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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 
<x,s(x)> 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 
<x,s(x)> could admit distinct interpretations in distinct 
applications. Given a set {<x,,s(x,)>} 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 <x,,s(x,)> is 
the measurement result of a quantity X; the function f 
must be then somehow applied to the terms <x,,s(x,)> to 
compute a measurement result <y,s(y)> for V; 
* “non-exact measurement results comparison”: all the 
<x,,s(x/)> 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 <x,,s(x,)> can be 
considered undistinguishable with each other. The formal 
identity <x,,s(x;)>=<xy,s(x ; )>, 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 
<x/,s(x,)>. For example, if <x;,s(x,)> 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, <x,s(x)>, admits two 
distinct, and actually opposite, meanings: