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being x and s(x) the average and standard deviation of
the (unknown) assumed probability distribution).
3. EVALUATION METHODS AND MEANINGS
The choice to express a non-exact measurement result as
still leaves open the decision about the
evaluation methods that can be adopted to obtain such a
result and the meaning to be attributed to it.
The GUM standpoint in reference to these decisions is
peculiar. With respect to the evaluation methods, the
GUM embodies a recommendation issued by the CIPM in
1981 (CIPM, 1981) and admits both statistical (“type A”)
and non-statistical (“type B”) methods. The condition to
make this pluralism operatively acceptable is that the
suggested techniques to formally deal with the results are
independent of the “type” of the evaluation method and
therefore the same in both cases. From the conceptual
point of view this position is an important step against the
radical objectivism of some classical interpretations of
measurement: some subjective information, in the form of
“degrees of belief (to quote the GUM) is present and
required even in the case of an “objective” operation as
measurement (Mari, Zingales, 1999).
Not so pluralistic is the position of the GUM in reference to
the meaning of s(x). Its basic interpretation is statistical, in
terms of the standard deviation of the (possibly unknown)
distribution of which x is the estimated average. The main
application suggested by the GUM for this so-called
“standard uncertainty” is to compute the “law of
uncertainty propagation” (what is classically called “error
propagation”) through functional relations. Only for
specific applications a further interpretation is recognized
as useful, in which s(x) (and more precisely ks(x), being k
a proportionality factor usually in the range [1,3]), in this
case called “expanded uncertainty”, is considered to
express the half width of the interval of which x is the
center point. This set-theoretic interpretation is however
deemed as explicitly dependent of the statistical one and
formally derived from it.
4. TWO CATEGORIES OF APPLICATIONS
The position of the GUM is conservative in this regard. A
more general standpoint recognizes that the same result
could admit distinct interpretations in distinct
applications. Given a set {} of such measurement
results, two basic categories of applications can be
identified:
* “non-exact derived measurement”: a quantity Y is known
as analytically dependent of the quantities X|,...,X„
through a function f, Y=f{Xi X„), and each is
the measurement result of a quantity X; the function f
must be then somehow applied to the terms to
compute a measurement result for V;
* “non-exact measurement results comparison”: all the
are repeated measurement results of the same
quantity X, and must be compared to each other via a
relation r, of which they are arguments, to establish
whether such a relation holds among them or not.
Uncertainty propagation is clearly related to the first
category, a case in which the statistical interpretation is
plausibly the preferred one. The GUM suggests to
compute y and s(y) with distinct procedures, only in
dependence on the terms x, and s(x,) respectively:
y = f(xi,...x n )
s(y) = [Errore.c, 2 s 2 (x,)] 1/2
df
being the “sensitivity coefficients” c, =-r— (in the case the
uXj
quantities X, are correlated further terms should be
introduced).
The most important application in the second category is
what could be called “non-exact equality”, aimed at
establishing whether two or more can be
considered undistinguishable with each other. The formal
identity =, i.e. x,=x y and s(x,)=s(xy), is
clearly a “too exact” criterion in this case, and different,
more general, principles have been proposed, typically
based on the set-theoretical interpretation of the terms
. For example, if is assumed as the
interval [xr-s(x,),x/+s(x/)] then two results could be judged
“compatible” with each other in the case their intersection
is non-null (cf. UNI, 1984).
It is worth to note that the GUM does not even mention
this second category of applications. For important
problems such as the definition of the procedures to
compare national standards and express the results of the
comparison an agreed position is still an open issue.
5. (NON-)EXACTNESS AS
(UN)CERTAINTY AND (NON-)SPECIFICITY
We suggest that the non-exactness of a measurement
result is formalized as a combination of two distinct
components, called “uncertainty” and “non-specificity”. An
example is helpful to introduce the meaning of such
concepts and their relations. Let us consider the following
two statements:
A = “this is a 120-page book”
B = “this is a book”
aimed at expressing the knowledge of an observer on a
given thing under examination. On the basis of the form of
A and B two conclusions can be immediately drawn:
* A entails B: if A is true then also B must be true (in set-
theoretical terms, A is a subset of B); therefore A is more
specific than B;
* regardless of the particular uncertainty assignment
chosen, A is at most as certain as B, and plausibly more
uncertain than it
Hence the same formal expression, , admits two
distinct, and actually opposite, meanings:
