<s>
The	O
standard	B-General_Concept
error	I-General_Concept
(	O
SE	O
)	O
of	O
a	O
statistic	O
(	O
usually	O
an	O
estimate	O
of	O
a	O
parameter	O
)	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
its	O
sampling	B-General_Concept
distribution	I-General_Concept
or	O
an	O
estimate	O
of	O
that	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
If	O
the	O
statistic	O
is	O
the	O
sample	O
mean	O
,	O
it	O
is	O
called	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
(	O
SEM	O
)	O
.	O
</s>
<s>
The	O
sampling	B-General_Concept
distribution	I-General_Concept
of	O
a	O
mean	O
is	O
generated	O
by	O
repeated	O
sampling	O
from	O
the	O
same	O
population	O
and	O
recording	O
of	O
the	O
sample	O
means	O
obtained	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
relationship	O
between	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
and	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
such	O
that	O
,	O
for	O
a	O
given	O
sample	O
size	O
,	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
equals	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
divided	O
by	O
the	O
square	O
root	O
of	O
the	O
sample	O
size	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
is	O
a	O
measure	O
of	O
the	O
dispersion	O
of	O
sample	O
means	O
around	O
the	O
population	O
mean	O
.	O
</s>
<s>
In	O
regression	O
analysis	O
,	O
the	O
term	O
"	O
standard	B-General_Concept
error	I-General_Concept
"	O
refers	O
either	O
to	O
the	O
square	O
root	O
of	O
the	O
reduced	B-General_Concept
chi-squared	I-General_Concept
statistic	I-General_Concept
or	O
the	O
standard	B-General_Concept
error	I-General_Concept
for	O
a	O
particular	O
regression	O
coefficient	O
(	O
as	O
used	O
in	O
,	O
say	O
,	O
confidence	O
intervals	O
)	O
.	O
</s>
<s>
Suppose	O
a	O
statistically	O
independent	O
sample	O
of	O
observations	O
is	O
taken	O
from	O
a	O
statistical	O
population	O
with	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
.	O
</s>
<s>
The	O
mean	O
value	O
calculated	O
from	O
the	O
sample	O
,	O
,	O
will	O
have	O
an	O
associated	O
standard	B-General_Concept
error	I-General_Concept
on	O
the	O
mean	O
,	O
,	O
given	O
by	O
:	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
population	O
being	O
sampled	O
is	O
seldom	O
known	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
is	O
usually	O
estimated	O
by	O
replacing	O
with	O
the	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
instead	O
:	O
</s>
<s>
As	O
this	O
is	O
only	O
an	O
estimator	O
for	O
the	O
true	O
"	O
standard	B-General_Concept
error	I-General_Concept
"	O
,	O
it	O
is	O
common	O
to	O
see	O
other	O
notations	O
here	O
such	O
as	O
:	O
</s>
<s>
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
population	O
(	O
)	O
,	O
</s>
<s>
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sample	O
(	O
)	O
,	O
</s>
<s>
the	O
estimator	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
mean	O
(	O
,	O
which	O
is	O
the	O
most	O
often	O
calculated	O
quantity	O
,	O
and	O
is	O
also	O
often	O
colloquially	O
called	O
the	O
standard	B-General_Concept
error	I-General_Concept
)	O
.	O
</s>
<s>
When	O
the	O
sample	O
size	O
is	O
small	O
,	O
using	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sample	O
instead	O
of	O
the	O
true	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
population	O
will	O
tend	O
to	O
systematically	O
underestimate	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
,	O
and	O
therefore	O
also	O
the	O
standard	B-General_Concept
error	I-General_Concept
.	O
</s>
<s>
See	O
unbiased	O
estimation	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
for	O
further	O
