Evolution of Philosophy and Description of Measurement
(Preliminary Rationale for VIM3)
Speaker/Author: Charles Ehrlich1
International Legal Metrology Group
Weights and Measures Division
Technology Services
National Institute of Standards and Technology
Gaithersburg, MD, 20899-2600, U.S.A.
(301) 975-4834 / charles.ehrlich@nist.gov
Wolfgang Wöger2
Drosselweg 1, D-50735
Cologne, Germany
wolfgang.woeger@ptb.de
René Dybkaer
Department of Standardization in Laboratory Medicine
H:S Frederiksberg Hospital
Copenhagen University Hospital, Denmark
Abstract
Different approaches to the philosophy and description of measurement have evolved
over time, and are still evolving. There is not always a clear demarcation between
approaches, but rather a blending of concepts and terminologies from one approach to
another. This sometimes causes confusion when trying to ascertain the objective of
measurement in the different approaches, since the same term may be used to describe
different concepts in the different approaches. Important examples include the concepts
and terms “value,” “true value,” “error,” “probability” and “uncertainty.” Constructing a
single vocabulary of metrology that is able to unambiguously encompass and harmonize
all of the approaches is therefore difficult, if not impossible. This paper examines the
evolution of common philosophies and ways of describing measurement, highlighting
some of the differences and providing some of the rationale for the entries and structure
of the March 2006 draft of the 3rd Edition of the International Vocabulary of Metrology,
Basic and General Concepts and Associated Terms, or VIM3.
1DISCLAIMER: Material discussed during this presentation does not represent the
current policy of the National Institute of Standards and Technology (NIST).
2Formerly Senior Scientist at Physikalisch-Technische Bundesanstalt (PTB),
Braunschweig, Germany2006 NCSL International Workshop and Symposium
Introduction
The concept of measurement covers a wide range of activities and purposes. Different
approaches to describing and characterizing measurement have been developed and have
evolved to address the various types and uses of measurements, and are still evolving.
Many terms have been used over time in the context of describing measurement, and the
evolution of the different approaches to measurement has led to sometimes subtle but
undoubtedly different uses of some terms.
A “vocabulary” is defined (e.g., ISO 1087-1) as “a terminological dictionary that contains
designations and definitions from one or more specific subject fields.” Ideally, every term
in a vocabulary should designate only one concept, in order to minimize confusion.
However, because of the different concepts that are sometimes associated with the same
term in the different approaches to measurement, it is virtually impossible to create a
vocabulary of measurement that designates only one concept with each term in the
vocabulary. This is a major difficulty that has been encountered in developing the 3rd
Edition of the International Vocabulary of Metrology, Basic and General Concepts and
Associated Terms (VIM3), where “metrology” is defined as “field of knowledge
concerned with measurement”.
This paper examines the evolution of the more common approaches to describing
measurement, highlighting some of the differences in the use of terms and providing
some of the rationale for how several of the terms are likely to be treated in the final
version of VIM3.
Common Elements of All Approaches to Measurement
There are a few fundamental concepts in most, if not all, approaches to describing
measurement. Probably the most fundamental concept pertains to the kinds of things that
can be measured (quantities). Another fundamental concept is the means used to express
the magnitude of that which has been measured (in terms of values). Just as fundamental
is the concept of measurement itself. The following definitions are taken from the March,
2006, draft of the VIM3:
A quantity (1.1) is a “property of a phenomenon, body or substance to which a number
can be assigned with respect to a reference” (which allows comparison with other
quantities of the same kind). A value of a quantity (quantity value, 1.l0) is a “number
and reference together expressing magnitude of a quantity”. Measurement (2.1) is the
“process of obtaining one or more quantity values that can reasonably be attributed to a
quantity”, usually through some type of experiment.
In VIM3 the term measurand (2.3) is defined as “quantity intended to be measured.”
This term has ‘evolved’ from the definition in the International Vocabulary of Basic and
General Terms in Metrology, 2nd Edition [1], VIM2, which is “particular quantity subject
to measurement,” that could be different than the quantity intended to be measured. This2006 NCSL International Workshop and Symposium
distinction must be kept in mind when considering the objective of measurement in the
different approaches, and will be discussed further later on.
