<s>
In	O
statistics	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
a	O
measure	O
of	O
the	O
amount	O
of	O
variation	O
or	O
dispersion	O
of	O
a	O
set	O
of	O
values	O
.	O
</s>
<s>
A	O
low	O
standard	B-General_Concept
deviation	I-General_Concept
indicates	O
that	O
the	O
values	O
tend	O
to	O
be	O
close	O
to	O
the	O
mean	O
(	O
also	O
called	O
the	O
expected	O
value	O
)	O
of	O
the	O
set	O
,	O
while	O
a	O
high	O
standard	B-General_Concept
deviation	I-General_Concept
indicates	O
that	O
the	O
values	O
are	O
spread	O
out	O
over	O
a	O
wider	O
range	O
.	O
</s>
<s>
Standard	B-General_Concept
deviation	I-General_Concept
may	O
be	O
abbreviated	O
SD	O
,	O
and	O
is	O
most	O
commonly	O
represented	O
in	O
mathematical	O
texts	O
and	O
equations	O
by	O
the	O
lower	O
case	O
Greek	O
letter	O
σ	B-Application
(	O
sigma	B-Application
)	O
,	O
for	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
,	O
or	O
the	O
Latin	O
letter	O
s	B-Language
,	O
for	O
the	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
a	O
random	O
variable	O
,	O
sample	O
,	O
statistical	O
population	O
,	O
data	B-General_Concept
set	I-General_Concept
,	O
or	O
probability	O
distribution	O
is	O
the	O
square	O
root	O
of	O
its	O
variance	O
.	O
</s>
<s>
It	O
is	O
algebraically	O
simpler	O
,	O
though	O
in	O
practice	O
less	O
robust	O
,	O
than	O
the	O
average	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
A	O
useful	O
property	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
that	O
,	O
unlike	O
the	O
variance	O
,	O
it	O
is	O
expressed	O
in	O
the	O
same	O
unit	O
as	O
the	O
data	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
a	O
population	O
or	O
sample	O
and	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
a	O
statistic	O
(	O
e.g.	O
,	O
of	O
the	O
sample	O
mean	O
)	O
are	O
quite	O
different	O
,	O
but	O
related	O
.	O
</s>
<s>
The	O
sample	O
mean	O
's	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
set	O
of	O
means	O
that	O
would	O
be	O
found	O
by	O
drawing	O
an	O
infinite	O
number	O
of	O
repeated	O
samples	O
from	O
the	O
population	O
and	O
computing	O
a	O
mean	O
for	O
each	O
sample	O
.	O
</s>
<s>
The	O
mean	O
's	O
standard	B-General_Concept
error	I-General_Concept
turns	O
out	O
to	O
equal	O
the	O
population	B-General_Concept
standard	I-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
and	O
is	O
estimated	O
by	O
using	O
the	O
sample	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>
For	O
example	O
,	O
a	O
poll	O
's	O
standard	B-General_Concept
error	I-General_Concept
(	O
what	O
is	O
reported	O
as	O
the	O
margin	O
of	O
error	O
of	O
the	O
poll	O
)	O
,	O
is	O
the	O
expected	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
estimated	O
mean	O
if	O
the	O
same	O
poll	O
were	O
to	O
be	O
conducted	O
multiple	O
times	O
.	O
</s>
<s>
Thus	O
,	O
the	O
standard	B-General_Concept
error	I-General_Concept
estimates	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
an	O
estimate	O
,	O
which	O
itself	O
measures	O
how	O
much	O
the	O
estimate	O
depends	O
on	O
the	O
particular	O
sample	O
that	O
was	O
taken	O
from	O
the	O
population	O
.	O
</s>
<s>
In	O
science	O
,	O
it	O
is	O
common	O
to	O
report	O
both	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
data	O
(	O
as	O
a	O
summary	O
statistic	O
)	O
and	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
estimate	O
(	O
as	O
a	O
measure	O
of	O
potential	O
error	O
in	O
the	O
findings	O
)	O
.	O
</s>
<s>
By	O
convention	O
,	O
only	O
effects	O
more	O
than	O
two	O
standard	B-General_Concept
errors	I-General_Concept
away	O
from	O
a	O
null	O
expectation	O
are	O
considered	O
"	B-General_Concept
statistically	I-General_Concept
significant	I-General_Concept
"	I-General_Concept
,	O
a	O
safeguard	O
against	O
spurious	O
conclusion	O
that	O
is	O
really	O
due	O
to	O
random	O
sampling	O
error	O
.	O
</s>
<s>
When	O
only	O
a	O
sample	O
of	O
data	O
from	O
a	O
population	O
is	O
available	O
,	O
the	O
term	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sample	O
or	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
can	O
refer	O
to	O
either	O
the	O
above-mentioned	O
quantity	O
as	O
applied	O
to	O
those	O
data	O
,	O
or	O
to	O
a	O
modified	O
quantity	O
that	O
is	O
an	O
unbiased	O
estimate	O
of	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
(	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
entire	O
population	O
)	O
.	O
</s>
<s>
For	O
a	O
finite	O
set	O
of	O
numbers	O
,	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
is	O
found	O
by	O
taking	O
the	O
square	O
root	O
of	O
the	O
average	O
of	O
the	O
squared	B-General_Concept
deviations	I-General_Concept
of	O
the	O
values	O
subtracted	O
from	O
their	O
average	O
value	O
.	O
</s>
<s>
First	O
,	O
calculate	O
the	O
deviations	B-General_Concept
of	O
each	O
data	O
point	O
from	O
the	O
mean	O
,	O
and	O
square	O
the	O
result	O
of	O
each	O
:	O
</s>
<s>
and	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
is	O
equal	O
to	O
the	O
square	O
root	O
of	O
the	O
variance	O
:	O
</s>
<s>
If	O
the	O
values	O
instead	O
were	O
a	O
random	O
sample	O
drawn	O
from	O
some	O
large	O
parent	O
population	O
(	O
for	O
example	O
,	O
they	O
were	O
8	O
students	O
randomly	O
and	O
independently	O
chosen	O
from	O
a	O
class	O
of	O
2million	O
)	O
,	O
then	O
one	O
divides	O
by	O
instead	O
of	O
in	O
the	O
denominator	O
of	O
the	O
last	O
formula	O
,	O
and	O
the	O
result	O
is	O
In	O
that	O
case	O
,	O
the	O
result	O
of	O
the	O
original	O
formula	O
would	O
be	O
called	O
the	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
and	O
denoted	O
by	O
s	B-Language
instead	O
of	O
Dividing	O
by	O
n−1	O
rather	O
than	O
by	O
n	O
gives	O
an	O
unbiased	O
estimate	O
of	O
the	O
variance	O
of	O
the	O
larger	O
parent	O
population	O
.	O
</s>
<s>
This	O
is	O
known	O
as	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
.	O
</s>
<s>
If	O
the	O
population	O
of	O
interest	O
is	O
approximately	O
normally	O
distributed	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
provides	O
information	O
on	O
the	O
proportion	O
of	O
observations	O
above	O
or	O
below	O
certain	O
values	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
average	O
height	O
for	O
adult	O
men	O
in	O
the	O
United	O
States	O
is	O
about	O
70inches	O
,	O
with	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
around	O
3inches	O
.	O
</s>
<s>
This	O
means	O
that	O
most	O
men	O
(	O
about	O
68%	O
,	O
assuming	O
a	O
normal	O
distribution	O
)	O
have	O
a	O
height	O
within	O
3inches	O
of	O
the	O
mean	O
(	O
67	O
–	O
73inches	O
)	O
one	O
standard	O
deviationand	O
almost	O
all	O
men	O
(	O
about	O
95%	O
)	O
have	O
a	O
height	O
within	O
6inches	O
of	O
the	O
mean	O
(	O
64	O
–	O
76inches	O
)	O
two	O
standard	B-General_Concept
deviations	I-General_Concept
.	O
</s>
<s>
If	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
were	O
zero	O
,	O
then	O
all	O
men	O
would	O
be	O
exactly	O
70inches	O
tall	O
.	O
</s>
<s>
If	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
were	O
20inches	O
,	O
then	O
men	O
would	O
have	O
much	O
more	O
variable	O
heights	O
,	O
with	O
a	O
typical	O
range	O
of	O
about	O
50	O
–	O
90inches	O
.	O
</s>
<s>
Three	O
standard	B-General_Concept
deviations	I-General_Concept
account	O
for	O
99.73	O
%	O
of	O
the	O
sample	O
population	O
being	O
studied	O
,	O
assuming	O
the	O
distribution	O
is	O
normal	O
or	O
bell-shaped	O
(	O
see	O
the	O
68	O
–	O
95	O
–	O
99.7	O
rule	O
,	O
or	O
the	O
empirical	O
rule	O
,	O
for	O
more	O
information	O
)	O
.	O
</s>
<s>
Using	O
words	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
the	O
square	O
root	O
of	O
the	O
variance	O
of	O
X	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
a	O
probability	O
distribution	O
is	O
the	O
same	O
as	O
that	O
of	O
a	O
random	O
variable	O
having	O
that	O
distribution	O
.	O
</s>
<s>
Not	O
all	O
random	O
variables	O
have	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
If	O
the	O
distribution	O
has	O
fat	O
tails	O
going	O
out	O
to	O
infinity	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
might	O
not	O
exist	O
,	O
because	O
the	O
integral	O
might	O
not	O
converge	O
.	O
</s>
<s>
The	O
normal	O
distribution	O
