54
• Second, as a consequence of this formalization, the information provided by all
uncertainty components in an uncertainty budget can be synthesized in a single
outcome: “The combined uncertainty should be characterized by the numerical
value obtained by applying the usual method for the combination of variances.
The combined uncertainty and its components should be expressed in the form
of standard deviations” (JCGM, 2008a, 2008b: 0.7).
Of course, such a position might be considered a pragmatic means of solving the
problem of separately reporting statistical and nonstatistical components of uncertainty, while simply sidestepping the traditional problem of separately identifying
random and systematic causes of errors, as maintained for example by Rabinovich
(2005: p. 286).
The subject is complex and widely discussed in the literature on the science and
philosophy of measurement, though for present purposes we need not discuss it
further here (for a short introduction see Box 3.1).
high uncertainty have nontrivial probabilities over many outcomes. Those outcomes need not have
numerical values. Distributions with high dispersion take on extreme numerical values. The distinction can be seen in stark relief by comparing a distribution that has maximal entropy with one
that has maximal variance. Given outcomes that take values 1–8, the distribution that maximizes
entropy places equal weight on each outcome. The distribution that maximizes variance takes
value 1 with probability 1/2 and value 8 with probability 1/2” (Page, 2018: p. 139, emphasis
added). The term “measurement uncertainty” is then taken by the GUM as idiomatic.
Box 3.1: The logic of error/uncertainty propagation
What traditionally has been called the law of propagation of errors can be
exemplified by the measurement of human body temperature by means of a
mercury thermometer. Several possible sources of error can be identified in
this measurement, including the finite resolution of the instrument (which is
not able to discriminate among temperatures closer to one another than a
given threshold), the effect of the temperature and the atmospheric pressure of
the environment on the instrument (so that the instrument output changes
when the temperature under measurement does not change), and the alteration
of the temperature under measurement due to the interaction between the
body and the instrument. Under the hypotheses that (1) each of these errors,
whether its source is of a statistical nature or not, can be formalized as a standard deviation (thus consistently with the CIPM recommendation mentioned
above); (2) such errors are statistically uncorrelated; and (3) their contribution
to the total error is not analytically known, the simplest case of the law of
propagation of errors is obtained, which prescribes computing the total error
as the square root of the sum of the squares of these standard deviations (i.e.,
of the variances associated with the errors).
3 Technical and cultural contexts for measurement systems
• Second, as a consequence of this formalization, the information provided by all
uncertainty components in an uncertainty budget can be synthesized in a single
outcome: “The combined uncertainty should be characterized by the numerical
value obtained by applying the usual method for the combination of variances.
The combined uncertainty and its components should be expressed in the form
of standard deviations” (JCGM, 2008a, 2008b: 0.7).
Of course, such a position might be considered a pragmatic means of solving the
problem of separately reporting statistical and nonstatistical components of uncertainty, while simply sidestepping the traditional problem of separately identifying
random and systematic causes of errors, as maintained for example by Rabinovich
(2005: p. 286).
The subject is complex and widely discussed in the literature on the science and
philosophy of measurement, though for present purposes we need not discuss it
further here (for a short introduction see Box 3.1).
high uncertainty have nontrivial probabilities over many outcomes. Those outcomes need not have
numerical values. Distributions with high dispersion take on extreme numerical values. The distinction can be seen in stark relief by comparing a distribution that has maximal entropy with one
that has maximal variance. Given outcomes that take values 1–8, the distribution that maximizes
entropy places equal weight on each outcome. The distribution that maximizes variance takes
value 1 with probability 1/2 and value 8 with probability 1/2” (Page, 2018: p. 139, emphasis
added). The term “measurement uncertainty” is then taken by the GUM as idiomatic.
Box 3.1: The logic of error/uncertainty propagation
What traditionally has been called the law of propagation of errors can be
exemplified by the measurement of human body temperature by means of a
mercury thermometer. Several possible sources of error can be identified in
this measurement, including the finite resolution of the instrument (which is
not able to discriminate among temperatures closer to one another than a
given threshold), the effect of the temperature and the atmospheric pressure of
the environment on the instrument (so that the instrument output changes
when the temperature under measurement does not change), and the alteration
of the temperature under measurement due to the interaction between the
body and the instrument. Under the hypotheses that (1) each of these errors,
whether its source is of a statistical nature or not, can be formalized as a standard deviation (thus consistently with the CIPM recommendation mentioned
above); (2) such errors are statistically uncorrelated; and (3) their contribution
to the total error is not analytically known, the simplest case of the law of
propagation of errors is obtained, which prescribes computing the total error
as the square root of the sum of the squares of these standard deviations (i.e.,
of the variances associated with the errors).
3 Technical and cultural contexts for measurement systems