Figure 1 demonstrates some simple common elements of all approaches to describing
measurement. The rectangular box represents the measurand, and the horizontal scale
represents the entire set of values that could possibly be attributed to that type of
measurand. Note that there is no measurement unit associated with the horizontal line,
because the quantity is an ordinal quantity (1.27), which is a “quantity, defined by a
conventional measurement procedure, for which a total ordering relation with other
quantities of the same kind is defined, but for which no algebraic operations among those
quantities are defined.” Also indicated in Figure 1 is a value (y) being attributed to the
measurand on the basis of a set of replicated measurements, illustrated schematically by a
histogram.
For those quantities where there are meaningful algebraic operations among the
quantities, a measurement unit (1.9) can be defined, which is a “scalar quantity, defined
and adopted by convention, with which any other quantity of the same kind can be
compared to express the ratio of the two quantities as a number.” This is indicated in
Figure 2, where the measurement unit is the reference to be associated with the numerical
value in the measured quantity value. The concept of a measurement unit is common to
all approaches to describing measurement (for other than ordinal quantities). The bell
curve in the figure illustrates a ‘gaussian’ fit to the histogram data. The curve is dashed to
indicate that replicate measurements are not always performed in a measurement (that is,
sometimes only a single measurement is performed), as will be elaborated below in the
discussion of the International Electrotechnical Commission (IEC) Approach.
The two main approaches to describing measurement that will be discussed in this paper
are sometimes called the ‘classical’ approach and the ‘uncertainty’ approach. Within each
of these approaches are sub-approaches. While the two main approaches are given
discrete names, there has in actuality been an evolution of these approaches that makes it
difficult to ascribe certain concepts to one approach or another. This evolution of
concepts will be discussed below. Also, since probability and statistics usually play an
important role in most aspects of measurement evaluation, both the ‘frequentist’ and
‘Bayesian’ theories of inference as used in measurement will be discussed, as
appropriate.
Classical Approach (CA) to Measurement
It is generally accepted that the key distinguishing premise of the classical approach to
measurement is that, for a specified measurand, there exists a unique value, called the
true value (1.11), that is consistent with the definition of the measurand. This is shown
schematically in Figure 3, where it is indicated that, in the general case, the value being
attributed to the measurand based on measurement is different from the true value. This
difference could be due to a variety of reasons, including mistakes in formulating the2006 NCSL International Workshop and Symposium
measurement model (such as not taking into consideration all significant factors and
influences), and blunders in carrying out the measurement procedure.
Another premise of the classical approach is that it is possible to determine the true value
of a measurand through measurement, at least in principle, if a ‘perfect’ measurement
were performed. The objective of measurement in the classical approach is then usually
considered to be to determine the true value of the measurand as ‘closely’ as possible, or
at least as closely as necessary, by eliminating or correcting for all (known) systematic
effects and mistakes, and also by performing enough repeated measurements to
adequately minimize effects and mistakes due to random causes.
In the classical approach it is recognized that it is not possible to perform a perfect
measurement and so there will be “errors”, both systematic and random, in the value
ultimately being attributed to the measurand based on measurement. This value is
frequently referred to as the ‘measurement result’ in the classical approach, and in other
approaches as well. Figure 4 illustrates the concept of an individual measurement error,
defined in the classical approach as the difference between an individual measurement
result and the true value. The individual measurement result (denoted by yi in the figure)
is illustrated with respect to the bell-curve, which is now solid to indicate that multiple
individual measurements are being considered. Also indicated in Figure 4 are “systematic
error,” defined as the difference between the unknown mean of the uncorrected
measurement result and the true value, and “random error,” defined as the difference
between an individual measurement result and the unknown mean of the uncorrected
measurement result. Note that the “mean of the uncorrected measurement result” here is
meant to be that of an infinite distribution, and so cannot be known exactly. This is
illustrated schematically in Figure 5, where two ‘systematic errors’ are shown, the lower
one (systematic errorb) with respect to the average of the histogram data, and the upper
one (systematic errora) with respect to the mean of the theoretical frequency distribution
for an ‘infinite’ set of data. The bell curve of the theoretical frequency distribution is
dashed to indicate that it is not knowable. The systematic errora line is also dashed to
indicate that its length cannot be known, since the mean of the theoretical frequency
distribution cannot be known. The question of whether or not the length of the systematic
errorb line can be known, as well as the lengths of the three ‘error lines’ in Figure 4, will
be discussed next.