discussion	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
error	I-General_Concept
on	O
the	O
mean	O
may	O
be	O
derived	O
from	O
the	O
variance	O
of	O
a	O
sum	O
of	O
independent	O
random	O
variables	O
,	O
given	O
the	O
definition	O
of	O
variance	O
and	O
some	O
simple	O
properties	O
thereof	O
.	O
</s>
<s>
where	O
we	O
've	O
approximated	O
the	O
standard	B-General_Concept
deviations	I-General_Concept
,	O
i.e.	O
,	O
the	O
uncertainties	O
,	O
of	O
the	O
measurements	O
themselves	O
with	O
the	O
best	O
value	O
for	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
population	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
error	I-General_Concept
is	O
,	O
by	O
definition	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
which	O
is	O
simply	O
the	O
square	O
root	O
of	O
the	O
variance	O
:	O
</s>
<s>
The	O
standard	B-General_Concept
error	I-General_Concept
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
Student	O
t-distribution	O
.	O
</s>
<s>
Small	O
samples	O
are	O
somewhat	O
more	O
likely	O
to	O
underestimate	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
and	O
have	O
a	O
mean	O
that	O
differs	O
from	O
the	O
true	O
population	O
mean	O
,	O
and	O
the	O
Student	O
t-distribution	O
accounts	O
for	O
the	O
probability	O
of	O
these	O
events	O
with	O
somewhat	O
heavier	O
tails	O
compared	O
to	O
a	O
Gaussian	O
.	O
</s>
<s>
To	O
estimate	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
a	O
Student	O
t-distribution	O
it	O
is	O
sufficient	O
to	O
use	O
the	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
"	O
s	O
"	O
instead	O
of	O
σ	O
,	O
and	O
we	O
could	O
use	O
this	O
value	O
to	O
calculate	O
confidence	O
intervals	O
.	O
</s>
<s>
If	O
the	O
sampling	B-General_Concept
distribution	I-General_Concept
is	O
normally	O
distributed	O
,	O
the	O
sample	O
mean	O
,	O
the	O
standard	B-General_Concept
error	I-General_Concept
,	O
and	O
the	O
quantiles	O
of	O
the	O
normal	O
distribution	O
can	O
be	O
used	O
to	O
calculate	O
confidence	O
intervals	O
for	O
the	O
true	O
population	O
mean	O
.	O
</s>
<s>
The	O
following	O
expressions	O
can	O
be	O
used	O
to	O
calculate	O
the	O
upper	O
and	O
lower	O
95%	O
confidence	O
limits	O
,	O
where	O
is	O
equal	O
to	O
the	O
sample	O
mean	O
,	O
is	O
equal	O
to	O
the	O
standard	B-General_Concept
error	I-General_Concept
for	O
the	O
sample	O
mean	O
,	O
and	O
1.96	O
is	O
the	O
approximate	O
value	O
of	O
the	O
97.5	O
percentile	O
point	O
of	O
the	O
normal	O
distribution	O
:	O
</s>
<s>
In	O
particular	O
,	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
a	O
sample	O
statistic	O
(	O
such	O
as	O
sample	O
mean	O
)	O
is	O
the	O
actual	O
or	O
estimated	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sample	O
mean	O
in	O
the	O
process	O
by	O
which	O
it	O
was	O
generated	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
it	O
is	O
the	O
actual	O
or	O
estimated	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sampling	B-General_Concept
distribution	I-General_Concept
of	O
the	O
sample	O
statistic	O
.	O
</s>
<s>
The	O
notation	O
for	O
standard	B-General_Concept
error	I-General_Concept
can	O
be	O
any	O
one	O
of	O
SE	O
,	O
SEM	O
(	O
for	O
standard	B-General_Concept
error	I-General_Concept
of	I-General_Concept
measurement	I-General_Concept
or	O
mean	O
)	O
,	O
or	O
SE	O
.	O
</s>
<s>
Standard	B-General_Concept
errors	I-General_Concept
provide	O
simple	O
measures	O
of	O
uncertainty	O
in	O
a	O
value	O
and	O
are	O
often	O
used	O
because	O
:	O
</s>
<s>
in	O
many	O
cases	O
,	O
if	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
several	O
individual	O
quantities	O
is	O
known	O
