 + r ⋅
V( x1 ) , x2 + r ⋅
V( x2 ) , ..., xJ + r ⋅
V( xJ ) )
which changes by varying value of r from 0 to 1. See Examples 7.6.1 and 7.6.2.
Vmax (VCy ) = max (
0 < r <1
1 J  ∂ ∂y

⋅ 
(
) ⋅ V( x j )  )
2 j =1  ∂x j ∂x j

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(7.41)
183
The function y is a linear function if it can be expressed using the
constants Cj :
y (x1 , x2 , ..., xJ ) = C1 · x1 + C2 · x2 + ... + CJ · xJ .
(7.42)
If the function y is linear and if components xj are uncorrelated, function
(7.40) gives not an approximate, but the exact value of the combined
variance. In this case, checking by function (7.41) is not required.
From (7.40) it follows that the combined variance V, of the sum or
difference of uncorrelated variables, is equal to the sum of the variances
of these variables, Vj :
V = V1 + V2 + ... + VJ .
(7.43)
Example. The voltage U1 is measured by subtracting the measured
voltage U3 from the measured voltage U2 : U1 = U2 − U3 . The correlation
between U2 and U3 is negligible. Compute the combined variance of the
evaluation U1, VC (U1).
U1 = U2 − U 3
2
2
 ∂U1 
 ∂U1 
VC (U1 ) = 
 ⋅ V(U 3 )
 ⋅ V(U2 ) + 
 ∂U2 
 ∂U 3 
∂ U1
∂ U1
= 1,
= −1
∂ U2
∂ U3
VC (U1 ) = V(U2 ) + V(U 3 )
The combined variance of an evaluation from uncorrelated component
values can be computed in the following steps. Examples are given in
Sections 7.6.1, 7.6.2, 7.6.3 and 7.6.4.
1) Write a function which gives an estimated value from component
variables.
2) For each of the component variables of the function from 1) find the
first and second partial derivative.
3) Compute the combined variance of the estimated value using (7.40).
4) Estimate the maximum possible variance of variance from 3) using
(7.41).
5) The combined variance from 3) is assumed as valid, considering
the nonlinearity of the function from 1), if its maximum possible variance
from 4) is eight or more times smaller than it.
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The variance of variance computed in 4) arises from an error of linear
approximation of a nonlinear function. The eight times larger variance
than the variance of this variance corresponds to the variance of variance
of an arithmetic mean of five values with the T-distribution. The relation
from 5) can be changed according to specific applications. See
6.22 Deviation of the standard deviation.
If the variance of variance is too large for a specific application and if
the component variables are uncorrelated, estimate the combined
variance by Monte Carlo simulation, or using (7.44) with a validation
check. In practice, there are rare cases of the too large variance of
variance, that is, with significant nonlinearity of the function.
If the component variables are correlated and if nonlinearity of the
function from 1) is not significant, use Monte Carlo simulation, or (7.49),
(7.50) and a procedure like the one above.
In the case with uncorrelated component variables and with
significantly nonlinear function y, the combined variance of the evaluation,
VC ( y ), is computed from the function (7.44).
2

 J J   2 2

J 
∂y 
∂y
∂y
∂3y 
 1

+
⋅
⋅
⋅
VC(y ) =    ⋅ V(xj ) +    ⋅ 
V(
x
)
V(
x
)
j
k 
 ∂xj 
 ∂xj ∂xk  ∂xj ∂x ∂x 2 



2
j =1 

j
k 



 j =1 k =1  
(7.44)
Partial derivatives of higher order are computed as follows.
∂
∂y
(y ) =
∂x j
∂x j
∂2 y
∂x j
2
=
∂ ∂y
(
)
∂x j ∂x j
∂2 y
∂ ∂y
∂ ∂y
(
)=
(
)
=
∂x j ∂xk
∂x j ∂xk
∂xk ∂x j
∂3 y
∂x j ∂xk
2
=
∂
∂ ∂y
(
(
))
∂x j ∂xk ∂xk
(7.45)
(7.46)
(7.47)
(7.48)
The accuracy of the variance given by (7.44) can be validated by
Monte Carlo simulation.
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185
In the case with correlated component variables and if nonlinearity of
the function y is insignificant, the combined variance of the evaluation,
VC ( y ), is computed from function (7.49). Standard covariances for the
variables x j and x k are denoted by V (x j , x k ). See 7.4 Experimental
standard covariance, as well as Examples 8.3.1, 9.12.3 and 9.12.4.

∂y
VC ( y ) =  


j =1  ∂x j

J
J
(
2

J −1

 ⋅ V( x j ) + 2 ⋅ 


j =1


J −1
)
=  c 2 ( x j ) ⋅ V( x j ) + 2 ⋅ 
j =1
J
 ∂y ∂y

 ∂x ⋅ ∂x ⋅ V( x j , xk )  =
j
k
k = j +1 


J
 ( c( x j ) ⋅ c( xk ) ⋅ V( x j , xk ))
j =1 k = j +1
(7.49)
The covariances V (x j , x k ) in (7.49) are for all pairs of J component
variables, without considering the order of the variables and excluding
pairs with two equal variables. The pairs are obtained by combinations
(without repetition) of second order from J elements, see
7.4 Experimental standard covariance.
Function (7.50) gives variance Vmax (VCy ), which is the maximum
possible variance of variance VC ( y ) from (7.49). If the variance of
variance, Vmax (VCy ), is several times smaller than the variance VC ( y ),
the nonlinearity of the function y is insignificant and (7.49) gives a valid
value. Partial derivatives of (7.50) are at the changing point
( x1 + r ⋅
V( x1 ) , x2 + r ⋅
V( x2 ) , ..., xJ + r ⋅
V( xJ ) )
which changes by varying the values of r from 0 to 1.
Vmax (VCy ) = max (
0< r <1
1 J  ∂ ∂y
 J−1 J  ∂ ∂y