Knowable Error?
Two important and related questions that arise in the classical approach are, first, whether
it is possible, in principle, to go about identifying and eliminating, or correcting for, all of
the errors in a measurement, and, second, if so, how? One possible way of addressing
these questions is to hypothesize that it is possible, at least in principle, to determine the
true value by carrying out a very large number of different types of measurements of the
same measurand, using different measurement procedures, measurement methods or even
measurement principles, a large number of times (so that various systematic errors will
‘average out’). This would require that a lot of information be obtained through
measurement (which may not always be practical, even if the philosophy is sound).2006 NCSL International Workshop and Symposium
Figure 6 illustrates this idea for just two different measurement principles, and Figure 7 is
meant to illustrate the advantage of using multiple measurement principles (indicated by
the four different curves). Using this idea in the classical approach, a probability is
usually assessed that the true value lies within a stated interval, as could be characterized
by the ‘widths’ of the large bell-shaped curves associated with the true value in both of
the Figures 6 and 7. Since this idea requires that an essentially infinite amount of
information be obtained in order to know the true value, it is recognized that, in practice,
a true value can never be known exactly using this idea. This is represented schematically
in Figure 7, where y-double-bar represents the average of the averages of the four curves.
The questions then remain whether it is possible, in principle, in a different way, to
identify and correct for all of the errors in a measurement, and, if so, how?
Error Analysis, Frequentist Theory in CA
One different way of trying to answer these questions is through the application of error
analysis, which is based on the frequentist theory of inference as used in measurement.
Error analysis is the attempt to estimate the total error using frequency-based statistics.
However, the systematic error cannot be estimated in a statistical way, since it is neither
observable nor behaves randomly in a measurement series under repeatability conditions.
Therefore error analysis, which includes statistical and nonstatistical procedures, leads to
inconsistencies in data analysis, especially in error propagation.
Bayesian Theory in CA
Another way of trying to answer these questions is to apply the Bayesian theory of
inference to data analysis. Here systematic and random errors are treated on the same
probabilistic basis, where probability is no longer understood as a relative frequency of
the occurrence of events, but as an information-based degree of belief about the truth of a
proposition, for example, about the true value. Using the Bayesian theory, it is still not
possible to determine a true value unless an essentially infinite amount of information is
obtained, so that it is again recognized that, in practice, a true value cannot be known.
Difficulties with the CA
So far no satisfactory way has been found to identify, let alone correct for, all of the
errors in a measurement. The implications are significant, as illustrated in Figure 8, where
a hypothetical three ‘known’ components of systematic error are shown (usually
estimated as ‘worst-cases’). Since it is virtually impossible to know for sure if there is
another component (say, due to a blunder, as indicated by the dashed line), the ‘total’
systematic error is unknown, as also indicated by a dashed line. If the total systematic
error is unknown, then the true value cannot be known. If the true value is not known,
then the error cannot be known (as again indicated by a dashed line). The random error,
when defined with respect to the average of the histogram data, is calculable, as indicated
by the solid line in the figure. However, when random error is defined with respect to the2006 NCSL International Workshop and Symposium
mean of the theoretical frequency distribution, it also becomes unknowable, as illustrated
by the dashed line for ‘random error’ in Figure 9.
Systematic and random errors can therefore typically only be estimated or guessed. No
generally-accepted means for combining them into an ‘overall error’ exists that would
provide some overall indication of how well it is thought that a measurement result
corresponds to the true value of the measurand (i.e., to give some indication of how
‘accurate’ the measurement result is thought to be, or how ‘close’ the measurement result
is thought to be to the true value of the measurand). The difficulty in the CA, of the lack
of a generally-accepted, good procedure for describing the perceived ‘quality’ of the
measurement result is one important reason that ‘modern’ metrology is moving away
from the philosophy and language of the CA. A solution to this difficulty is addressed in
the uncertainty approach (UA) to measurement (as will be described shortly). There are
also other reasons, but they will not be discussed here.
VIM3 RATIONALE: There are many measurement situations, typically of a relatively
simple nature, where it is likely possible to be able to identify and correct for all of the
significant systematic errors, as well as to obtain a sufficient number of replicate
measurements for the purpose, such that description of the measurement result using the
language and philosophy of the classical approach is a seemingly reasonable thing to do,
and many people still do it. This is one of the main reasons that it was decided to keep
many of the terms and concepts from the classical approach in the main body of VIM3,
and not relegate them to an Annex. Another reason, as mentioned earlier and that will be
elaborated further below, is that there is not always a clear demarcation between
approaches. As an example, it is not clear to which measurement approach to ascribe the
premise of a lack of uniqueness of a true value of a measurand.
Uniqueness(?) of True Value
Obviously no measurand can be completely specified, meaning that there will always be
a set of true values that are consistent with the definition of the measurand. The important
question is whether the range (defined as the difference between the upper and lower
limit) of the set of true values is significant when compared with the range of measured
quantity values. This is illustrated schematically in Figure 10, where the interval of the
set of true values consistent with the definition of the measurand is indicated by a pair of
vertical dotted lines. The corresponding range is shown bracketing the measured quantity
value to indicate that, even if a series of replicated, ‘perfect’ measurements of the
measurand were possible, there would still be a set of measured values having that same
range. The dotted bell curve illustrates a situation where the range of the set of actual
measured quantity values is broader than the range of the set of true values.
It is in general desirable to have a measurement situation where the measurand can be
progressively better defined such that the range of the set of true values is relatively
insignificant with respect to the range of measured quantity values that can be obtained
when using the (best) available measuring system, as illustrated in Figure 11. Under these
conditions the measurand can be regarded as having an ‘essentially unique’ true value2006 NCSL International Workshop and Symposium
(i.e., ‘the’ true value), and the ‘customary’ language and mathematics of measurement
can be employed.
However, this situation is not always found, either because the measurand cannot, or
needs not, be specified very specifically, or because the measurement system is so ‘good’
that it is always capable of producing measured quantity values within the range of true
values for that type of measurand, as illustrated in Figure 12. Under these conditions it is
necessary to think differently about the way of describing measurement, irrespective of
the measurement approach. For example, in the classical approach it would no longer be
possible to talk about ‘the true value’ of a measurand, or ‘the systematic error’ associated
with a measurement result, since such unique values would no longer have meaning. This
measurement situation will also be addressed further in the discussion about the
uncertainty approach.
Before leaving the discussion of the classical approach, it is worth noting that the
classical approach is also sometimes called the ‘traditional approach’ or ‘true value
approach.’ However, the latter is a misnomer, since the concept of true value is actually
also used in ‘modern’ approaches, such as the ‘uncertainty approach,’ as will be
discussed next.
Uncertainty Approach (UA) to Measurement
The concept of measurement uncertainty had its beginnings in addressing the difficulties
described above with the CA, namely the questions of 1) whether it is possible, both in
principle and in practice, to know the true value and error, 2) whether or not the true
value is unique, and 3) how to combine information about random error and systematic
error in a generally accepted way that gives information about the overall perceived
‘quality’ of the measurement. Further, if the true value, or set of true values, is not
knowable in principle, then the questions arise of whether the concept of true value is
necessary, useful or even harmful! All of these issues and perspectives will be addressed
below.
While different approaches exist within the UA, the two most prominent approaches are
those put forward in the Guide to the Expression of Uncertainty in Measurement (GUM,
1993 and 1995) [2] and in IEC 60359 (Electrical and Electronic Measurement Equipment
– Expression of Performance) [3]. The IEC approach is parallel and complementary to
the GUM, but uses a more operational or pragmatic philosophy, focusing primarily on
single measurements made with measuring instruments. Both of these approaches, along
with their impact on VIM3, will be described.
GUM Approach to UA
The GUM approach to the UA provides a more refined means than the CA for describing
the perceived quality of a measurement. One of the main premises of the GUM approach2006 NCSL International Workshop and Symposium
is that it is possible to characterize the quality of a measurement by accounting for both
random and systematic ‘effects’ on an equal footing, and a means for doing this is
provided. Another basic premise of the GUM approach is that it is not possible to know
the true value of a measurand (GUM 3.3.1): “The result of a measurement after
correction for recognized systematic effects is still only an estimate of the value of the
measurand because of the uncertainty arising from random effects and from imperfect
correction of the result for systematic effects.” A third basic premise of the GUM
approach is that it is not possible to know the error of a measurement result (GUM 3.2.1,
Note): “Error is an idealized concept and errors cannot be known exactly.”