has	O
tails	O
going	O
out	O
to	O
infinity	O
,	O
but	O
its	O
mean	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
do	O
exist	O
,	O
because	O
the	O
tails	O
diminish	O
quickly	O
enough	O
.	O
</s>
<s>
The	O
Pareto	O
distribution	O
with	O
parameter	O
has	O
a	O
mean	O
,	O
but	O
not	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
(	O
loosely	O
speaking	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
infinite	O
)	O
.	O
</s>
<s>
The	O
Cauchy	O
distribution	O
has	O
neither	O
a	O
mean	O
nor	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
and	O
where	O
the	O
integrals	B-Algorithm
are	O
definite	B-Algorithm
integrals	I-Algorithm
taken	O
for	O
x	O
ranging	O
over	O
the	O
set	O
of	O
possible	O
values	O
of	O
the	O
random	O
variableX	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
parametric	B-General_Concept
family	I-General_Concept
of	I-General_Concept
distributions	I-General_Concept
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
can	O
be	O
expressed	O
in	O
terms	O
of	O
the	O
parameters	O
.	O
</s>
<s>
One	O
can	O
find	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
an	O
entire	O
population	O
in	O
cases	O
(	O
such	O
as	O
standardized	B-General_Concept
testing	I-General_Concept
)	O
where	O
every	O
member	O
of	O
a	O
population	O
is	O
sampled	O
.	O
</s>
<s>
In	O
cases	O
where	O
that	O
cannot	O
be	O
done	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
σ	B-Application
is	O
estimated	O
by	O
examining	O
a	O
random	O
sample	O
taken	O
from	O
the	O
population	O
and	O
computing	O
a	O
statistic	O
of	O
the	O
sample	O
,	O
which	O
is	O
used	O
as	O
an	O
estimate	O
of	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
Such	O
a	O
statistic	O
is	O
called	O
an	O
estimator	O
,	O
and	O
the	O
estimator	O
(	O
or	O
the	O
value	O
of	O
the	O
estimator	O
,	O
namely	O
the	O
estimate	O
)	O
is	O
called	O
a	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
and	O
is	O
denoted	O
by	O
s	B-Language
(	O
possibly	O
with	O
modifiers	O
)	O
.	O
</s>
<s>
Unlike	O
in	O
the	O
case	O
of	O
estimating	O
the	O
population	O
mean	O
,	O
for	O
which	O
the	O
sample	O
mean	O
is	O
a	O
simple	O
estimator	O
with	O
many	O
desirable	O
properties	O
(	O
unbiased	O
,	O
efficient	O
,	O
maximum	O
likelihood	O
)	O
,	O
there	O
is	O
no	O
single	O
estimator	O
for	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
with	O
all	O
these	O
properties	O
,	O
and	O
unbiased	O
estimation	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
a	O
very	O
technically	O
involved	O
problem	O
.	O
</s>
<s>
Most	O
often	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
estimated	O
using	O
the	O
corrected	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
(	O
using	O
N−1	O
)	O
,	O
defined	O
below	O
,	O
and	O
this	O
is	O
often	O
referred	O
to	O
as	O
the	O
"	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
"	O
,	O
without	O
qualifiers	O
.	O
</s>
<s>
However	O
,	O
other	O
estimators	O
are	O
better	O
in	O
other	O
respects	O
:	O
the	O
uncorrected	O
estimator	O
(	O
using	O
N	O
)	O
yields	O
lower	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
,	O
while	O
using	O
N−	O
1.5	O
(	O
for	O
the	O
normal	O
distribution	O
)	O
almost	O
completely	O
eliminates	O
bias	O
.	O
</s>
<s>
The	O
formula	O
for	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
(	O
of	O
a	O
finite	O
population	O
)	O
can	O
be	O
applied	O
to	O
the	O
sample	O
,	O
using	O
the	O
size	O
of	O
the	O
sample	O
as	O
the	O
size	O
of	O
the	O
population	O
(	O
though	O
the	O
actual	O
population	O
size	O
from	O
which	O
the	O
sample	O
is	O
drawn	O
may	O
be	O
much	O
larger	O
)	O
.	O
</s>
<s>
This	O
estimator	O
,	O
denoted	O
by	O
sN	O
,	O
is	O
known	O
as	O
the	O
uncorrected	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
or	O
sometimes	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sample	O
(	O
considered	O
as	O
the	O
entire	O
population	O
)	O
,	O
and	O
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
where	O
are	O
the	O
observed	O
values	O
of	O
the	O
sample	O
items	O
,	O
and	O
is	O
the	O
mean	O
value	O
of	O
these	O
observations	O
,	O
while	O
the	O
denominatorN	O
stands	O
for	O
the	O
size	O
of	O
the	O
sample	O
:	O
this	O
is	O
the	O
square	O
root	O
of	O
the	O
sample	O
variance	O
,	O
which	O
is	O
the	O
average	O
of	O
the	O
squared	B-General_Concept
deviations	I-General_Concept
about	O
the	O
sample	O
mean	O
.	O
</s>
<s>
Thus	O
for	O
very	O
large	O
sample	O
sizes	O
,	O
the	O
uncorrected	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
generally	O
acceptable	O
.	O
</s>
<s>
This	O
estimator	O
also	O
has	O
a	O
uniformly	O
smaller	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
than	O
the	O
corrected	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
Here	O
taking	O
the	O
square	O
root	O
introduces	O
further	O
downward	O
bias	O
,	O
by	O
Jensen	O
's	O
inequality	O
,	O
due	O
to	O
the	O
square	O
root	O
's	O
being	O
a	O
concave	O
function	O
.	O
</s>
<s>
An	O
unbiased	O
estimator	O
for	O
the	O
variance	O
is	O
given	O
by	O
applying	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
,	O
using	O
N−1	O
instead	O
of	O
N	O
to	O
yield	O
the	O
unbiased	O
sample	O
variance	O
,	O
denoted	O
s2	O
:	O
</s>
<s>
N−1	O
corresponds	O
to	O
the	O
number	O
of	O
degrees	O
of	O
freedom	O
in	O
the	O
vector	O
of	O
deviations	B-General_Concept
from	O
the	O
mean	O
,	O
</s>
<s>
often	O
)	O
,	O
yielding	O
the	O
corrected	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
denoted	O
by	O
s	B-Language
:	O
</s>
<s>
As	O
explained	O
above	O
,	O
while	O
s2	O
is	O
an	O
unbiased	O
estimator	O
for	O
the	O
population	O
variance	O
,	O
s	B-Language
is	O
still	O
a	O
biased	O
estimator	O
for	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
,	O
though	O
markedly	O
less	O
biased	O
than	O
the	O
uncorrected	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
This	O
estimator	O
is	O
commonly	O
used	O
and	O
generally	O
known	O
simply	O
as	O
the	O
"	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
"	O
.	O
</s>
<s>
For	O
unbiased	O
estimation	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
there	O
is	O
no	O
formula	O
that	O
works	O
across	O
all	O
distributions	O
,	O
unlike	O
for	O
mean	O
and	O
variance	O
.	O
</s>
<s>
Instead	O
,	O
s	B-Language
is	O
used	O
as	O
a	O
basis	O
,	O
and	O
is	O
scaled	O
by	O
a	O
correction	O
factor	O
to	O
produce	O
an	O
unbiased	O
estimate	O
.	O
</s>
<s>
For	O
the	O
normal	O
distribution	O
,	O
an	O
unbiased	O
estimator	O
is	O
given	O
by	O
s/c4	O
,	O
where	O
the	O
correction	O
factor	O
(	O
which	O
depends	O
on	O
N	O
)	O
is	O
given	O
in	O
terms	O
of	O
the	O
Gamma	O
function	O
,	O
and	O
equals	O
:	O
</s>
<s>
This	O
arises	O
because	O
the	O
sampling	O
distribution	O
of	O
the	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
follows	O
a	O
(	O
scaled	O
)	O
chi	O
distribution	O
,	O
and	O
the	O
correction	O
factor	O
is	O
the	O
mean	O
of	O
the	O
chi	O
distribution	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
we	O
obtain	O
by	O
sampling	O
a	O
distribution	O
is	O
itself	O
not	O
absolutely	O
accurate	O
,	O
both	O
for	O
mathematical	O
reasons	O
(	O
explained	O
here	O
by	O
the	O
confidence	O
interval	O
)	O
and	O
for	O
practical	O
reasons	O
of	O
measurement	O
(	O
measurement	O
error	O
)	O
.	O
</s>
<s>
A	O
small	O
population	O
of	O
N	O
=	O
2	O
has	O
only	O
1	O
degree	O
of	O
freedom	O
for	O
estimating	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
A	O
larger	O
population	O
of	O
N	O
=	O
10	O
has	O
9	O
degrees	O
of	O
freedom	O
for	O
estimating	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
These	O
same	O
formulae	O
can	O
be	O
used	O
to	O
obtain	O
confidence	O
intervals	O
on	O
the	O
variance	O
of	O
residuals	O
from	O
a	O
least	B-Algorithm
squares	I-Algorithm
fit	I-Algorithm
under	O
standard	O
normal	O
theory	O
,	O
where	O
k	O
is	O
now	O
the	O
number	O
of	O
degrees	O
of	O
freedom	O
for	O
error	O
.	O
</s>
<s>
For	O
a	O
set	O
of	O
N	O
>	O
4	O
data	O
spanning	O
a	O
range	O
of	O
values	O
R	O
,	O
an	O
upper	O
bound	O
on	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
s	B-Language
is	O
given	O
by	O
s	B-Language
=	O
0.6R	O
.	O
</s>
<s>
An	O
estimate	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
for	O
N	O
>	O
100	O
data	O
taken	O
to	O
be	O
approximately	O
normal	O
follows	O
from	O
the	O
heuristic	O
that	O
95%	O
of	O
the	O
area	O
under	O
the	O
normal	O
curve	O
lies	O
roughly	O
two	O
standard	B-General_Concept
deviations	I-General_Concept
to	O
either	O
side	O
of	O
the	O
mean	O
,	O
so	O
that	O
,	O
with	O
95%	O
probability	O
the	O
total	O
range	O
of	O
values	O
R	O
represents	O
four	O
standard	B-General_Concept
deviations	I-General_Concept
so	O
that	O
s	B-Language
≈	O
R/4	O
.	O
</s>
<s>
This	O
so-called	O
range	O
rule	O
is	O
useful	O
in	O
sample	O
size	O
estimation	O
,	O
as	O
the	O
range	O
of	O
possible	O
values	O
is	O
easier	O
to	O
estimate	O
than	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
Other	O
divisors	O
K(N )	O
of	O
the	O
range	O
such	O
that	O
s	B-Language
≈	O
R/K	O
( N	O
)	O
are	O
available	O
for	O
other	O
values	O
of	O
N	O
and	O
for	O
non-normal	O
distributions	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
invariant	O
under	O
changes	O
in	O
location	O
,	O
and	O
scales	O
directly	O
with	O
the	O
scale	O
of	O
the	O
random	O
variable	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sum	O
of	O
two	O
random	O
variables	O
can	O
be	O
related	O
to	O
their	O
individual	O
standard	B-General_Concept
deviations	I-General_Concept
and	O
the	O
covariance	O
between	O
them	O
:	O
</s>
<s>
The	O
calculation	O
of	O
the	O
sum	B-General_Concept
of	I-General_Concept
squared	I-General_Concept
deviations	I-General_Concept
can	O
be	O
related	O
to	O
moments	O
calculated	O
directly	O
from	O
the	O
data	O
.	O
</s>
<s>
The	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
can	O
be	O
computed	O
as	O
:	O
</s>
<s>
which	O
means	O
that	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
equal	O
to	O
the	O
square	O
root	O
of	O
the	O
difference	O
between	O
the	O
average	O
of	O
the	O
squares	O
of	O
the	O
values	O
and	O
the	O
square	O
of	O
the	O
average	O
value	O
.	O
</s>
<s>
See	O
computational	O
formula	O
for	O
the	O
variance	O
for	O
proof	O
,	O
and	O
for	O
an	O
analogous	O
result	O
for	O
the	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
A	O
large	O
standard	B-General_Concept
deviation	I-General_Concept
indicates	O
that	O
the	O
data	O
points	O
can	O
spread	O
far	O
from	O
the	O
mean	O
and	O
a	O
small	O
standard	B-General_Concept
deviation	I-General_Concept
indicates	O
that	O
they	O
are	O
clustered	O
closely	O
around	O
the	O
mean	O
.	O
</s>
<s>
Their	O
standard	B-General_Concept
deviations	I-General_Concept
are	O
7	O
,	O
5	O
,	O
and	O
1	O
,	O
respectively	O
.	O
</s>
<s>
The	O
third	O
population	O
has	O
a	O
much	O
smaller	O
standard	B-General_Concept
deviation	I-General_Concept
than	O
the	O
other	O
two	O
because	O
its	O
values	O
are	O
all	O
close	O
to	O
7	O
.	O
</s>
<s>
These	O
standard	B-General_Concept
deviations	I-General_Concept
have	O
the	O
same	O
units	O
as	O
the	O
data	O
points	O
themselves	O
.	O
</s>
<s>
If	O
,	O
for	O
instance	O
,	O
the	O
data	B-General_Concept
set	I-General_Concept
{	O
0	O
,	O
6	O
,	O
8	O
,	O
14}	O
represents	O
the	O
ages	O
of	O
a	O
population	O
of	O
four	O
siblings	O
in	O
years	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
5	O
years	O
.	O
</s>
<s>
It	O
has	O
a	O
mean	O
of	O
1007	O
meters	O
,	O
and	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
5	O
meters	O
.	O
</s>
<s>
Standard	B-General_Concept
deviation	I-General_Concept
may	O
serve	O
as	O
a	O
measure	O
of	O
uncertainty	O
.	O
</s>
<s>
In	O
physical	O
science	O
,	O
for	O
example	O
,	O
the	O
reported	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
a	O
group	O
of	O
repeated	O
measurements	O
gives	O
the	O
precision	O
of	O
those	O
measurements	O
.	O
</s>
<s>
When	O
deciding	O
whether	O
measurements	O
agree	O
with	O
a	O
theoretical	O
prediction	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
those	O
measurements	O
is	O
of	O
crucial	O
importance	O
:	O
if	O
the	O
mean	O
of	O
the	O
measurements	O
is	O
too	O
far	O
away	O
from	O
the	O
prediction	O
(	O
with	O
the	O
distance	O
