<s>
In	O
probability	O
theory	O
and	O
statistics	O
,	O
variance	O
is	O
the	O
expectation	O
of	O
the	O
squared	B-General_Concept
deviation	I-General_Concept
of	O
a	O
random	O
variable	O
from	O
its	O
population	O
mean	O
or	O
sample	O
mean	O
.	O
</s>
<s>
Variance	O
has	O
a	O
central	O
role	O
in	O
statistics	O
,	O
where	O
some	O
ideas	O
that	O
use	O
it	O
include	O
descriptive	B-General_Concept
statistics	I-General_Concept
,	O
statistical	O
inference	O
,	O
hypothesis	O
testing	O
,	O
goodness	O
of	O
fit	O
,	O
and	O
Monte	B-Algorithm
Carlo	I-Algorithm
sampling	I-Algorithm
.	O
</s>
<s>
The	O
variance	O
is	O
the	O
square	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
the	O
second	O
central	B-General_Concept
moment	I-General_Concept
of	O
a	O
distribution	O
,	O
and	O
the	O
covariance	O
of	O
the	O
random	O
variable	O
with	O
itself	O
,	O
and	O
it	O
is	O
often	O
represented	O
by	O
,	O
,	O
,	O
,	O
or	O
.	O
</s>
<s>
An	O
advantage	O
of	O
variance	O
as	O
a	O
measure	O
of	O
dispersion	O
is	O
that	O
it	O
is	O
more	O
amenable	O
to	O
algebraic	O
manipulation	O
than	O
other	O
measures	O
of	O
dispersion	O
such	O
as	O
the	O
expected	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
;	O
for	O
example	O
,	O
the	O
variance	O
of	O
a	O
sum	O
of	O
uncorrelated	O
random	O
variables	O
is	O
equal	O
to	O
the	O
sum	O
of	O
their	O
variances	O
.	O
</s>
<s>
A	O
disadvantage	O
of	O
the	O
variance	O
for	O
practical	O
applications	O
is	O
that	O
,	O
unlike	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
its	O
units	O
differ	O
from	O
the	O
random	O
variable	O
,	O
which	O
is	O
why	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
more	O
commonly	O
reported	O
as	O
a	O
measure	O
of	O
dispersion	O
once	O
the	O
calculation	O
is	O
finished	O
.	O
</s>
<s>
The	O
great	O
body	O
of	O
available	O
statistics	O
show	O
us	O
that	O
the	O
deviations	B-General_Concept
of	O
a	O
human	O
measurement	O
from	O
its	O
mean	O
follow	O
very	O
closely	O
the	O
Normal	O
Law	O
of	O
Errors	O
,	O
and	O
,	O
therefore	O
,	O
that	O
the	O
variability	O
may	O
be	O
uniformly	O
measured	O
by	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
corresponding	O
to	O
the	O
square	O
root	O
of	O
the	O
mean	B-Algorithm
square	I-Algorithm
error	I-Algorithm
.	O
</s>
<s>
When	O
there	O
are	O
two	O
independent	O
causes	O
of	O
variability	O
capable	O
of	O
producing	O
in	O
an	O
otherwise	O
uniform	O
population	O
distributions	O
with	O
standard	B-General_Concept
deviations	I-General_Concept
and	O
,	O
it	O
is	O
found	O
that	O
the	O
distribution	O
,	O
when	O
both	O
causes	O
act	O
together	O
,	O
has	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
It	O
is	O
therefore	O
desirable	O
in	O
analysing	O
the	O
causes	O
of	O
variability	O
to	O
deal	O
with	O
the	O
square	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
as	O
the	O
measure	O
of	O
variability	O
.	O
</s>
<s>
The	O
variance	O
of	O
a	O
random	O
variable	O
is	O
the	O
expected	O
value	O
of	O
the	O
squared	B-General_Concept
deviation	I-General_Concept
from	I-General_Concept
the	I-General_Concept
mean	I-General_Concept
of	O
,	O
:	O
</s>
<s>
The	O
variance	O
is	O
typically	O
designated	O
as	O
,	O
or	O
sometimes	O
as	O
or	O
,	O
or	O
symbolically	O
as	O
or	O
simply	O
(	O
pronounced	O
"	O
sigma	B-Application
squared	O
"	O
)	O
.	O
</s>
<s>
This	O
equation	O
should	O
not	O
be	O
used	O
for	O
computations	O
using	O
floating	B-Algorithm
point	I-Algorithm
arithmetic	I-Algorithm
,	O
because	O
it	O
suffers	O
from	O
catastrophic	B-Algorithm
cancellation	I-Algorithm
if	O
the	O
two	O
components	O
of	O
the	O
equation	O
are	O
similar	O
in	O
magnitude	O
.	O
</s>
<s>
For	O
other	O
numerically	O
stable	O
alternatives	O
,	O
see	O
Algorithms	B-Algorithm
for	I-Algorithm
calculating	I-Algorithm
variance	I-Algorithm
.	O
</s>
<s>
The	O
variance	O
of	O
a	O
set	O
of	O
equally	O
likely	O
values	O
can	O
be	O
equivalently	O
expressed	O
,	O
without	O
directly	O
referring	O
to	O
the	O
mean	O
,	O
in	O
terms	O
of	O
squared	B-General_Concept
deviations	I-General_Concept
of	O
all	O
pairwise	O
squared	O
distances	O
of	O
points	O
from	O
each	O
other	O
:	O
</s>
<s>
A	O
fair	O
six-sided	B-Language
die	I-Language
can	O
be	O
modeled	O
as	O
a	O
discrete	O
random	O
variable	O
,	O
,	O
with	O
outcomes	O
1	O
through	O
6	O
,	O
each	O
with	O
equal	O
probability	O
1/6	O
.	O
</s>
<s>
Similar	O
decompositions	O
are	O
possible	O
for	O
the	O
sum	B-General_Concept
of	I-General_Concept
squared	I-General_Concept
deviations	I-General_Concept
(	O
sum	O
of	O
squares	O
,	O
)	O
:	O
</s>
<s>
Unlike	O
the	O
expected	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
,	O
the	O
variance	O
of	O
a	O
variable	O
has	O
units	O
that	O
are	O
the	O
square	O
of	O
the	O
units	O
of	O
the	O
variable	O
itself	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
describing	O
data	O
sets	O
via	O
their	O
standard	B-General_Concept
deviation	I-General_Concept
or	O
root	B-General_Concept
mean	I-General_Concept
square	I-General_Concept
deviation	I-General_Concept
is	O
often	O
preferred	O
over	O
using	O
the	O
variance	O
.	O
</s>
<s>
In	O
the	O
dice	B-Language
example	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
,	O
slightly	O
larger	O
than	O
the	O
expected	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
of1.5	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
and	O
the	O
expected	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
can	O
both	O
be	O
used	O
as	O
an	O
indicator	O
of	O
the	O
"	O
spread	O
"	O
of	O
a	O
distribution	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
more	O
amenable	O
to	O
algebraic	O
manipulation	O
than	O
the	O
expected	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
,	O
and	O
,	O
together	O
with	O
variance	O
and	O
its	O
generalization	O
covariance	O
,	O
is	O
used	O
frequently	O
in	O
theoretical	O
statistics	O
;	O
however	O
the	O
expected	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
tends	O
to	O
be	O
more	O
robust	O
as	O
it	O
is	O
less	O
sensitive	O
to	O
outliers	O
arising	O
from	O
measurement	O
anomalies	O
or	O
an	O
unduly	O
heavy-tailed	O
distribution	O
.	O
</s>
<s>
This	O
formula	O
for	O
the	O
variance	O
of	O
the	O
mean	O
is	O
used	O
in	O
the	O
definition	O
of	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
sample	O
mean	O
,	O
which	O
is	O
used	O
in	O
the	O
central	O
limit	O
theorem	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
additional	O
correlated	O
observations	O
are	O
not	O
as	O
effective	O
as	O
additional	O
independent	O
observations	O
at	O
reducing	O
the	O
uncertainty	B-General_Concept
of	I-General_Concept
the	I-General_Concept
mean	I-General_Concept
.	O
</s>
<s>
Most	O
simply	O
,	O
the	O
sample	O
variance	O
is	O
computed	O
as	O
an	O
average	O
of	O
squared	B-General_Concept
deviations	I-General_Concept
about	O
the	O
(	O
sample	O
)	O
mean	O
,	O
by	O
dividing	O
by	O
n	O
.	O
However	O
,	O
using	O
values	O
other	O
than	O
n	O
improves	O
the	O
estimator	O
in	O
various	O
ways	O
.	O
</s>
<s>
Four	O
common	O
values	O
for	O
the	O
denominator	O
are	O
n	O
,	O
n−1	O
,	O
n+1	O
,	O
and	O
n−	O
1.5	O
:	O
n	O
is	O
the	O
simplest	O
(	O
population	O
variance	O
of	O
the	O
sample	O
)	O
,	O
n−1	O
eliminates	O
bias	O
,	O
n+1	O
minimizes	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
for	O
the	O
normal	O
distribution	O
,	O
and	O
n−	O
1.5	O
mostly	O
eliminates	O
bias	O
in	O
unbiased	O
estimation	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
for	O
the	O
normal	O
distribution	O
.	O
</s>
<s>
Firstly	O
,	O
if	O
the	O
true	O
population	O
mean	O
is	O
unknown	O
,	O
then	O
the	O
sample	O
variance	O
(	O
which	O
uses	O
the	O
sample	O
mean	O
in	O
place	O
of	O
the	O
true	O
mean	O
)	O
is	O
a	O
biased	O
estimator	O
:	O
it	O
underestimates	O
the	O
variance	O
by	O