In the GUM approach it is explicitly recognized that it is not possible to know, for sure,
how ‘close’ a value obtained through measurement is to the true value of a measurand
(i.e., to know the error). Instead a methodology for constructing a quantity, called the
standard measurement uncertainty, is established that can be used to characterize a set of
values that are thought, on a probabilistic basis, to correspond to the true value, based on
the information obtained from the measurement. The objective of measurement in the
GUM approach then becomes to establish a probability density function, usually
Gaussian (normal) in shape, that can be used to calculate probabilities, based on the
belief that no mistakes have been made, that various values obtained through
measurement actually correspond to the ‘essentially unique’ (true) value of the
measurand. Note that the GUM does not explicitly state the objective of measurement
this way, but it can be inferred through its description of standard uncertainty (see, e.g.,
GUM 6.1.2). Another way of viewing the objective of measurement in the GUM approach
is that it is to establish an interval within which the ‘essentially unique’ (true) value of
the measurand is thought to lie, with a given level of confidence (probability as degree of
belief), based on the information used from the measurement. The term “true” has been
put in parenthesis here as an alert that the GUM discourages use of the term (but not of
the concept) “true value,” and instead treats “true value” and “value” as equivalent, and
thus omits the term “true”. This, however, causes terminological difficulties that are
treated in VIM3, and are discussed below.
VIM3 RATIONALE (measurement uncertainty): The concept of measurement
uncertainty is defined in VIM3 (2.12) as “parameter characterizing the dispersion of the
quantity values being attributed to a measurand, based on the information used.” As
stated above, this important concept is introduced in the UA to provide a quantitative
means of combining information arising from both random and systematic effects (if they
can be distinguished at all!) in measurement into a single parameter that can be used to
characterize the dispersion of the values being attributed to a measurand, based on the
information used from the measurement. The VIM3 definition is modified from the
VIM2 [1] (and GUM [2]) definition because of the way that the term “measurement
result” has been redefined in VIM3 (see next rationale below).
VIM3 RATIONALE (measurement result): The GUM uses the VIM2 definition of
“measurement result” (value attributed to a measurand, obtained by measurement), which
is the same as the estimate mentioned above. However, it was decided by the developers
of VIM3 to emphasize the importance of including measurement uncertainty in reporting2006 NCSL International Workshop and Symposium
the outcome of a measurement by incorporating into the definition of measurement result
the notion that “a complete statement of a measurement result includes information about
the uncertainty of measurement,” as stated in Note 2 of the VIM2 definition of
measurement result. Accordingly, measurement result is defined in VIM3 as “set of
quantity values being attributed to a measurand together with any other available relevant
information,” which implicitly includes information not about just a single value, but
rather about the measurement uncertainty as well. The “other available relevant
information,” when available, pertains to being able to state probabilities.
VIM3 RATIONALE (measured value): Since the term “measurement result” is defined
in VIM3 in the more general sense given above, it was decided to introduce a separate
term in VIM3 that could be associated with the concept of each of the individual values
of the set of values being attributed to the measurand based on measurement. Any
individual value representing (or that belongs to) the measurement result is called a
measured quantity value (2.9) in VIM3.
VIM3 RATIONALE (definitional uncertainty): Another basic premise of the GUM
approach is that no measurand can be completely specified, as has already been discussed
earlier in the context of lack of uniqueness of the true value. In the GUM approach this
premise is implemented such that, at some level, there is always an ‘intrinsic’ uncertainty
that is the minimum uncertainty with which an incompletely defined measurand can be
determined (GUM D.3.4). An explicit term covering this concept (definitional
uncertainty, 2.13) has been introduced into VIM3. The implication of this concept, as
discussed above, is that there is no single true value for an incompletely defined
measurand. However, a very important point to remember concerning the GUM approach
is that it “is primarily concerned with the expression of uncertainty in the measurement of
a well-defined physical quantity – the measurand – that can be characterized by an
essentially unique value.” (GUM 1.2) ‘Essentially unique’ means that the definitional
uncertainty can be regarded as negligible when compared with the range of the interval
given by the measurement uncertainty. Therefore, when using the GUM ‘mathematical
machinery’ and language, it is important to make sure that this ‘negligibility’ condition
applies. If it does not, then use of different approximations and language might be
required. This will be elaborated further below.
