Information Technology Reference
In-Depth Information
The values of quantitative characteristics measured with these scales are
referred to as physical ones.
To measure qualitative characteristics the following scales are used, in
particular, names, ordinal (rank).
The research conclusions made can be adequate to a reality if and when they do
not depend on what unit of measurement is preferred by a researcher, i.e. these
conclusions should be invariant with regard to admissible transformation of the
characteristic values measured in any scale. Let us give admissible
transformations of function
Ф
x
in the scales below [17—20]:
)
Absolute scale
Ф
(
x
)
=
x
Ratio scale
Ф
(
x
)
=
ax
(
a
>
0
Scale of intervals
Ф
x
)
=
ax
+
b
(
a
>
0
b
∈
R
)
Scale of differences
Ф
(
x
)
=
x
+
b
(
b
∈
R
)
Scale of names
Ф
x
— all one-to-one transformations
)
Ordinal (rank) scale
Ф
x
— all strictly increasing transformations
)
When experts use ordinal scales to measure qualitative characteristics, then for
definition of aggregating indicators average values of score expert evaluations are
used often enough [21—27]. There are some methods of average computing, in
particular, arithmetic mean, geometrical mean, harmonic mean, mean square
value, mode, median. Let us consider application of an arithmetic mean in an
ordinal scale, being most often used. Let us assume that two entrants got marks 4
and 3, accordingly, for the one entrance examination, and marks 4 and 5 for the
other entrance examination. Their total scores and arithmetic mean values of two
examinations are identical and equal to 8 and 4, accordingly. The conclusion is
that they have equal rights to matriculation. As examination scores are allotted
according to an ordinal scale, let us use strictly increasing transformation of this
scale
Ф
:
()
()
()
. According to the transformation made
(which is admissible), the total score and the arithmetic mean of marks of one
entrant remained unchanged, and the same of the second entrant became equal to
10 and 5, accordingly. Thus, the second entrant has preferable rights for
matriculation than the first one. With the admissible transformation completed, the
result stability is broken that means incorrectness of arithmetical operations in the
ordinal and nominal scales [28].
Having in mind that used of averages in various scales is spread enough, the
problems [29—32] of achieving average values were set and solved, with results
of the average comparison being stable with regard to the admissible
transformations of characteristic values measured in the specific scale. Let u
provide definitions of mean values according to Kolmogorov and Cauchy.
For numbers
;
;
Ф
3
=
3
Ф
4
=
4
Ф
5
=
7
x
,
x
,
...
x
, a mean value according to Kolmogorov is the
1
2
n
following function
() ()
( )
,
⎛
F
x
+
F
x
+
...
+
F
x
⎞
−
1
F
⎜
⎝
1
2
n
⎟
⎠
n
()
()
where
is strictly monotone function;
is inverse function to
.
F
x
()
F
x
F
−
x