measured	O
in	O
standard	B-General_Concept
deviations	I-General_Concept
)	O
,	O
then	O
the	O
theory	O
being	O
tested	O
probably	O
needs	O
to	O
be	O
revised	O
.	O
</s>
<s>
This	O
makes	O
sense	O
since	O
they	O
fall	O
outside	O
the	O
range	O
of	O
values	O
that	O
could	O
reasonably	O
be	O
expected	O
to	O
occur	O
,	O
if	O
the	O
prediction	O
were	O
correct	O
and	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
appropriately	O
quantified	O
.	O
</s>
<s>
While	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
does	O
measure	O
how	O
far	O
typical	O
values	O
tend	O
to	O
be	O
from	O
the	O
mean	O
,	O
other	O
measures	O
are	O
available	O
.	O
</s>
<s>
An	O
example	O
is	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
,	O
which	O
might	O
be	O
considered	O
a	O
more	O
direct	O
measure	O
of	O
average	O
distance	O
,	O
compared	O
to	O
the	O
root	B-General_Concept
mean	I-General_Concept
square	I-General_Concept
distance	I-General_Concept
inherent	O
in	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
The	O
practical	O
value	O
of	O
understanding	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
a	O
set	O
of	O
values	O
is	O
in	O
appreciating	O
how	O
much	O
variation	O
there	O
is	O
from	O
the	O
average	O
(	O
mean	O
)	O
.	O
</s>
<s>
Standard	B-General_Concept
deviation	I-General_Concept
is	O
often	O
used	O
to	O
compare	O
real-world	O
data	O
against	O
a	O
model	O
to	O
test	O
the	O
model	O
.	O
</s>
<s>
By	O
using	O
standard	B-General_Concept
deviations	I-General_Concept
,	O
a	O
minimum	O
and	O
maximum	O
value	O
can	O
be	O
calculated	O
that	O
the	O
averaged	O
weight	O
will	O
be	O
within	O
some	O
very	O
high	O
percentage	O
of	O
the	O
time	O
(	O
99.9	O
%	O
or	O
more	O
)	O
.	O
</s>
<s>
Particle	O
physics	O
conventionally	O
uses	O
a	O
standard	O
of	O
"	O
5	O
sigma	B-Application
"	O
for	O
the	O
declaration	O
of	O
a	O
discovery	O
.	O
</s>
<s>
A	O
five-sigma	O
level	O
translates	O
to	O
one	O
chance	O
in	O
3.5	O
million	O
that	O
a	O
random	O
fluctuation	O
would	O
yield	O
the	O
result	O
.	O
</s>
<s>
Thus	O
,	O
while	O
these	O
two	O
cities	O
may	O
each	O
have	O
the	O
same	O
average	O
maximum	O
temperature	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
daily	O
maximum	O
temperature	O
for	O
the	O
coastal	O
city	O
will	O
be	O
less	O
than	O
that	O
of	O
the	O
inland	O
city	O
as	O
,	O
on	O
any	O
particular	O
day	O
,	O
the	O
actual	O
maximum	O
temperature	O
is	O
more	O
likely	O
to	O
be	O
farther	O
from	O
the	O
average	O
maximum	O
temperature	O
for	O
the	O
inland	O
city	O
than	O
for	O
the	O
coastal	O
one	O
.	O
</s>
<s>
In	O
finance	O
,	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
often	O
used	O
as	O
a	O
measure	O
of	O
the	O
risk	O
associated	O
with	O
price-fluctuations	O
of	O
a	O
given	O
asset	O
(	O
stocks	O
,	O
bonds	O
,	O
property	O
,	O
etc	O
.	O
</s>
<s>
Standard	B-General_Concept
deviation	I-General_Concept
provides	O
a	O
quantified	O
estimate	O
of	O
the	O
uncertainty	O
of	O
future	O
returns	O
.	O
</s>
<s>
Stock	O
A	O
over	O
the	O
past	O
20	O
years	O
had	O
an	O
average	O
return	O
of	O
10	O
percent	O
,	O
with	O
a	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
20	O
percentage	O
points	O
(	O
pp	O
)	O
and	O
Stock	O
B	O
,	O
over	O
the	O
same	O
period	O
,	O
had	O
average	O
returns	O
of	O
12	O
percent	O
but	O
a	O
higher	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
30	O
pp	O
.	O
</s>
<s>
On	O
the	O
basis	O
of	O
risk	O
and	O
return	O
,	O
an	O
investor	O
may	O
decide	O
that	O
Stock	O
A	O
is	O
the	O
safer	O
choice	O
,	O
because	O
Stock	O
B	O
's	O
additional	O
two	O
percentage	O
points	O
of	O
return	O
is	O
not	O
worth	O
the	O
additional	O
10	O
pp	O
standard	B-General_Concept
deviation	I-General_Concept
(	O
greater	O
risk	O
or	O
uncertainty	O
of	O
the	O
expected	O
return	O
)	O
.	O
</s>
<s>
When	O
considering	O
more	O
extreme	O
possible	O
returns	O
or	O
outcomes	O
in	O
future	O
,	O
an	O
investor	O
should	O
expect	O
results	O
of	O
as	O
much	O
as	O
10	O
percent	O
plus	O
or	O
minus	O
60	O
pp	O
,	O
or	O
a	O
range	O
from	O
70	O
percent	O
to	O
−50	O