⋅   ( ) ⋅ V(xj ) +    ( ) ⋅ V(xj , xk ) ) (7.50)
2 j =1  ∂xj ∂xj
x
x
∂
∂
 j =1 k= j +1  j k

Example. Let the evaluation y is obtained from three correlated
component variables a, b and c, then the combined variance of y, VC (y ),
is computed from (7.49) as follows.
y = y (a, b, c)
2
2
2
 ∂y 
 ∂y 
 ∂y 
VC ( y ) =   ⋅ V(a ) +   ⋅ V(b ) +   ⋅ V(c ) +
 ∂a 
 ∂b 
 ∂c 
∂y ∂y
∂y ∂y
 ∂y ∂y

+ 2⋅
⋅
⋅ V(a, b ) +
⋅
⋅ V(a, c ) +
⋅
⋅ V(b, c ) 
∂a ∂c
∂b ∂c
 ∂a ∂b

The rule (or law) of propagation of deviation, and the rule (or law) of
propagation of measurement uncertainty are expressed by function (7.49).
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8 Tests and functions fitting
8.1 Detection of outliers
Results of statistical analyses are valid only if the analyzed values are
without outliers. Therefore, if there is a significant possibility that outliers
will occur, their detection and rejection should be a mandatory part of the
statistical analysis.
A good practice is to consider as outliers and reject the values which
differ from an arithmetic mean of a sample, x , more than the triple
standard deviation of the sample, s.
According to this practice, detection of outliers should be carried out
by firstly rejecting the values which are supposed to be outliers. It is
followed by computing the mean x and the standard deviation s. Finally, it
should be determined whether the supposed outliers are out of the
interval x ± 3 · s, and if they are, the supposition is true, so x and s can be
assumed as valid. If among the supposed outliers they are certain values
which are not out of the interval, the procedure should be repeated
without rejection of these values. In the case of values with the normal
distribution, such practice is justified by the fact that the probability that a
valid value is out of x ± 3 · s is equal to 0.27 %. ● In the case of 30 values
with the T-distribution, the probability is 0.55 %; ● for 10 values, it is
1.5 %; ● for 5 values, it is 4.0 %.
The text continues on the next page.
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203
If the previous manner of detection of outliers is not sufficiently
reliable, the following procedure is recommended for a sample from a
population with the normal distribution. This test determines, with a
requested probability, whether a value with the largest difference from the
arithmetic mean is an outlier. The test is performed in the following steps.
1) Check if the sample is from a population with the normal
distribution. Checking can be done according to the instructions in 8.6
Estimating the type and parameters of a distribution. If the sample is from
the normal population, the test can be applied.
2) Let n be the number of values in the sample, including the tested
extreme value. Let P be the requested probability of the test result.
Determine a coverage factor k (ν, P) for the T-distribution with the
parameters: a degrees of freedom, ν, and the probability P. This factor
can be obtained from Table 6.3. The parameter ν is given by (8.1), and for
ν > 100, it can be assumed that ν = ∞.
3) Determine the truth (that is, validity) of the inequality (8.2). The
extreme value is denoted by xextreme. An arithmetic mean of the sample
without the extreme value is x ', and the standard deviation of the mean
is s x '. If the inequality is true, with the probability P or larger, xextreme is
an outlier.
ν = n−2
xextreme − x ' > k ( ν, P ) ⋅ s x '
(8.1)
(8.2)
4) The extreme value is rejected if it is determined to be an outlier.
The test is then repeated for each of the following extreme values, until
finding a value which is not an outlier.
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8.1.1 Example
The following values were obtained as the results of measurements of
pH, and then ordered in the ascending array: 3.75; 3.86; 4.19; 4.19; 4.24;
4.37; 4.37; 4.50; 4.53; 4.53; 4.55; 4.58; 4.76; 4.77 and 5.23. Determine
whether the value of 5.23 is an outlier, which can be supposed on the
basis of the histogram in Figure 8.1.
Solution
1) The population of the results is with the normal distribution because
errors of the results are a consequence of a large number of independent
individual errors with a small share. (See Section 6.12.)
2) The number of the results is n = 15.
From (8.1) we obtain that ν = n − 2 = 15 − 2 = 13.
The requested probability of the test is P = 0.95.
From Table 6.3 we determine the coverage factor
k (ν, P ) = k (13, 0.95) = 2.16.
3) The tested extreme value is xextreme = 5.23.
The arithmetic mean of the array without xextreme is x ' = 4.371.
The deviation of the arithmetic mean is s x ' = 0.302.
Determine the truth of the inequality (8.2):
xextreme − x ' = 5.23 − 4.371 = 0.859,
k ( ν, P ) ⋅ s x ' = 2.16 ⋅ 0.302 = 0.652,
since 0.859 > 0.652, the inequality (8.2) is true. It can be concluded that
with a probability larger than 0.95, the result 5.23 is an outlier.
Figure 8.1. The histogram of the measurement results of pH.
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8.2 Fitting
Fitting is defining a function, or a particular value, on the basis of a
sample, so that it meets specific requirements of the goodness of fit to
the sample.
Fitting is most often performed by a variant of the method of least
squares. See Section 8.3 and Example 9.12.2 (from the GUM).
Regression analysis (that is, analysis of differences) is used to find a
function which describes a relationship between correlated values.
Parameters of this regression function (or smoothing function, or
regression model) are determined by fitting. The regression function is
used to estimate or predict values.
Linear regression function is a linear function which describes a
relationship between correlated values.
[Ivković] PP 49, PP 50 [Njegić] PP 182
8.3 Method of the least sum of squares
Method of least squares, LSM, is a procedure of fitting of a function,
or a particular value, so that the sum of squares of differences between
the determined values and input values in a sample is minimized.
The method of least squares gives the maximum likelihood values
when input values are with a symmetrically decreasing distribution.
An example of fitting of a particular value by the method of least
squares is computation of the arithmetic mean.
The example of fitting of a function by the method of least squares is
given below.
If we have n pairs of correlated values (xi , yi ), i = 1, 2, ..., n, their
mutual relationship can be described by a function, more or less
accurately. A linear function is commonly used:
y'(x) = a + b · x .
(8.3)
The parameters a and b of the linear function can be determined by
the method of least squares, that is, by minimizing the sum S which is
given by formula (8.4). S is the sum of squares of differences between the
values given by the function y'(xi ) and the input values yi .
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S=
n
n
i =1
i =1
 [ y'( xi ) − y i ] 2 =  [(a + b ⋅ xi ) − y i ] 2
(8.4)
The sum S given by (8.4) is minimized when the system of equations
given by (8.5) is fulfilled. Therefore, the parameters of the linear function
are determined by solving the system for a and b. The system (8.5)
always has a unique solution.







n
∂Z
=
∂a
 {2 ⋅ [(a + b ⋅ xi ) − y i ]}
=0
i =1
n
∂Z
=
∂b
 {2 ⋅ [(a + b ⋅ xi ) − y i ] ⋅ xi } = 0
(8.5)
i =1
The system (8.5) can be solved by using the following quantities.
n
 xi
x =
i =1
n
the arithmetic mean of xi
(8.6)
the arithmetic mean of yi
(8.7)
n
 yi
y =
i =1
n
n
x2 =
 xi 2
i =1
n
(8.8)
n
 ( xi ⋅ y i )
x⋅y =
i =1
n
Using these quantities, the system (8.5) can be rewritten as:
(8.9)
 y = a + b ⋅ x
(8.10)

2
 x ⋅ y = a ⋅ x + b ⋅ x .
The parameters a and b are obtained by solving the system (8.10):

x ⋅y − x ⋅y
b=

x2 − x2
(8.11)

 a = y − b⋅x .
See Example 8.3.1 and Example 9.12.2 (from the GUM). In both
examples, the input values are equivalent and the solutions are the same.
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The following model describes prediction of the value y by using the
linear function y'(x):
y(x) = y'(x) + ε(x) = a + b · x + ε(x).
(8.12)
The summand ε(x) represents the differences between the input values
and the values predicted by the function y'(x), that is, negative of an error
of prediction by the function y'(x).
If there are n input values and assuming that the differences ε(x) have
the normal distribution, the standard variance of the differences, Vε, is given
by (8.13). The degrees of freedom of the variance, νVε , is given by (8.14).
n
 (a + b ⋅ xi − y i ) 2
Vε =
i =1
n−2
νVε = n − 2
(8.13)
(8.14)
The parameters a and b have the standard variances Va and Vb given
by (8.15) and (8.16). Their standard covariance, V (a, b), is given by (8.17).
Vε
Va =
(8.15)
x2
)
n ⋅ (1 −
x2
Vb =
Vε
n ⋅ ( x 2 − x 2)
V(a, b ) = −
x ⋅ Vε
n ⋅ ( x 2 − x 2)
(8.16)
(8.17)
A value predicted by the linear function y'(x) has the standard variance
Vy' given by function (8.18) and derived in the normal manner explained
in 7.6 Combined variance. The degrees of freedom of the variance, νVy'
is given by (8.19).
Vy' ( x ) = Va + x 2 ⋅ Vb + 2 ⋅ x ⋅ V(a, b )
(8.18)
νVy' = n − 2
(8.19)
For a given x, it is possible to determine the interval which has a
requested probability P that within it be a value from the population of
input values. Such probability and interval are the level and interval of
confidence in the function of x.
For the requested probability P, and for n input values, assuming that
ν = n − 2 , from Table 6.3 for the T-distribution, we obtain the
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corresponding coverage factor k. A half-width of the confidence interval,
∆(x), is given by (8.20), while the confidence interval is given by (8.21).
∆ (x) = k ⋅
Vy' (x )
(8.20)
[ y'(x) − ∆(x), y'(x) + ∆(x) ]
(8.21)
[GUM] H.3.4 [Panchenko] 29.1, 29.2, 30.1 [Ivković] PP 49, PP 50, II.6 [Njegić] PP 64
[Pantić] PP 134 [Anscombe] [Britannica] Statistics [Perović, communication]
8.3.1 Example
Using a linear function to approximate the calibration function of a
measuring device. Then, for the correction addend predicted by this
calibration function, estimate the standard deviation and the intervals of
confidence. From Table 8.1 use the reference values of the standard and the
indications of the measuring device. The reference values of the standard
have negligible errors. Neglect the bias of the standard deviations because
the number of input pairs is large. The experimental correction addends
have the normal distribution, so the method of least squares can be used.
Solution
Using the values xi and yi from Table 8.1 as the input pairs, the
requested parameters is computed by using the method of least squares.
n = 11 from Table 8.1, the number of the input pairs (xi , yi )
n
 xi
x=
i =1
= 4.008
n
from (8.6), the arithmetic mean of xi
n
 yi
y=
i =1
= − 0.1625
n
from (8.7), the arithmetic mean of yi
n
x2 =
 xi 2
i =1
= 18.56
n
from (8.8)
n
 ( xi ⋅ y i )
x⋅y =
i =1
n
= − 0.6458 from (8.9)
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b=
x⋅y − x ⋅y
x2 − x2
= 0.002183
from (8.11), the parameter of function
from (8.11), the parameter of function
a = y − b ⋅ x = −0.1712
y'(x ) = a + b ⋅ x = −0.1712 + 0.002183 ⋅ x
the calibration function
n
 (a + b ⋅ x i − y i ) 2
i =1
= 1,223 ⋅ 10 −5
n−2
from (8.13), the standard variance for ε(x)
Vε =
Vε
Va =
x2
n ⋅ (1 −
Vb =
x2
= 8,281⋅ 10 −6
)
Vε
n ⋅( x
2
2
−x )
V (a, b ) = −
from (8.15), the standard variance for a
= 4,461⋅ 10 −7
x ⋅ Vε
2
2
from (8.16), the standard variance for b
= −1,788 ⋅ 10 −6
n ⋅( x − x )
from (8.17), the standard covariance of a and b
Vy' ( x ) = Va + x 2 ⋅Vb + 2 ⋅ x ⋅ V(a, b) =
= 8,281 ⋅10 −6 + x 2 ⋅ 4,461 ⋅ 10−7 + 2 ⋅ x ⋅ ( −1,7883 ⋅ 10−6 )
from (8.18), the standard variance for the predicted correction addend y'(x)
sy' ( x ) =
Vy' ( x )
the standard deviation for the predicted correction addend
νsy ' = νVy ' = n − 2 = 9
the degrees of freedom of the variance sy' (x)
∆ ( x ) = k ⋅ s y' ( x )
from (8.20), the half-width of the confidence interval for the requested
probability P.
[GUM] H.3
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Table 8.1 With Example 8.3.1: values for obtaining the calibration function
Reading
number
i
Input values
1
2
3
4
5
6
7
8
9
10
11
Reference
value of
standard
Ri
1.350
1.843
2.346
2.844
3.343
3.834
4.357
4.845
5.344
5.849
6.351
Indication
of
measuring
device
xi
1.521
2.012
2.512
3.003
3.507
3.999
4.513
5.002
5.503
6.010
6.511
Experimental
correction
addend
yi = Ri − xi
−0.171
−0.169
−0.166
−0.159
−0.164
−0.165
−0.156
−0.157
−0.159
−0.161
−0.160
Prediction of obtained
function
Error of
Predicted
predicted
correction
correction
addend
addend
y'(xi )
−εi = y'(xi ) − yi
−0.16788
−0.16681
−0.16572
−0.16465
−0.16355
−0.16248
−0.16135
−0.16029
−0.15919
−0.15809
−0.15699
0.00312
0.00219
0.00028
−0.00565
0.00045
0.00252
−0.00535
−0.00329
−0.00019
0.00291
0.00301
Figure 8.2. With Example 8.3.1: input pairs (xi , yi ) and the calibration
function, y'(x), with the intervals of confidence for different confidence
levels, P.
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8.4 P-value
Goodness-of-fit evaluation is an indicator of mutual closeness of
functions or values.
Let the unfit evaluation of a sample and an expected function, T,
gives a function with the distribution t(x). Then the P-value of the
goodness of fit of the sample and the expected function, q, is given by the
following expression (8.22) and represented in the following diagram. For
an example, see 8.7 Pearson chi-square test for the goodness of fit of
distributions.
q=
∞

