Information Technology Reference
In-Depth Information
2
The
mean square deviation
σ
(
X
,
x
0
)
of
X
with respect to a
reference value
x
0
∈
R
is a sum, for
x
varying in
X
, given by:
1
n
2
x
∈
X
(
x
−
x
0
)
2
=
x
∈
X
ν
(
x
)(
x
−
x
0
)
2
σ
(
X
,
x
0
)=
.
(7.11)
2
The
variance
of
X
,
σ
(
X
)
is the
mean square deviation
of
X
with respect to the
mean
μ
(
X
)
:
2
2
σ
(
X
)=
σ
(
X
,
μ
(
X
))
.
(7.12)
It can be shown that variance reaches the minimum value among all possible mean
square deviations.
The
standard deviation
σ
(
X
)
of a statistical distribution
X
is the square root of
variance:
2
σ
(
X
)=
σ
(
X
)
.
(7.13)
A fundamental (elementary) result, due to the Russian mathematician Chebichev,
states that in any statistical distribution
X
of mean
,for
any positive real
k
, the fraction of the values of
X
which, in absolute value, differ
from
μ
and standard deviation
σ
k
2
.
μ
more than or exactly equal to
k
σ
is a fraction less than or equal to 1
/
Formally, setting
(
x
≥
y
)=
1ifthe
≥
relation holds between
x
,
y
and zero otherwise,
we have:
∑
x
∈
X
(
x
−
μ
≥
k
σ
)
k
2
≤
1
/
.
(7.14)
|
X
|
=
√
2, at least half of the values lie in the interval
This means that, using
k
μ
−
2
σ
and
σ
.
The
entropy
H
μ
+
2
(
X
)
of a statistical distribution
X
is a sum, for
x
varying in
X
,
given by:
)=
−
x
∈
X
ν
(
x
)
log ν
(
x
)
.
H
(
X
(7.15)
This value, which is the basis of Shannon's information theory, corresponds to a
measure of the
disorder
of a population
X
.
7.6.1
Statistics and Probability
A
sample
of a population
X
is a subpopulation of
X
, that is, a multiset where any
element occurs with a multiplicity less or equal to its multiplicity in
X
.Asampleof
X
is
typical
if the frequency of any element of the sample is the same frequency that
it has in
X
.
Statistics
is the science of discovering properties of populations from its
samples.
In statistical distribution the notion of multiplicity, which is the crucial concept
of multisets, provides naturally the notion of frequency, which can be seen as a sort
of relative multiplicity. Frequencies measure the occurrence of values by means of
fractions, and the sum of all frequencies of a statistical distribution is equal to 1.