percent	O
,	O
which	O
includes	O
outcomes	O
for	O
three	O
standard	B-General_Concept
deviations	I-General_Concept
from	O
the	O
average	O
return	O
(	O
about	O
99.7	O
percent	O
of	O
probable	O
returns	O
)	O
.	O
</s>
<s>
Squaring	O
the	O
difference	O
in	O
each	O
period	O
and	O
taking	O
the	O
average	O
gives	O
the	O
overall	B-General_Concept
variance	I-General_Concept
of	O
the	O
return	O
of	O
the	O
asset	O
.	O
</s>
<s>
Finding	O
the	O
square	O
root	O
of	O
this	O
variance	O
will	O
give	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
investment	O
tool	O
in	O
question	O
.	O
</s>
<s>
Population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
is	O
used	O
to	O
set	O
the	O
width	O
of	O
Bollinger	O
Bands	O
,	O
a	O
technical	O
analysis	O
tool	O
.	O
</s>
<s>
Financial	O
time	O
series	O
are	O
known	O
to	O
be	O
non-stationary	O
series	O
,	O
whereas	O
the	O
statistical	O
calculations	O
above	O
,	O
such	O
as	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
apply	O
only	O
to	O
stationary	O
series	O
.	O
</s>
<s>
If	O
our	O
three	O
given	O
values	O
were	O
all	O
equal	O
,	O
then	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
would	O
be	O
zero	O
and	O
P	O
would	O
lie	O
on	O
L	O
.	O
So	O
it	O
is	O
not	O
unreasonable	O
to	O
assume	O
that	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
related	O
to	O
the	O
distance	O
of	O
P	O
to	O
L	O
.	O
That	O
is	O
indeed	O
the	O
case	O
.	O
</s>
<s>
A	O
little	O
algebra	O
shows	O
that	O
the	O
distance	O
between	O
P	O
and	O
M	O
(	O
which	O
is	O
the	O
same	O
as	O
the	O
orthogonal	O
distance	O
between	O
P	O
and	O
the	O
line	O
L	O
)	O
is	O
equal	O
to	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
vector	O
(	O
x1	O
,	O
x2	O
,	O
x3	O
)	O
,	O
multiplied	O
by	O
the	O
square	O
root	O
of	O
the	O
number	O
of	O
dimensions	O
of	O
the	O
vector	O
(	O
3	O
in	O
this	O
case	O
)	O
.	O
</s>
<s>
An	O
observation	O
is	O
rarely	O
more	O
than	O
a	O
few	O
standard	B-General_Concept
deviations	I-General_Concept
away	O
from	O
the	O
mean	O
.	O
</s>
<s>
Chebyshev	O
's	O
inequality	O
ensures	O
that	O
,	O
for	O
all	O
distributions	O
for	O
which	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
defined	O
,	O
the	O
amount	O
of	O
data	O
within	O
a	O
number	O
of	O
standard	B-General_Concept
deviations	I-General_Concept
of	O
the	O
mean	O
is	O
at	O
least	O
as	O
much	O
as	O
given	O
in	O
the	O
following	O
table	O
.	O
</s>
<s>
where	O
μ	O
is	O
the	O
expected	O
value	O
of	O
the	O
random	O
variables	O
,	O
σ	B-Application
equals	O
their	O
distribution	O
's	O
standard	B-General_Concept
deviation	I-General_Concept
divided	O
by	O
n1/2	O
,	O
and	O
n	O
is	O
the	O
number	O
of	O
random	O
variables	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
therefore	O
is	O
simply	O
a	O
scaling	O
variable	O
that	O
adjusts	O
how	O
broad	O
the	O
curve	O
will	O
be	O
,	O
though	O
it	O
also	O
appears	O
in	O
the	O
normalizing	O
constant	O
.	O
</s>
<s>
If	O
a	O
data	O
distribution	O
is	O
approximately	O
normal	O
,	O
then	O
the	O
proportion	O
of	O
data	O
values	O
within	O
z	O
standard	B-General_Concept
deviations	I-General_Concept
of	O
the	O
mean	O
is	O
defined	O
by	O
:	O
</s>
<s>
If	O
a	O
data	O
distribution	O
is	O
approximately	O
normal	O
then	O
about	O
68	O
percent	O
of	O
the	O
data	O
values	O
are	O
within	O
one	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
mean	O
(	O
mathematically	O
,	O
μ±σ	O
,	O
where	O
μ	O
is	O
the	O
arithmetic	O
mean	O
)	O
,	O
about	O
95	O
percent	O
are	O
within	O
two	O
standard	B-General_Concept
deviations	I-General_Concept
( μ±2σ	O
)	O
,	O
and	O
about	O
99.7	O
percent	O
lie	O
within	O
three	O
standard	B-General_Concept
deviations	I-General_Concept
( μ±3σ	O
)	O
.	O
</s>
<s>
The	O
mean	O
and	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
a	O
set	B-General_Concept
of	I-General_Concept
data	I-General_Concept
are	O
descriptive	B-General_Concept
statistics	I-General_Concept
usually	O
reported	O
together	O
.	O
</s>
<s>
In	O
a	O
certain	O
sense	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
a	O
"	O
natural	O
"	O