VIM3 RATIONALE (value, true value): As already noted, in the GUM approach the
term “true” in “true value” is considered to be redundant (GUM D.3.5), and so a “true
value” is just called a “value”. It is important to recognize that this does not mean that
the concept of true value is discouraged or ignored in the GUM. Rather, the concept of
“true value” (defined in VIM3 as “value consistent with the definition of a quantity”) has
only been renamed “value”, or “the value,” in the GUM. This sometimes causes serious
confusion, especially since the same term “value” is also frequently used in the GUM in
the more general, super-ordinate sense of “number and reference together expressing
magnitude of a quantity.” Another reason for potential confusion is that, if true value is
unknowable, then the need for the concept can be questioned (this will also be discussed
later in connection with the IEC approach). However, as discussed earlier, in the GUM
approach, the concept of true value is necessary for describing the objective of2006 NCSL International Workshop and Symposium
measurement. The concept of true value is also necessary for formulating a measurement
model.
The GUM Approach is illustrated schematically in Figure 13, where the objective(s) of
measurement are given at the top. Note that the vertical axis is no longer the number of
times that a possible quantity value that could be attributed to the measurand is obtained
by replicated measurements. Rather, the vertical axis is now the probability that
individual ‘estimates’ of the value of the measurand actually correspond to the
(essentially unique true) value of the measurand, where probability here means degree of
belief under the assumption that no mistakes have occurred. The curve is now a
probability density function (PDF) that is constructed on the basis of both replicate
measurements (using so-called Type A evaluation) and other information obtained from
measurement, such as values obtained from reference data tables and professional
experience (using so-called Type B evaluation).
The combined standard uncertainty, expanded uncertainty and coverage interval are also
illustrated in Figure 13, where the coverage interval is defined in VIM3 (2.19) as
“interval containing the set of true quantity values of a measurand with a stated
probability, based on the information available.” As indicated above, the GUM does not
use the word “true” in connection with the concept of true value, and so “value” is
indicated in the figure. Also indicated is the ‘intrinsic’ uncertainty associated with the
fact that the (true) value is not unique (but only ‘essentially unique’) in the GUM
Approach.
Incorporation of the terminology associated with the VIM3 rationales discussed above is
illustrated schematically in Figure 14. The objective(s) of measurement are again given at
the top of the figure, where the new terminology has also been incorporated. It is
important to notice that nothing has changed in going from Figure 13 to Figure 14 other
than the terminology, which is meant to emphasize that VIM3 is not intended to change
the philosophy of the GUM approach, but only to clarify and possibly harmonize some of
the terminology.
Figure 15 demonstrates the situation where the definitional uncertainty is not small
compared to the rest of the measurement uncertainty, in which case the objective(s) of
measurement are stated differently in recognition that probabilities must now be stated
with respect to a set of true values, and not to an essentially unique true value. This
measurement regime, and use of probability, is not treated in the GUM. However, the
GUM indicates (e.g., Figure D.2) that definitional uncertainty is to be included in the
calculation of measurement uncertainty.
The PDF from Figure 14 (solid curve) is reproduced as the solid curve in Figure 15. A
broadened PDF (dashed curve) and larger coverage interval are presented in Figure 15 in
order to emphasize the necessity of now incorporating the definitional uncertainty into
the probability considerations. Because of the new definition of measurand in VIM3, as
“quantity intended to be measured,” if it is thought (but not known) that the quantity
actually being measured is different from the measurand, then, using the GUM approach,2006 NCSL International Workshop and Symposium
the corresponding uncertainty is a part of the measurement uncertainty, and similar
considerations concerning use of ‘probability’ would apply.
