<s>
In	O
statistics	O
,	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
is	O
the	O
use	O
of	O
n−1	O
instead	O
of	O
n	O
in	O
the	O
formula	O
for	O
the	O
sample	O
variance	O
and	O
sample	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
,	O
where	O
n	O
is	O
the	O
number	O
of	O
observations	O
in	O
a	O
sample	O
.	O
</s>
<s>
It	O
also	O
partially	O
corrects	O
the	O
bias	O
in	O
the	O
estimation	O
of	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
However	O
,	O
the	O
correction	O
often	O
increases	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
in	O
these	O
estimations	O
.	O
</s>
<s>
In	O
some	O
literature	O
,	O
the	O
above	O
factor	O
is	O
called	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
.	O
</s>
<s>
One	O
can	O
understand	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
as	O
the	O
degrees	O
of	O
freedom	O
in	O
the	O
residuals	O
vector	O
(	O
residuals	O
,	O
not	O
errors	O
,	O
because	O
the	O
population	O
mean	O
is	O
unknown	O
)	O
:	O
</s>
<s>
For	O
a	O
more	O
intuitive	O
explanation	O
of	O
the	O
need	O
for	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
,	O
see	O
.	O
</s>
<s>
Generally	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
is	O
an	O
approach	O
to	O
reduce	O
the	O
bias	O
due	O
to	O
finite	O
sample	O
size	O
.	O
</s>
<s>
Such	O
finite-sample	O
bias	O
correction	O
is	O
also	O
needed	O
for	O
other	O
estimates	O
like	O
skew	B-General_Concept
and	O
kurtosis	B-Error_Name
,	O
but	O
in	O
these	O
the	O
inaccuracies	O
are	O
often	O
significantly	O
larger	O
.	O
</s>
<s>
For	O
instance	O
a	O
correct	O
correction	O
for	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
depends	O
on	O
the	O
kurtosis	B-Error_Name
(	O
normalized	O
central	O
4th	O
moment	O
)	O
,	O
but	O
this	O
again	O
has	O
a	O
finite	O
sample	O
bias	O
and	O
it	O
depends	O
on	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
i.e.	O
</s>
<s>
There	O
are	O
three	O
caveats	O
to	O
consider	O
regarding	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
:	O
</s>
<s>
It	O
does	O
not	O
yield	O
an	O
unbiased	O
estimator	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
The	O
corrected	O
estimator	O
often	O
has	O
a	O
higher	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
(	O
MSE	O
)	O
than	O
the	O
uncorrected	O
estimator	O
.	O
</s>
<s>
Firstly	O
,	O
while	O
the	O
sample	O
variance	O
(	O
using	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
)	O
is	O
an	O
unbiased	O
estimator	O
of	O
the	O
population	O
variance	O
,	O
its	O
square	O
root	O
,	O
the	O
sample	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
,	O
is	O
a	O
biased	O
estimate	O
of	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
;	O
because	O
the	O
square	O
root	O
is	O
a	O
concave	O
function	O
,	O
the	O
bias	O
is	O
downward	O
,	O
by	O
Jensen	O
's	O
inequality	O
.	O
</s>
<s>
There	O
is	O
no	O
general	O
formula	O
for	O
an	O
unbiased	O
estimator	O
of	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
,	O
though	O
there	O
are	O
correction	O
factors	O
for	O
particular	O
distributions	O
,	O
such	O
as	O
the	O
normal	O
;	O
see	O
unbiased	O
estimation	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
for	O
details	O
.	O
</s>
<s>
Secondly	O
,	O
the	O
unbiased	O
estimator	O
does	O
not	O
minimize	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
(	O
MSE	O
)	O
,	O
and	O
generally	O
has	O
worse	O
MSE	O
than	O
the	O
uncorrected	O
estimator	O
(	O
this	O
varies	O
with	O
excess	O
kurtosis	B-Error_Name
)	O
.	O
</s>
<s>
The	O
optimal	O
value	O
depends	O
on	O
excess	O
kurtosis	B-Error_Name
,	O
as	O
discussed	O
in	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
:	O
variance	O
;	O
for	O
the	O
normal	O
distribution	O
this	O
is	O
optimized	O
by	O
dividing	O
by	O
n+1	O
(	O
instead	O
of	O
n−1	O
or	O
n	O
)	O
.	O
</s>
<s>
Thirdly	O
,	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
is	O
only	O
necessary	O
when	O
the	O
population	O
mean	O
is	O
unknown	O
,	O
and	O
one	O
is	O
estimating	O
both	O
population	O
mean	O
and	O
population	O
variance	O
from	O
a	O
given	O
sample	O
,	O
using	O
the	O
sample	O
mean	O
to	O
estimate	O
the	O
population	O
mean	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
population	O
mean	O
is	O
known	O
,	O
then	O
the	O
deviations	O
of	O
the	O
observations	O
from	O
the	O
population	O
mean	O
have	O
n	O
degrees	O
of	O
freedom	O
(	O
because	O
the	O
mean	O
is	O
not	O
being	O
estimated	O
–	O
the	O
deviations	O
are	O
not	O
residuals	O
but	O
errors	O
)	O
and	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
is	O
not	O
applicable	O
.	O
</s>
<s>
Then	O
and	O
,	O
but	O
with	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
,	O
,	O
which	O
is	O
an	O
unbiased	O
estimate	O
(	O
if	O
all	O
possible	O
samples	O
of	O
n	O
=	O
2	O
are	O
taken	O
and	O
this	O
method	O
is	O
used	O
,	O
the	O
average	O
estimate	O
will	O
be	O
12.4	O
,	O
same	O
as	O
the	O
sample	O
variance	O
with	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
.	O
)	O
</s>
<s>
This	O
correction	O
is	O
so	O
common	O
that	O
the	O
term	O
"	O
sample	O
variance	O
"	O
and	O
"	O
sample	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
"	O
are	O
frequently	O
used	O
to	O
mean	O
the	O
corrected	O
estimators	O
(	O
unbiased	O
sample	O
variation	O
,	O
less	O
biased	O
sample	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
)	O
,	O
using	O
n−1	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviations	I-General_Concept
will	O
then	O
be	O
the	O
square	O
roots	O
of	O
the	O
respective	O
variances	O
.	O
</s>
<s>
Since	O
the	O
square	O
root	O
introduces	O
bias	O
,	O
the	O
terminology	O
"	O
uncorrected	O
"	O
and	O
"	O
corrected	O
"	O
is	O
preferred	O
for	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
estimators	O
:	O
</s>
<s>
sn	O
is	O
the	O
uncorrected	O
sample	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
(	O
i.e.	O
</s>
<s>
s	O
is	O
the	O
corrected	O
sample	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
(	O
i.e.	O
</s>
<s>
As	O
a	O
background	O
fact	O
,	O
we	O
use	O
the	O
identity	O
which	O
follows	O
from	O
the	O
definition	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
and	O
linearity	O
of	O
expectation	O
.	O
</s>