T
t (x) ⋅ d x = 1 −
T

0
t (x) ⋅ d x
(8.22)
As the unfit evaluation T decreases, the corresponding P-value
increases, so the P-value is an indicator of the probability of the fit of the
sample and the expected function (a larger P-value corresponds to a
greater fit).
The P-value is an indicator of the probability of a fit, but it is neither the
probability of a fit, nor the probability that a hypothesis is true, nor the
probability of wrongly rejecting a hypothesis. Also, the probability of a fit is
not proportional to the P-value (except in special cases).
Contrary to the statement in the second paragraph, in some cases the
fit evaluation is taken for the evaluation T, then the P-value is an indicator
of the probability of an unfit (a larger P-value corresponds to a greater
unfit).
Computation of the P-value is schematized and relatively simple.
Therefore, the P-value is used in a wide variety of goodness-of-fit tests.
Ronald Fisher advocated using the P-value, and it is widely accepted
in many sciences. However, it is very often misunderstood and wrongly
used.
[Berger] [Sellke]
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9 Measurement uncertainty
9.1 Measurement result and its uncertainty
Measurement uncertainty (uncertainty, or uncertainty of
measurement) is a parameter of measurement result that describes the
accuracy of the result by an indicator of dispersion of values that can be
reasonably taken as the measurement result.
A measurement result can be used to compute the level and interval
of confidence, if the result is the best estimated value of a measurand,
and if that result is associated with the measurement uncertainty which is
the standard deviation of that best estimated value, as well as the
degrees of freedom of that deviation.
The best estimated value of a measurand, the measurement
uncertainty of the evaluation, and the degrees of freedom of this
uncertainty, can be used as component values in evaluating the
uncertainty and degrees of freedom of another value in which the first
value is used. This allows traceability.
In order to establish uniform estimating and expressing of
measurement results, in 1993 ISO published the first edition of Guide to
the expression of uncertainty in measurement, GUM. The Guide is
applicable at various levels of accuracy, from a shop floor to the
fundamental sciences. The basis of the Guide is the internationally
adopted Recommendation of CIPM (from 1981) which accepts the
Recommendation of BIPM Working Group on the Statement of
Uncertainties (from 1980). Today, publishing and correction of the Guide
are within the competence of the JCGM. The JCGM member
organizations are: BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and
OIML.
The application of measurement uncertainty and traceability are
mandatory for accredited laboratories, as well as organizations that have
the quality management system according to ISO 9001:2008.
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The summarized overview of the procedures given in the GUM for
estimating of a measurement result and its measurement uncertainty is
as follows.
● A measurement result is obtained based on the results of repeated
measurements by determining the best estimated value of a measurand.
● For this best estimated value, the standard deviation, called the
standard measurement uncertainty, is determined.
● For this standard uncertainty, the degrees of freedom is determined.
These evaluations can be obtained by using classical statistical
methods (including convolution) which are regarded as the primary
methods, or where appropriate, by using Monte Carlo simulation.
Standard measurement uncertainty (standard uncertainty) is the
standard deviation of a measurement result which is the best estimated
value of a measurand. Depending on the method for estimating the
uncertainties, they are grouped into two categories: ● the standard
uncertainties which are computed by statistical analysis of results of
repeated measurements, type A; and ● the standard uncertainties which
are estimated by statistical analysis of reliable information, type B.
The squared standard uncertainty, that is, the standard variance, is
the unbiased evaluation of the squared deviation of a population of
measurement results, that is, the variance of a population of these
results. See 7.2 Experimental standard variance.
The standard deviation is the biased evaluation of the deviation of a
population of measurement results. The bias of the uncertainty increases
as its degrees of freedom decreases. In the case of the normal
distribution, the relative error of the evaluation is in the interval from 0 %
to +8.5 % for the degrees of freedom from ∞ to 3. See 6.20 Experimental
standard deviation. When the standard uncertainty with a small degrees
of freedom is taken as the evaluation of the deviation of the population of
measurement results, the bias of this uncertainty should be corrected.
The standard uncertainty of the best estimated value of a measurand
arises only from random effects and uncertainties of corrections.
Component standard uncertainty (or standard uncertainty
component, or component of standard uncertainty) is a standard
uncertainty of a component value.
[GUM] 2.2.3, 0, PP v, Foreword, Annex A, 3.2, 4.1.4, 0.7 2) 3) 4), 2.3.1, 4.2.3, 4.2.6,
7.2.1 d), 4.1.5, 4.1.6, 2.3.2, 2.3.3, 8, C.2.21 [VIM 2] 3.9 [VIM 3] 2.26 [IEC 17025] 5.4.6.1,
5.4.6.2, 5.10.4.1 [ILAC] 4.8 [ILAC, OIML] 1., 2. [NVLAP] 5.4.6, Annex B.1 [ISO 10012]
7.3 [ISO 9001] 7.6 [GUM-S1] Introduction, 1
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9.2 Type A standard measurement uncertainty
Type A standard measurement uncertainty (type A standard
uncertainty, or type A standard uncertainty of the result of a
measurement) is the standard deviation, of the best estimated value of a