Since they were discussed earlier in connection with the CA, it is interesting to consider
how the Bayesian and frequentist theories of inference relate to the GUM approach. In a
sense, it can be said that the GUM approach, and in fact the UA in general, are
consequences of the Bayesian theory of describing one’s state of knowledge about a
measurand. Using the Bayesian theory in the GUM approach, measurement can be
thought to consist of incrementally improving one’s state of knowledge and belief about a
true value based on all of the accumulated information that is available through
measurement. Using the Bayesian theory, the measurement uncertainty based on
probability density functions associated with a particular measurand will continually
change according to additional information obtained through measurement. The
frequentist theory of inference can be useful for determining certain Type A components
of measurement uncertainty, but is not capable of treating most Type B components. An
example of the difficulty of the frequentist theory of inference within the GUM approach
is that the frequentist theory is not able to be used to assess the uncertainty of a single
measured value when using a measuring instrument, such as a voltmeter. The reason is
that the uncertainty here derives from ‘nonstatistical’ information obtained from the
instrument’s calibration certificate. This type of single measurement comprises a large
fraction of the types of measurements routinely made daily throughout the world.
IEC Approach to the UA
The other major approach to describing and characterizing measurement that will be
discussed here is that used by the International Electrotechnical Commission (IEC), as
presented primarily through their IEC 60359 (Electrical and Electronic Measurement
Equipment – Expression of Performance) [3]. The IEC philosophy questions the
existence, in principle, of a true value of a quantity. The objective of measurement in this
view is not to determine a true value of a measurand with a given probability, but
concentrates instead on compatibility of measurement results.
The IEC approach uses a more operational or pragmatic philosophy than the GUM
approach. Most notably, the IEC approach treats the concept of true value as both
unknowable and unnecessary, discouraging and in fact eliminating at least explicit use of
the concept of true value, even in stating the objective of measurement. In the IEC
approach, as presented in the Introduction and Annex A of IEC 60359 [3], the stated
objective of measurement is to obtain measurement results that are compatible with each
other, within their respective measurement uncertainties. The philosophy is that, from an
operational perspective, this is all that can really be done in measurement. This is
illustrated schematically in Figure 16, where the four horizontal lines represent sets of
measured values (central values, along with their measurement uncertainties) for four
separate measurements of the same specified quantity being measured (which might be
different than the measurand). From the IEC perspective, it can even be argued that the2006 NCSL International Workshop and Symposium
concept of true value is potentially harmful, since it leads to thinking about something
that is not relevant!
VIM3 RATIONALE: As a result of this key difference in philosophy between the IEC
approach and the GUM approach to the UA, it is necessary to generalize several of the
key concepts and definitions in VIM3 to accommodate both approaches whenever
possible. For reasons discussed earlier, the important concept of true value (VIM3 1.11)
is kept in VIM3, but is not explicitly used in the context of definitions that also apply to
IEC. For example, the definition of measured value (VIM3 2.9) has been generalized to
“quantity value representing a measurement result,” instead of “quantity value
representing the set of true values of a quantity …” so that true value does not need to be
explicitly mentioned, but can be still be inferred for the classical and GUM approaches.
Similarly, measurement result (VIM3 2.10) has been defined in VIM3 as “set of
quantity values being attributed to a measurand together with any other available relevant
information”, rather than as “information about the set of quantity values being attributed
to a measurand,” in order to accommodate the IEC view that a measurement result is just
a set of values, with every element of the set having equal status. The probabilistic aspect
of the GUM approach is left to the end of the definition as “any other available relevant
information,” which can be ignored for the IEC approach. One last example that will be
given is definitional uncertainty (VIM3 2.13), now defined in VIM3 as “minimum
measurement uncertainty resulting from the inherently finite amount of detail in the
definition of the measurand,” rather than “parameter characterizing the estimated
dispersion of the true values of a quantity …,” in order to remove explicit reference to
true value.
Another key aspect of the IEC approach is that it focuses very heavily on providing
guidance for estimating measurement uncertainty for situations where single
measurements are made using measuring instruments, and where the measuring
instrument is operating not only under reference conditions, but anywhere within its rated
operating conditions or even extreme operating conditions. The key aspect of the IEC
approach in this regard, as described in IEC 60359 [3], is to construct a calibration
diagram for establishing the measurement uncertainty that can be ascribed to a single
indication of a measuring instrument under various operating conditions. A calibration
diagram is illustrated in Figure 17, where the horizontal axis (or ‘reading axis’)
corresponds to the indications or ‘reading’ of a measuring instrument (in ‘units of
output’), and the vertical axis (or ‘measurement axis’) corresponds to measured values (in
‘units of measurement’) as obtained using measurement standards. The ‘boundary of
measured values’ around the calibration curve is obtained during the course of calibration
of the measuring instrument, and is used to assess the measurement uncertainty to
associate with a given subsequent indicated value for an unknown measurand (‘reading’
of the measuring instrument), as illustrated in the figure.
