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If
, the average according to Kolmogorov is an arithmetic mean, if
F
x
=
x
()
()
it is a harmonic mean.
It is proven [18] that according to Kolmogorov, among all mean values only an
arithmetic mean is possible to use in a scale of intervals, and only power means
and a geometrical mean are possible to use in a scale of ratios.
For numbers
F
x
=
ln
x
, it is an geometrical mean, if
F
x
=
1
/
x
x
,
x
,
...
x
, according to Cauchy, a mean value is the function
1
2
n
(
)
(
)
(
)
(
)
.
It is proven [18] that according to Cauchy, among all mean values only terms of
a variation series, in particular, a median, can be used in an ordinal scale. Use of
terms of a variation series to determine an aggregating indicator is often
noninformational for a number of particular indicators owing to very coarse
estimate. For example, they are used in educational process, where knowledge is
estimated in marks from two to five.
To estimate qualitative and quantitative characteristics, experts use verbal
scales often enough. Values in verbal scales are words expressing characteristic
appearance intensity degree. These words are referred to as levels or gradations.
Let us consider only those verbal scales with which it is possible to define a linear
order, i.e. “less — more” ratio.
Problems of definition of sets of verbal scale levels and quantitative values of
qualitative characteristics appearance within the limits of these levels are main
ones in expert evaluations [33]. For the purpose of employment of known
mathematical models of information processing, numerical points are put in
correspondence to levels of verbal scales. The result of this approach is that the
verbal scale is mapped to a verbally-numerical scale. Definition of values of the
points put in correspondence to levels of verbal scales is a separate problem, the
solution of which influences the stability of the results obtained within the limits
of a mathematical model, so the justification of use of these values is needed. For
example, marks “2”, “3”, “4”, “5”, which are put in correspondence to verbal
values E ("unsatisfactory”), C ("satisfactory”), B (“good”), A (“excellent”),
compose a verbally-numerical scale in their aggregate. Certainly, it is ought to
remember that the numbers put in correspondence to verbal levels of qualitative
characteristics are elements of an ordinal scale and all restrictions mentioned
above are applicable to them.
However, if within the scope of a specific problem use of a certain verbally-
numerical scale is justified, in actual practice experts face essential difficulties
caused by intermittent transitions between levels, not allowing to catch and
estimate intermediate conditions of the characteristic under evaluation [34]. To
estimate intermediate conditions, process of artificial fuzzification of numerical
points corresponding to levels of verbal scales is applied. For example, in
educational process when evaluating the pupils' knowledge without any
limitations imposed on generality of "good" knowledge, not only mark "4”, but
also the whole range of marks [3.5; 4.5] is quite often used. Such process of points
fuzzification simulates smoothness of estimating activity of experts, but does not
facilitate process of exposing real objects with evaluations arranged near the
boundaries of fuzzy areas.
f
x
,
x
,
...
x
, if
min
x
,
x
,
...
x
f
x
,
x
,
...
x
max
x
,
x
,
...
x
1
2
n
1
2
n
1
2
n
1
2
n
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