measurand, computed by statistical analysis of results of repeated
measurements. For this uncertainty, we should always also give its
degrees of freedom. Unless otherwise indicated, it is understood that the
best estimated value of a measurand is the maximum likelihood value of
the measurand and that this evaluation has the normal distribution.
Type A standard uncertainty is denoted by uA , and its degrees of
freedom by νA .
The maximum likelihood value of a measurand is most often
computed as the experimental arithmetic mean of results of repeated
measurements. This mean must be corrected for all significant systematic
errors. For this corrected mean, the deviation of the arithmetic mean, sx ,
is computed according to 6.21 Deviation of the arithmetic mean, and then
it is taken as the type A standard uncertainty, uA :
u A = sx .
(9.1)
The degrees of freedom of the type A standard uncertainty is
computed according to 6.24 Degrees of freedom. When from n results of
repeated measurements we compute the experimental arithmetic mean
which is then used to compute the deviation of the arithmetic mean, the
degrees of freedom of this deviation, as well as, of this uncertainty, is:
νA = n – 1.
(9.2)
See Section 6.24.1, example d).
[GUM] A.2, 0.7, 8, 3.1.2, 3.2.4, 4.2.1, 4.2.3, 4.2.6, 3.2.3, 3.4.4, 3.3.5
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9.3 Type B standard measurement uncertainty
Type B standard measurement uncertainty (type B standard
uncertainty) is the standard deviation, of the best estimated value of a
measurand, estimated by statistical analysis of reliable information. For
this uncertainty, we should also give its degrees of freedom if it is deemed
useful. Unless otherwise indicated, it is understood that the best
estimated value of a measurand is the maximum likelihood value of the
measurand and that this evaluation has the normal distribution.
Type B standard uncertainty is denoted by uB , and its degrees of
freedom by νB .
The type B standard uncertainty is estimated if there is not enough
measurement data to compute the type A standard uncertainty.
The best estimated value of a measurand for which we give the type B
standard uncertainty is often obtained by taking a single measurement
result or by estimate on the basis of a small amount of information.
The type B standard uncertainty is estimated on the basis of an
approximated probability density function obtained by analysis of the
probability that an event will occur. The evaluations of distribution,
deviation, and degrees of freedom of deviation, must be estimated by
scientific judgment based on available information. Among other means,
information can be obtained from the following sources:
● previous measurement results
● experience or general knowledge of the properties of materials and
devices for measurement
● specifications provided by manufacturers of materials and
devices for measurement
● calibration reports or other certificates
● handbooks with measurement uncertainties assigned to the given
values.
Estimate of the type B standard uncertainty needs to be comprehensive
so that the obtained evaluation is approximately as accurate as the type A
standard uncertainty. This aim is easily achieved when the type A
standard uncertainty is estimated for an evaluation obtained from a small
number of averaged results. Section 6.22 Deviation of the standard
deviation, Table 6.1, shows that the deviation of the standard deviation of
an arithmetic mean is not negligible in practical cases.
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The degrees of freedom of the standard uncertainty of type B can be
computed according to: ● 6.24 Degrees of freedom; ● 6.25 Effective degrees
of freedom; ● 6.28 Standard deviation and its degrees of freedom from the
level and interval of confidence; or ● 6.26 Degrees of freedom from the
deviation of the standard deviation of the arithmetic mean. If it is considered
that the type B standard uncertainty is absolutely reliable, we assign to it
the infinite degrees of freedom. See the following Examples 9.3.1 and 9.3.2.
[GUM] 0.7, 8, 3.1.2, 3.2.4, 4.3, 7.2.1 d), 7.2.7 c), G.4.2, 3.3.5, 4.3.2, E.4.3, G.4.2, G.6.4
9.3.1 Example
An ambient temperature is measured by a digital thermometer placed
in an appropriate position. The thermometer is from a reliable
manufacturer, with the resolution of 0.1 ºC. However, information about
its accuracy is not available. The ambient temperature is achieved
naturally (without using air conditioning).
For the two cases given below, estimate the environment temperature,
its standard uncertainty and the degrees of freedom of the uncertainty.
a) The thermometer constantly gives 25.0 ºC.
b) The thermometer alternatively gives 25.0 ºC and 25.1 ºC, in
approximately equal durations.
Solutions
a) The evaluation of the environment temperature ta is given by:
ta = thermometer_indication − systematic_error − absolute_error_of_resolution.
The furthermore evaluations are as follows. The systematic error has
the normal distribution and its maximum likelihood arithmetic mean is
equal to zero. The absolute error due to resolution has the uniform
distribution and has the maximum likelihood value equal to zero.
According to 6.12 Normal distribution, the resulting distribution of the
estimated temperature is approximately normal. Therefore, we conclude
that the maximum likelihood environment temperature is:
ta = 25.0 − 0 − 0 = 25.0 ºC.
Low-accuracy thermometers normally have a maximum systematic
error equal to their resolution. This maximum error probably matches the
double standard deviation, so the estimated standard deviation of the
systematic error is:
s1 = 0.1 ºC / 2 = 0.050 ºC.
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The standard deviation of the error due to resolution is computed from
(6.95):
s2 =
0.1 °C / 2
3
= 0.029 °C.
The standard uncertainty of the estimated environment temperature is
