313 
Mari L, Savino M., 
Italian experience 
837. 
e. Cambridge Univ. 
Hilary of basic and 
nal Organization of 
UNCERTAINTY IN MEASUREMENT: A SURVEY 
Luca Mari 
Libero Istituto Universitario C.Cattaneo 
C.so Matteotti 22,21053 Castellanza (Va) - Italy 
email: lmari@liuc.it 
ISPRS Commission VI, Working Group 3 
;sion of uncertainty 
Organization of 
"versky A., 1971. 
emic Press, New 
[1990). 
3n and assignment 
pp.79-90. 
litative analysis of 
ts, accepted for 
f. Addison-Wesley, 
»ychophysics, and 
Ratoosh (eds.), 
es, J.Wiley, New 
KEY WORD: Foundations of Measurement, Measurement Uncertainty 
ABSTRACT: 
The paper analyzes the concept of non-exactness of measurement results, by clearly distinguishing between (i) the way 
the results are expressed to make their uncertainty explicit; (ii) the way the chosen expression is interpreted as a suitable 
combination of non-specificity and uncertainty; (iii) the way the interpreted results are formally dealt with. In this 
perspective the merits and flaws of the ISO Guide to the expression of uncertainty in measurement are highlighted. 
1. WHY NON-EXACTNESS IS AN ISSUE 
IN MEASUREMENT 
Measurement is a means to set a bridge between the 
empirical world (to whom the measured thing belongs) 
and the linguistic world (to whom the measurement result 
belongs), aimed at enabling to faithfully re-interpret within 
the empirical world the information that has been obtained 
by handling symbols. The crucial point is the faithfulness 
of such a re-interpretation. In terms of the following 
diagram: 
direct handling 
“things” p “empirical results” 
“symbols” — — ► “symbolic results” 
processing 
the issue is whether the procedures a and b would lead to 
the same result. 
The fact is that the two worlds are inherently different. 
According to Bridgman, «there are certain human 
activities which apparently have perfect sharpness. The 
realm of mathematics and of logic is such a realm, par 
excellence. Here we have yes-no sharpness. But this yes- 
no sharpness is found only in the realm of things we say, 
as distinguished from the realm of things we do. Nothing 
that happens in the laboratory corresponds to the 
statement that a given point is either on a given line or it is 
not» (Bridgman, 1959). 
The non-exactness (in the following the difference 
between non-exactness and uncertainty will be 
maintained and discussed) of measurement results 
accounts for such a distinction, although «by forcing the 
physical experience into the straight jacket of 
mathematics, with its yes-no sharpness, one is discarding 
an essential aspect of all physical experience and to that 
extent renouncing the possibility of exactly reproducing 
that experience. In this sense, the commitment of physics 
to the use of mathematics itself constitutes, paradoxically, 
a renunciation of the possibility of rigor» (Bridgman, 
1959). 
2. THE EXPRESSION OF NON-EXACT 
MEASUREMENT RESULTS 
Taking into account the linguistic side of the problem, the 
first decision to be made is related to the form a 
measurement result should be given to make its non 
exactness explicit. According to the ISO Guide to the 
expression of uncertainty in measurement (GUM) (ISO, 
1993; a useful synthesis of the Guide can be found in 
Taylor, Kuyatt, 1997): 
meas. result = <measurand value, uncertainty value> 
= <x,s(x)> 
where the first term of the couple could be obtained as the 
average of the population of the instrument readings and 
the second term as its estimated standard deviation. 
Two notes in this regard: 
* in presence of non-exactness the concepts of 
measurement result and measurand value cannot dealt 
with as equivalent: the measurement conveys information 
not only on the measurand value, but also on its 
uncertainty; in other terms, in this case a measurement 
result is not complete without the indication of a degree of 
uncertainty; 
* different, and somehow more general, representations 
could be chosen, for example subsets, or fuzzy subsets, 
or probability distributions. All of these are more widely 
applicable than the representation suggested by the 
GUM, that can be employed only for algebraically strong 
scales (although the generality is understood, because of 
the applicability of the Chebyshev inequality, of 
expressing measurement results as couples <x,s(x)>,