Returning to the fundamental IEC philosophy that the concept of true value is
unnecessary, and that all that really matters is that measurement results are compatible
with each other, one might ask what to do when measurement results are not compatible
with each other, as illustrated schematically by ‘measurement number 5’ in Figure 18? In2006 NCSL International Workshop and Symposium
this case it is necessary to investigate whether any mistakes have been made in
performing all of the measurements. If no mistakes can be found, then it is assumed that
the quantity that was measured was different for some of the measurements. In this case it
becomes necessary to somehow ‘average all of the measurements’ and create an
uncertainty that encompasses all of the measurement results.
Conventional Value Hybrid Approach; Knowable Error!
Before concluding, it is useful here to discuss a hybrid approach to the CA and UA that is
frequently employed as a practical solution for handling the conceptual and
terminological problems described earlier concerning the inability to know error, yet not
abandon use of the concept and term, since they are still so widely used. This hybridapproach, which will be called here the ‘Conventional Value Hybrid Approach’, or
CVHA, is typically used in measurement situations where a decision must be made
concerning whether a measured quantity conforms to a particular requirement, such as a
specified machine tolerance or a legal regulation. The ‘hybrid’ aspect of the CVHA is
that, while error is used, measurement uncertainty is also taken into account, as will be
explained.
The CVHA is a two-step approach, where in the first step a measurement standard is
calibrated and assigned an (essentially-unique) ‘conventional’ quantity value, and then, in
the second step, a second measurement is performed on the calibrated measurement
standard. Error is assessed in the second step with respect to the conventional value that
was assigned to the measurement standard in the first step. This error can be expressed as
a rational quantity since it is defined with respect to the conventional value, and not the
true value, of the measurement standard.
Figures 19 and 20 schematically illustrate the two step process of the CVHA. Figure 19
shows the conventional value being assigned (through measurement, using a “high-level”
measuring system) to the measurement standard. In this first step the systematic error,
and hence the error, as defined with respect to the true value, cannot be known (the
systematic error is set to zero by convention). The curve represents a fit to a set of
histogram data that are obtained when using the measuring system to measure the
measurement standard. Note that a measurement uncertainty of the conventional value
can be determined, but this is not illustrated in the figure.
Figure 20 illustrates the second step of the process, where the quantity associated with the
measurement standard (to which a conventional value has been assigned) is now
measured with a “lower-level” measuring system. The measured values obtained when
using this system are denoted schematically by the “fit to histogram data2” on the right
side of the figure, and an individual measured value (yi2) is also indicated. Note that the
measurement scale has been shifted in Figure 20, such that the difference between the
conventional value and true value is meant to be the same in the two figures, and the “fit
to histogram data1” in the two figures is also meant to be the same. Figure 20 illustrates
that, typically in the CVHA, the measured value using the “lower-level” measuring2006 NCSL International Workshop and Symposium
system is not expected to be as “close” to the true value as the conventional value is and,
further, the width of the “fit to histogram data2” is not expected to be as small as that of
the “fit to histogram data1”. More importantly in Figure 20, however, is the illustration
that systematic error and error can be defined in the second step of the CVHA both with
respect to true value (in which case they are unknowable) or with respect to conventional
value (in which case they are knowable). Note that systematic error here is also defined
with respect to the mean of the histogram data and not the mean of the theoretical
frequency distribution, as discussed earlier.
The advantage of the CVHA is that it can be used in measurement situations where the
uncertainty associated with the conventional value is small with respect to the typical
“knowable error,” and so it is possible to perform relatively straightforward
measurements using the lower-level systems, and make equally straightforward
conformity assessment decisions, without having to perform a possibly complicated
uncertainty analysis. This approach has been used for many years and covers many types
of measurement situations where, in fact, the “knowable error” is frequently treated as the
measurand.
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发表于 2017-11-26 15:28:29
发表于 2017-11-26 22:48:22
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