computed from (6.41):
uB a = s a =
s 12 + s 2 2 =
0.050 2 + 0.029 2 = 0.058 °C.
The thermometer constantly indicates 25.0 ºC, and this is equivalent to
an infinite number of results, n, from which the evaluations are obtained.
The degrees of freedom of the uncertainty uB a is computed from (6.64):
νB a = n − 1 = ∞ − 1 = ∞.
b) The estimated environment temperature tb , is given by:
tb = thermometer_indication − systematic_error − absolute_error_of_evaluation_of_indication.
The furthermore evaluations are as follows. The systematic error and
the absolute error of evaluation of indication have the normal distributions,
and the arithmetic means are equal to zero. According to 6.12 Normal
distribution, the resulting distribution of the estimated temperature is
approximately normal.
Considering the principle of operation of the analog-to-digital
converter, the thermometer indication is the arithmetic mean of the two
values which the thermometer alternatively gives:
25.0 °C + 25.1 °C
= 25.05 °C.
2
So the maximum likelihood environment temperature is:
thermometer_indication =
tb = 25.05 − 0 − 0 = 25.05 ºC.
(Like the case a.) Low-accuracy thermometers normally have a
magnitude of the maximum systematic error equal to their resolution. This
maximum error probably matches the double standard deviation, so the
estimated standard deviation of the systematic error is:
s3 = 0.1 ºC / 2 = 0.050 ºC.
The thermometer alternatively gives two values that are seemingly of
equal durations, probably until the durations start to differ more than
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20 % (0.2). Therefore, the absolute error of evaluation of indication has
the approximate uniform distribution with the half-width equal to
(0.2 / 2) · resolution = 0.01 ºC, so the standard deviation of this error is
computed from (6.95):
s4 =
0.01 °C
3
= 0.0058 °C.
The standard uncertainty of the estimated environment temperature is
computed from (6.41):
uB b = s b =
s 32 + s 42 =
0.050 2 + 0.0058 2 = 0.050 °C.
The thermometer constantly indicates 25.05 ºC, and this is equivalent to
an infinite number of results, n, from which the evaluations are obtained.
The degrees of freedom of the uncertainty uB b is computed from (6.64):
νB b = n − 1 = ∞ − 1 = ∞.
9.3.2 Example
It is estimated that out of 12 measurement results of length of one-type
elements, half of the results (which are not individually known) have
values in the interval from 10.07 mm to 10.15 mm. The results are from a
population with the normal distribution (because errors of the results are a
consequence of a large number of independent individual errors with a
small share). The evaluation of the interval has a negligible systematic error.
Estimate: 1) the most frequent length of the elements, 2) the standard
uncertainty of the evaluation of this length and 3) the degrees of freedom
of this uncertainty.
Solution
1) The most frequent length is the mode. The estimated confidence
limits are a = 10.07 mm and b = 10.15 mm. The results of measurement
of the length have the T-distribution, so the value of the mode, l, is in the
midpoint of the interval from a to b:
l=
a+b
10.07 mm + 10.15 mm
=
= 10.11 mm.
2
2
3) The estimate of the standard uncertainty is carried out from total
n = 12 results. Based on this, we compute the degrees of freedom of this
uncertainty from (6.80):
νB = n − 1 = 12 − 1 = 11.
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9.5 Expanded standard measurement uncertainty
Expanded standard measurement uncertainty (expanded
standard uncertainty, expanded uncertainty) U is the combined
standard uncertainty uC multiplied by a coverage factor k :
U = k · uC .
(9.3)
The combined standard uncertainty is multiplied by the coverage
factor in order to get the interval given by (9.4) in which the value of a
measurand is with the required probability.
MEASUREMENT_RESULT ± U
(9.4)
The interval is the confidence interval, and the probability is the
confidence level. See 6.27 Confidence level and confidence interval.
Typically, the coverage factor will be in the interval from 2 to 3, which
in the case of the normal distribution gives the confidence level from
95.5 % to 99.7 %. See Table 6.2.
In the case of the T-distribution, the coverage factor for the required
confidence level can be determined from Table 6.3, which gives the
relations among the confidence level, degrees of freedom, and coverage
factor. If the degrees of freedom is not an integer, the recommendation of
[GUM] is to truncate its decimals.
In the case of an asymmetric distribution, in order to determine the
coverage factor, we can use that real distribution to obtain an accurate
confidence interval with the requested confidence level. Then, the
confidence interval can be defined using two coverage factors, k1 and k2 :
[MEASUREMENT_RESULT – k1 · uC , MEASUREMENT_RESULT + k2 · uC ]. (9.5)
However, based on the central limit theorem, the asymmetric distribution
can often be approximated by the normal distribution with the arithmetic
mean and deviation equal to the maximum likelihood value and deviation
of the real distribution.
The expanded uncertainty is given in some commercial, industrial and
legal documents. In this case, a report of a measurement result gives the
expanded standard uncertainty, confidence level and coverage factor,
and can also give a type of distribution of the results.
[GUM] 6, 0.7 5), G.1.4, G.6
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9.6 Relative standard measurement uncertainties
Relative standard measurement uncertainty (relative standard
uncertainty) uR( y) is equal to the quotient of the standard uncertainty of
a measurement result, u( y), and the absolute value of the measurement
result, │y │:
u R (y ) =
u( y )
.
y
(9.6)
The symbols of relative standard uncertainties are:
● uR , relative standard uncertainty
● uA R , relative type A standard uncertainty
● uB R , relative type B standard uncertainty
● uC R , relative combined standard uncertainty
● UR , relative expanded standard uncertainty.
[GUM] J
9.7 Evaluations from non-normally distributed results
Table 9.1 gives an overview of common distributions of measurement
results and the corresponding evaluations of the maximum likelihood
values of a measurand, as well as the type A standard uncertainties of
these maximum likelihood values. The formulas in the table are valid only
when measurement results are corrected for all significant systematic
errors.
A non-normal distribution of results can often be approximated by the
normal distribution with the arithmetic mean and deviation equal to the
maximum likelihood value and deviation of the real distribution (based on
the central limit theorem). See Example 6.42.2 Example of estimate of a
length of gauge block and its statistical parameters.
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Table 9.1 The common distributions of n results and the evaluations
from the results [GUM] 4.2.3, 4.4.5, 4.4.6, F.2.4.4 [NASA] 3.2.5
[Yuan] [Panchenko] 4.1 [Njegić] 3.1.
The distribution of
measurement results
The maximum likelihood
value of a measurand 9)
Lognormal distribution
from (6.16) and (6.17):
The type A standard
uncertainty of the
maximum likelihood value
xmode =
= most_often_value =
= x
from (6.56) and (6.49):
max L ( x )
Normal distribution
u=
sx
=
n
n
 ( xi − x )
=
T-distribution
2
i =1
n ⋅ (n − 1)
from (6.30):
x =
Uniform distribution
Triangular distribution
U-distribution
9)
x1 + x2 + ... + xn
n
Note
from (6.56) and (6.95):
In the case of the
uniform distribution and
U-distribution, x is the
best estimated value of a
measurand according to
the criterion of the
smallest absolute value
of the maximum possible
error.
u=
from (6.56) and (6.100):
u=
a
6⋅n
from (6.56) and (6.105):
u=
250
a
3⋅n
a
2⋅n
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9.11 Procedure for estimating the measurement
result and associated data
According to [GUM], the steps that should be followed for estimating
the measurement result and associated data may be summarized as
follows.
1) Write the functional relationship, symbolically denoted as f, which
gives the value of measurand, Y, from its component values Xi :
Y = f ( X1 , X2 , …, XN ).
(9.8)
The function f should contain all quantities that significantly increase the
uncertainty of the measurement result. See Examples 2.14.1, 4.7.1 and
9.13.2.
The functional relationship can be extremely complex or can never be
written down explicitly. In such case, it should be given as a computer
program, or an algorithm, that can be computed numerically.
2) Determine the best estimated values xi of the component values Xi ;
preferably on the basis of measurement results, or by other means. See
6.9 Best estimate.
3) Determine the standard uncertainty u( xi ) of each estimated value xi .
Depending on the kind of available data, the standard uncertainty is
estimated according to: 9.2 Type A standard measurement uncertainty;
9.3 Type B standard measurement uncertainty; or according to this same
procedure.
4) Determine the standard covariances of any component values that
are correlated. See 7.4 Experimental standard covariance.
5) Determine the degrees of freedom, νi , of each standard uncertainty
u( xi ). See 6.24 Degrees of freedom; 6.25 Effective degrees of freedom;
6.26 Degrees of freedom from the deviation of the standard deviation of
the arithmetic mean; or 6.28 Standard deviation and its degrees of
freedom from the level and interval of confidence and sample size.
6) Compute the measurement result, y, of the value of measurand, Y,
from the functional relationship symbolically denoted as f in (9.8), or from
other relationship. For this, use the evaluations xi , instead of the
component values Xi . Estimate the distribution of the measurement result
according to 8.6 Estimating the type and parameters of a distribution.
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7) Compute the combined standard uncertainty u C ( y), of the
measurement result, y, from the standard uncertainties and standard
covariances of the estimated values xi . See 9.4 Combined standard
measurement uncertainty. If the measurement determines several
measurement results from the same estimated values xi , also compute
the covariances of these results. See 7.4 Experimental standard
covariance, and also Example 9.13 Resistance, reactance and
impedance, computed from the same component measurement results.
If it is necessary to give an expanded standard uncertainty U on the
basis of the required level of confidence, P, select coverage factor k, then
compute the uncertainty U. See 9.5 Expanded standard measurement
uncertainty.
8) Estimate the degrees of freedom, νC , of the combined standard
uncertainty u C ( y). See 6.25 Effective degrees of freedom; 6.24 Degrees
of freedom; 6.26 Degrees of freedom from the deviation of the standard
deviation of the arithmetic mean; or 6.28 Standard deviation and its
degrees of freedom from the level and interval of confidence and sample
size.
[GUM] 0.7, 8, 7.2.7 Note, 4.1.2, 4.1.3, 5.2.3, 5.2.5, 4.2.6, 7.2.1 d)
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9.14 Metrological traceability
Metrological traceability (traceability) is the distinction of a
measurement result which means that its specification of combined
standard uncertainty includes uncertainties of all its component values
beginning from a source of traceability.
The combined standard uncertainty of a traceable value is computed
based on a relation between component values and the traceable value,
normally, as given in 9.4 Combined standard measurement uncertainty.
The value is traceable if its combined standard uncertainty is
computed based on standard uncertainties of all its component values
which are traceable. For such value, it is said that it is traceable to
quantities of a specified source of traceability, to which its component
values are traceable.
Example. A meter is calibrated by a measurement standard whose
value is traceable to a national measurement standard. The result of this
meter, with its combined standard uncertainty, is therefore, traceable to
the national measurement standard.
The combined standard uncertainty of the traceable value is due to the
uncertainties of all its component values beginning from the source of
traceability.
The combined standard uncertainty of the traceable value enables
accurate computation of the level and interval of confidence. See
9.1 Measurement result and its uncertainty.
Traceability chain is an unbroken sequence of calibrations and
traceable results of these calibrations.
The traceability chain is defined through a calibration hierarchy.
Source of traceability is a quantity that is at the beginning of the
traceability chain. The source of traceability has the least measurement
uncertainty in the traceability chain. The measurement uncertainties of
traceable values increase by starting from the source of traceability, see
also 5.11 Calibration.
For the traceable value, the source of traceability must be declared.
“Traceable to SI” means “metrologically traceable to SI units”.
A laboratory can choose the source of traceability which is suitable for
its needs.
[NIST] [VIM 3] 2.41, 2.42, 2.43. [VIM 2] 6.10.
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References
[Allan] David W. Allan; Should the classical variance be used as a
basic measure in standards metrology; IEEE Transactions on
Instrumentation and Measurement, Vol. 1M-36, No. 2, 1987.
[Anscombe] F. J. Anscombe; Graphs in statistical analysis; The
American Statistician, Vol. 27, No. 1, 1973.
[Aslan] B. Aslan, G. Zech; Comparison of different goodness-of-fit
tests; In: M. R. Whalley, L. Lyons (editors), Proceedings of Conference on
Advanced Statistical Techniques in Particle Physics, Durham, 2002.
[Audoin] Claude Audoin, Bernard Guinot; The Measurement of Time:
Time, Frequency and the Atomic Clock; Cambridge University Press,
Cambridge, 2001.
[Berger] James O. Berger; Could Fisher, Jeffreys and Neyman have
agreed on testing?; Duke University, Durham, 2002.
[BIPM] Internet site of the BIPM: http://www.bipm.org/; 2000 to 2019.
[Box] G. E. P. Box, D. R. Cox; An Analysis of Transformations;
Journal of the Royal Statistical Society, Series B (Methodological), Vol.
26, No. 2, 1964.
[Box 2005] George E. P. Box, Stuart Hunter, William G. Hunter;
Statistics for experimenters: design, innovation, and discovery (2nd ed.);
John Wiley & Sons, 2005.
[Britannica] Britannica Encyclopaedia (CD ROM); 1997.
[Bronshtein] I. N. Bronshtein et al; Handbook of Mathematics;
Springer-Verlag, Berlin Heidelberg, 2015.
[Chamberlain] Richard Chamberlain; Computer systems validation
for the pharmaceutical and medical device industries (2nd ed.); Alaren
Press, Libertyville, 1994.
[CODATA] Peter J. Mohr, Barry N. Taylor, David B. Newell; CODATA
recommended values of the fundamental physical constants: 2014; Rev.
Mod. Phys., Vol. 88, No. 3, 2016.
[Cohen] Morris R. Cohen, Ernest Nagel; An introduction to logic and
scientific method; Mr. Sunil Sachdev (Allied Publishers Limited), New
Delhi, 1998.
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Goran Kostić
Metrology Companion
the new SI e-book on concepts and computations in measurements
1st English e-edition, 2nd English updated edition in total
Leskovac, 2019
●
English editors
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Technology, University of Novi Sad, Novi Sad, Serbia
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University of Novi Sad, Novi Sad, Serbia
●
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●
Cover design
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Publisher
Goran Kostić, Jovana Cvijića 5, RS -16 000 Leskovac, Serbia
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Editor
Goran Kostić
●
Production
Produced by the publisher.
Circulation, 1000.
Release date, August 2019.
●
Copyright
© 2018, 2019 by the publisher, all rights reserved.
●
ISBN 978-86-900284-2-9
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References
ISBN 978-86-900284-2-9
10 11 12 13 14 15 16 17 18 19 20 21
[...]
9
9 Measurement uncertainty
8
8 Tests and functions fitting
7
7 Variances
In front of you is the Companion with
concisely described subjects that are
indispensable in all measurements.
The Companion consists of about
minimal and sufficient set of concepts
and methods required to compute a
measurement result and its
measurement uncertainty. The text is
in the form that enables direct
applications in computations, manual
or spreadsheet, and in writing
software.
6
6 Outlines of the statistics
5
5 Measurement standards and
calibrations
The essence of measurement is the
same in all areas. The purpose of
this publication is to be a companion
to people dealing with
measurements, that is experiments
or observations, from production to
research, from physics and
chemistry to biology and medicine.
The publication is also intended for
students.
4
4 Measurement results, accuracy
and errors
Goran Kostić
3
3 Devices for measurement
2
2 Measurements
From a shop floor to the fundamental
sciences, statistical analysis of
results of repeated measurements is
required. The purpose of the
analysis is to determine the value of
a measured quantity, as well as the
accuracy of the determined value. It
is requested that accuracy is
described by the standard
measurement uncertainty to enable
traceability and ensure computing of
the confidence level and the
confidence interval.
1
1 Quantities and units of
measurement
From the Preface
Goran Kostić
Chapters
THE NEW SI E-BOOK ON CONCEPTS AND
COMPUTATIONS IN MEASUREMENTS