60
and if more analytical information were required, the whole uncertainty budget
could be reported.
This relation (or at least its right-hand-side term) is to be considered the measurement result, contrary to the tradition still witnessed in the definition of given in the second edition of the VIM: “value attributed to a
measurand, obtained by measurement” (ISO, 1993: 3.1). In other words, from this
perspective the measurement uncertainty is assumed to be a constitutive component
of the measurement result, and not just an additional, complementary entity. Indeed,
“when a measurement result of a quantity is reported, the estimated value of the
measurand […] and the uncertainty associated with that value, are necessary”
(BIPM, 2019: p. 127). In the clear words of Lira (2002: p. 43),
we will […] refrain from referring to the “uncertainty of a measurement result”. There are
two reasons for this. First, a measurement result should be understood as comprising both
the estimated value of the measurand and its associated uncertainty. Second, once the estimated value is obtained, there is nothing uncertain about it. […] Hence, expressions such as
the uncertainty in knowing the measurand or the uncertainty associated with an estimated
value are more appropriate, even if longer sentences result.
This paves the way for extending the very concept of measurement uncertainty to
the evaluation of quantities for which the expected value and the standard deviation
of the underlying distribution are not sufficiently representative. An example would
be where the distribution is strongly asymmetric. More generally this could encompass ordinal or nominal properties (Mari, Narduzzi, Nordin, & Trapmann, 2020),
for which standard deviations are not meaningful. One solution is to acknowledge
that entire probability distributions of values could be reported to convey the available information, on each uncertainty component and then the measurand, as in
16
measurand distribution of values of a property
=
Attributing to the measurand a single value or a distribution of values may in fact be
considered the two extreme options, where other strategies are possible for reporting the information acquired by the measurement, so as to convey more information
than a single value
17
but less information than an entire distribution. In particular, a
16 As an example, let us consider the task to determine the character written in a given ink pattern,
called “optical character recognition” (OCR) in the context of information technology. If the recognition of a given character from a given pattern is not certain, more than one character could be
attributed to the pattern, and in the most general case the result of the recognition is a probability
distribution over the chosen alphabet (Mari et al., 2020). Hence in this case it is a list of distributions, and not of standard uncertainties, that has to be propagated. Due to the analytical complexity
of the problem, the GUM framework includes a numerical procedure for such a propagation of
distributions, based on a Monte Carlo method (JCGM, 2008b).
17 For measurands that are quantities, the value is a number that multiplies a unit. In this case the
number may actually convey some information about the intended quality of the result through its
number of significant digits, so that, for example, “1.23” can be interpreted as including all numbers in the range (1.2250 …, 1.2349 …). This offers a justification for the admission that “the
measurement result may be expressed as a single measured quantity value. In many fields, this is
the common way of expressing a measurement result” (JCGM, 2012: 2.9, Note 2). Of course, this
is less informative than the standard deviation format, except if it is also assumed that the distribution in the range is of a particular kind, such as a uniform distribution.
3 Technical and cultural contexts for measurement systems
and if more analytical information were required, the whole uncertainty budget
could be reported.
This relation (or at least its right-hand-side term) is to be considered the measurement result, contrary to the tradition still witnessed in the definition of
measurand, obtained by measurement” (ISO, 1993: 3.1). In other words, from this
perspective the measurement uncertainty is assumed to be a constitutive component
of the measurement result, and not just an additional, complementary entity. Indeed,
“when a measurement result of a quantity is reported, the estimated value of the
measurand […] and the uncertainty associated with that value, are necessary”
(BIPM, 2019: p. 127). In the clear words of Lira (2002: p. 43),
we will […] refrain from referring to the “uncertainty of a measurement result”. There are
two reasons for this. First, a measurement result should be understood as comprising both
the estimated value of the measurand and its associated uncertainty. Second, once the estimated value is obtained, there is nothing uncertain about it. […] Hence, expressions such as
the uncertainty in knowing the measurand or the uncertainty associated with an estimated
value are more appropriate, even if longer sentences result.
This paves the way for extending the very concept of measurement uncertainty to
the evaluation of quantities for which the expected value and the standard deviation
of the underlying distribution are not sufficiently representative. An example would
be where the distribution is strongly asymmetric. More generally this could encompass ordinal or nominal properties (Mari, Narduzzi, Nordin, & Trapmann, 2020),
for which standard deviations are not meaningful. One solution is to acknowledge
that entire probability distributions of values could be reported to convey the available information, on each uncertainty component and then the measurand, as in
16
measurand distribution of values of a property
=
Attributing to the measurand a single value or a distribution of values may in fact be
considered the two extreme options, where other strategies are possible for reporting the information acquired by the measurement, so as to convey more information
than a single value
17
but less information than an entire distribution. In particular, a
16 As an example, let us consider the task to determine the character written in a given ink pattern,
called “optical character recognition” (OCR) in the context of information technology. If the recognition of a given character from a given pattern is not certain, more than one character could be
attributed to the pattern, and in the most general case the result of the recognition is a probability
distribution over the chosen alphabet (Mari et al., 2020). Hence in this case it is a list of distributions, and not of standard uncertainties, that has to be propagated. Due to the analytical complexity
of the problem, the GUM framework includes a numerical procedure for such a propagation of
distributions, based on a Monte Carlo method (JCGM, 2008b).
17 For measurands that are quantities, the value is a number that multiplies a unit. In this case the
number may actually convey some information about the intended quality of the result through its
number of significant digits, so that, for example, “1.23” can be interpreted as including all numbers in the range (1.2250 …, 1.2349 …). This offers a justification for the admission that “the
measurement result may be expressed as a single measured quantity value. In many fields, this is
the common way of expressing a measurement result” (JCGM, 2012: 2.9, Note 2). Of course, this
is less informative than the standard deviation format, except if it is also assumed that the distribution in the range is of a particular kind, such as a uniform distribution.
3 Technical and cultural contexts for measurement systems