then	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
some	O
function	O
of	O
the	O
quantities	O
can	O
be	O
easily	O
calculated	O
;	O
</s>
<s>
as	O
the	O
sample	O
size	O
tends	O
to	O
infinity	O
the	O
central	O
limit	O
theorem	O
guarantees	O
that	O
the	O
sampling	B-General_Concept
distribution	I-General_Concept
of	O
the	O
mean	O
is	O
asymptotically	O
normal	O
.	O
</s>
<s>
In	O
scientific	O
and	O
technical	O
literature	O
,	O
experimental	O
data	O
are	O
often	O
summarized	O
either	O
using	O
the	O
mean	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sample	O
data	O
or	O
the	O
mean	O
with	O
the	O
standard	B-General_Concept
error	I-General_Concept
.	O
</s>
<s>
However	O
,	O
the	O
mean	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
are	O
descriptive	B-General_Concept
statistics	I-General_Concept
,	O
whereas	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
is	O
descriptive	O
of	O
the	O
random	O
sampling	O
process	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sample	O
data	O
is	O
a	O
description	O
of	O
the	O
variation	O
in	O
measurements	O
,	O
while	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
is	O
a	O
probabilistic	O
statement	O
about	O
how	O
the	O
sample	O
size	O
will	O
provide	O
a	O
better	O
bound	O
on	O
estimates	O
of	O
the	O
population	O
mean	O
,	O
in	O
light	O
of	O
the	O
central	O
limit	O
theorem	O
.	O
</s>
<s>
Put	O
simply	O
,	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
sample	O
mean	O
is	O
an	O
estimate	O
of	O
how	O
far	O
the	O
sample	O
mean	O
is	O
likely	O
to	O
be	O
from	O
the	O
population	O
mean	O
,	O
whereas	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sample	O
is	O
the	O
degree	O
to	O
which	O
individuals	O
within	O
the	O
sample	O
differ	O
from	O
the	O
sample	O
mean	O
.	O
</s>
<s>
If	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
is	O
finite	O
,	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
of	O
the	O
sample	O
will	O
tend	O
to	O
zero	O
with	O
increasing	O
sample	O
size	O
,	O
because	O
the	O
estimate	O
of	O
the	O
population	O
mean	O
will	O
improve	O
,	O
while	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sample	O
will	O
tend	O
to	O
approximate	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
as	O
the	O
sample	O
size	O
increases	O
.	O
</s>
<s>
The	O
formula	O
given	O
above	O
for	O
the	O
standard	B-General_Concept
error	I-General_Concept
assumes	O
that	O
the	O
population	O
is	O
infinite	O
.	O
</s>
<s>
When	O
the	O
sampling	O
fraction	O
(	O
often	O
termed	O
f	O
)	O
is	O
large	O
(	O
approximately	O
at	O
5%	O
or	O
more	O
)	O
in	O
an	O
enumerative	O
study	O
,	O
the	O
estimate	O
of	O
the	O
standard	B-General_Concept
error	I-General_Concept
must	O
be	O
corrected	O
by	O
multiplying	O
by	O
a	O
''	O
finite	O
population	O
correction''	O
(	O
a.k.a.	O
</s>
<s>
The	O
unbiased	O
standard	B-General_Concept
error	I-General_Concept
plots	O
as	O
the	O
ρ	O
=	O
0	O
diagonal	O
line	O
with	O
log-log	O
slope	O
−½	O
.	O
</s>
<s>
If	O
values	O
of	O
the	O
measured	O
quantity	O
A	O
are	O
not	O
statistically	O
independent	O
but	O
have	O
been	O
obtained	O
from	O
known	O
locations	O
in	O
parameter	O
space	O
x'''	O
,	O
an	O
unbiased	O
estimate	O
of	O
the	O
true	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
(	O
actually	O
a	O
correction	O
on	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
part	O
)	O
may	O
be	O
obtained	O
by	O
multiplying	O
the	O
calculated	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
sample	O
by	O
the	O
factor	O
f''	O
:	O
</s>
<s>
See	O
also	O
unbiased	O
estimation	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
for	O
more	O
discussion	O
.	O
</s>