measure	O
of	O
statistical	O
dispersion	O
if	O
the	O
center	O
of	O
the	O
data	O
is	O
measured	O
about	O
the	O
mean	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
from	O
the	O
mean	O
is	O
smaller	O
than	O
from	O
any	O
other	O
point	O
.	O
</s>
<s>
Using	O
calculus	O
or	O
by	O
completing	O
the	O
square	O
,	O
it	O
is	O
possible	O
to	O
show	O
that	O
σ(r )	O
has	O
a	O
unique	O
minimum	O
at	O
the	O
mean	O
:	O
</s>
<s>
Variability	O
can	O
also	O
be	O
measured	O
by	O
the	O
coefficient	O
of	O
variation	O
,	O
which	O
is	O
the	O
ratio	O
of	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
to	O
the	O
mean	O
.	O
</s>
<s>
We	O
can	O
obtain	O
this	O
by	O
determining	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
sampled	O
mean	O
.	O
</s>
<s>
Assuming	O
statistical	O
independence	O
of	O
the	O
values	O
in	O
the	O
sample	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
mean	O
is	O
related	O
to	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
distribution	O
by	O
:	O
</s>
<s>
In	O
order	O
to	O
estimate	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
mean	O
it	O
is	O
necessary	O
to	O
know	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
entire	O
population	O
beforehand	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
a	O
series	O
of	O
10	O
measurements	O
of	O
a	O
previously	O
unknown	O
quantity	O
is	O
performed	O
in	O
a	O
laboratory	O
,	O
it	O
is	O
possible	O
to	O
calculate	O
the	O
resulting	O
sample	O
mean	O
and	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
but	O
it	O
is	O
impossible	O
to	O
calculate	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
mean	O
.	O
</s>
<s>
However	O
,	O
one	O
can	O
estimate	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
entire	O
population	O
from	O
the	O
sample	O
,	O
and	O
thus	O
obtain	O
an	O
estimate	O
for	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
mean	O
.	O
</s>
<s>
The	O
following	O
two	O
formulas	O
can	O
represent	O
a	O
running	O
(	O
repeatedly	O
updated	O
)	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
Given	O
the	O
results	O
of	O
these	O
running	O
summations	O
,	O
the	O
values	O
N	O
,	O
s1	O
,	O
s2	O
can	O
be	O
used	O
at	O
any	O
time	O
to	O
compute	O
the	O
current	O
value	O
of	O
the	O
running	O
standard	B-General_Concept
deviation	I-General_Concept
:	O
</s>
<s>
Similarly	O
for	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
</s>
<s>
In	O
a	O
computer	O
implementation	O
,	O
as	O
the	O
two	O
sj	O
sums	O
become	O
large	O
,	O
we	O
need	O
to	O
consider	O
round-off	B-Algorithm
error	I-Algorithm
,	O
arithmetic	B-Algorithm
overflow	I-Algorithm
,	O
and	O
arithmetic	B-Algorithm
underflow	I-Algorithm
.	O
</s>
<s>
The	O
method	O
below	O
calculates	O
the	O
running	O
sums	O
method	O
with	O
reduced	O
rounding	B-Algorithm
errors	I-Algorithm
.	O
</s>
<s>
Applying	O
this	O
method	O
to	O
a	O
time	O
series	O
will	O
result	O
in	O
successive	O
values	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
corresponding	O
to	O
n	O
data	O
points	O
as	O
n	O
grows	O
larger	O
with	O
each	O
new	O
sample	O
,	O
rather	O
than	O
a	O
constant-width	O
sliding	O
window	O
calculation	O
.	O
</s>
<s>
And	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
equations	O
remain	O
unchanged	O
.	O
</s>
<s>
The	O
incremental	O
method	O
with	O
reduced	O
rounding	B-Algorithm
errors	I-Algorithm
can	O
also	O
be	O
applied	O
,	O
with	O
some	O
additional	O
complexity	O
.	O
</s>
<s>
The	O
term	O
standard	B-General_Concept
deviation	I-General_Concept
was	O
first	O
used	O
in	O
writing	O
by	O
Karl	O
Pearson	O
in	O
1894	O
,	O
following	O
his	O
use	O
of	O
it	O
in	O
lectures	O
.	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
index	O
(	O
SDI	O
)	O
is	O
used	O
in	O
external	O
quality	O
assessments	O
,	O
particularly	O
for	O
medical	O
laboratories	O
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
can	O
be	O
illustrated	O
with	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
ellipse	O
(	O
see	O
Multivariate	O
normal	O
distribution	O
§	O
Geometric	O
interpretation''	O
)	O
.	O
</s>