a	O
factor	O
of	O
(	O
n−1	O
)	O
/	O
n	O
;	O
correcting	O
by	O
this	O
factor	O
(	O
dividing	O
by	O
n−1	O
instead	O
of	O
n	O
)	O
is	O
called	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
.	O
</s>
<s>
Secondly	O
,	O
the	O
sample	O
variance	O
does	O
not	O
generally	O
minimize	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
between	O
sample	O
variance	O
and	O
population	O
variance	O
.	O
</s>
<s>
Correcting	O
for	O
bias	O
often	O
makes	O
this	O
worse	O
:	O
one	O
can	O
always	O
choose	O
a	O
scale	O
factor	O
that	O
performs	O
better	O
than	O
the	O
corrected	O
sample	O
variance	O
,	O
though	O
the	O
optimal	O
scale	O
factor	O
depends	O
on	O
the	O
excess	O
kurtosis	B-Error_Name
of	O
the	O
population	O
(	O
see	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
:	O
variance	O
)	O
,	O
and	O
introduces	O
bias	O
.	O
</s>
<s>
For	O
the	O
normal	O
distribution	O
,	O
dividing	O
by	O
n+1	O
(	O
instead	O
of	O
n−1	O
or	O
n	O
)	O
minimizes	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
.	O
</s>
<s>
In	O
many	O
practical	O
situations	O
,	O
the	O
true	B-Architecture
variance	I-Architecture
of	O
a	O
population	O
is	O
not	O
known	O
a	O
priori	O
and	O
must	O
be	O
computed	O
somehow	O
.	O
</s>
<s>
Directly	O
taking	O
the	O
variance	O
of	O
the	O
sample	O
data	O
gives	O
the	O
average	O
of	O
the	O
squared	B-General_Concept
deviations	I-General_Concept
:	O
</s>
<s>
The	O
use	O
of	O
the	O
term	O
n−1	O
is	O
called	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
,	O
and	O
it	O
is	O
also	O
used	O
in	O
sample	O
covariance	O
and	O
the	O
sample	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
(	O
the	O
square	O
root	O
of	O
variance	O
)	O
.	O
</s>
<s>
The	O
square	O
root	O
is	O
a	O
concave	O
function	O
and	O
thus	O
introduces	O
negative	O
bias	O
(	O
by	O
Jensen	O
's	O
inequality	O
)	O
,	O
which	O
depends	O
on	O
the	O
distribution	O
,	O
and	O
thus	O
the	O
corrected	O
sample	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
(	O
using	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
)	O
is	O
biased	O
.	O
</s>
<s>
The	O
unbiased	O
estimation	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
a	O
technically	O
involved	O
problem	O
,	O
though	O
for	O
the	O
normal	O
distribution	O
using	O
the	O
term	O
n−	O
1.5	O
yields	O
an	O
almost	O
unbiased	O
estimator	O
.	O
</s>
<s>
where	O
κ	O
is	O
the	O
kurtosis	B-Error_Name
of	O
the	O
distribution	O
and	O
μ4	O
is	O
the	O
fourth	O
central	B-General_Concept
moment	I-General_Concept
.	O
</s>
<s>
The	O
F-test	B-General_Concept
of	I-General_Concept
equality	I-General_Concept
of	I-General_Concept
variances	I-General_Concept
and	O
the	O
chi	B-General_Concept
square	I-General_Concept
tests	I-General_Concept
are	O
adequate	O
when	O
the	O
sample	O
is	O
normally	O
distributed	O
.	O
</s>
<s>
Several	O
non	O
parametric	O
tests	O
have	O
been	O
proposed	O
:	O
these	O
include	O
the	O
Barton	O
–	O
David	O
–	O
Ansari	O
–	O
Freund	O
–	O
Siegel	O
–	O
Tukey	O
test	O
,	O
the	O
Capon	O
test	O
,	O
Mood	B-General_Concept
test	I-General_Concept
,	O
the	O
Klotz	O
test	O
and	O
the	O
Sukhatme	O
test	O
.	O
</s>
<s>
Resampling	B-General_Concept
methods	O
,	O
which	O
include	O
the	B-Application
bootstrap	I-Application
and	O
the	O
jackknife	B-General_Concept
,	O
may	O
be	O
used	O
to	O
test	O
the	O
equality	O
of	O
variances	O
.	O
</s>
<s>
The	O
result	O
is	O
a	O
positive	B-Algorithm
semi-definite	I-Algorithm
square	I-Algorithm
matrix	I-Algorithm
,	O
commonly	O
referred	O
to	O
as	O
the	O
variance-covariance	O
matrix	O
(	O
or	O
simply	O
as	O
the	O
covariance	O
matrix	O
)	O
.	O
</s>
<s>
If	O
is	O
a	O
vector	O
-	O
and	O
complex-valued	O
random	O
variable	O
,	O
with	O
values	O
in	O
then	O
the	O
covariance	O
matrix	O
is	O
where	O
is	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
This	O
matrix	O
is	O
also	O
positive	O
semi-definite	O
and	O
square	O
.	O
</s>
