<s>
In	O
statistics	O
,	O
robust	B-General_Concept
measures	I-General_Concept
of	I-General_Concept
scale	I-General_Concept
are	O
methods	O
that	O
quantify	O
the	O
statistical	O
dispersion	O
in	O
a	O
sample	O
of	O
numerical	O
data	O
while	O
resisting	O
outliers	O
.	O
</s>
<s>
The	O
most	O
common	O
such	O
robust	O
statistics	O
are	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
(	O
IQR	B-General_Concept
)	O
and	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
(	O
MAD	O
)	O
.	O
</s>
<s>
These	O
are	O
contrasted	O
with	O
conventional	O
or	O
non-robust	O
measures	O
of	O
scale	O
,	O
such	O
as	O
sample	O
variance	O
or	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
which	O
are	O
greatly	O
influenced	O
by	O
outliers	O
.	O
</s>
<s>
To	O
illustrate	O
robustness	O
,	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
can	O
be	O
made	O
arbitrarily	O
large	O
by	O
increasing	O
exactly	O
one	O
observation	O
(	O
it	O
has	O
a	O
breakdown	O
point	O
of	O
0	O
,	O
as	O
it	O
can	O
be	O
contaminated	O
by	O
a	O
single	O
point	O
)	O
,	O
a	O
defect	O
that	O
is	O
not	O
shared	O
by	O
robust	O
statistics	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
common	O
robust	B-General_Concept
measures	I-General_Concept
of	I-General_Concept
scale	I-General_Concept
is	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
(	O
IQR	B-General_Concept
)	O
,	O
the	O
difference	O
between	O
the	O
75th	O
percentile	O
and	O
the	O
25th	O
percentile	O
of	O
a	O
sample	O
;	O
this	O
is	O
the	O
25%	O
trimmed	O
range	B-General_Concept
,	O
an	O
example	O
of	O
an	O
L-estimator	O
.	O
</s>
<s>
Other	O
trimmed	O
ranges	O
,	O
such	O
as	O
the	O
interdecile	B-General_Concept
range	I-General_Concept
(	O
10%	O
trimmed	O
range	B-General_Concept
)	O
can	O
also	O
be	O
used	O
.	O
</s>
<s>
For	O
a	O
Gaussian	O
distribution	O
,	O
IQR	B-General_Concept
is	O
related	O
to	O
as	O
.	O
</s>
<s>
Another	O
familiar	O
robust	B-General_Concept
measure	I-General_Concept
of	I-General_Concept
scale	I-General_Concept
is	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
(	O
MAD	O
)	O
,	O
the	O
median	O
of	O
the	O
absolute	O
values	O
of	O
the	O
differences	O
between	O
the	O
data	O
values	O
and	O
the	O
overall	O
median	O
of	O
the	O
data	O
set	O
;	O
for	O
a	O
Gaussian	O
distribution	O
,	O
MAD	O
is	O
related	O
to	O
as	O
(	O
the	O
derivation	O
can	O
be	O
found	O
here	O
)	O
.	O
</s>
<s>
Robust	B-General_Concept
measures	I-General_Concept
of	I-General_Concept
scale	I-General_Concept
can	O
be	O
used	O
as	O
estimators	O
of	O
properties	O
of	O
the	O
population	O
,	O
either	O
for	O
parameter	O
estimation	O
or	O
as	O
estimators	O
of	O
their	O
own	O
expected	O
value	O
.	O
</s>
<s>
For	O
example	O
,	O
robust	O
estimators	O
of	O
scale	O
are	O
used	O
to	O
estimate	O
the	O
population	B-General_Concept
variance	I-General_Concept
or	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
,	O
generally	O
by	O
multiplying	O
by	O
a	O
scale	B-Algorithm
factor	I-Algorithm
to	O
make	O
it	O
an	O
unbiased	O
consistent	O
estimator	O
;	O
see	O
scale	O
parameter	O
:	O
estimation	O
.	O
</s>
<s>
For	O
example	O
,	O
dividing	O
the	O
IQR	B-General_Concept
by	O
2	O
erf−1( 	O
1/2	O
)	O
(	O
approximately	O
1.349	O
)	O
,	O
makes	O
it	O
an	O
unbiased	O
,	O
consistent	O
estimator	O
for	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
if	O
the	O
data	O
follow	O
a	O
normal	O
distribution	O
.	O
</s>
<s>
In	O
other	O
situations	O
,	O
it	O
makes	O
more	O
sense	O
to	O
think	O
of	O
a	O
robust	B-General_Concept
measure	I-General_Concept
of	I-General_Concept
scale	I-General_Concept
as	O
an	O
estimator	O
of	O
its	O
own	O
expected	O
value	O
,	O
interpreted	O
as	O
an	O
alternative	O
to	O
the	O
population	B-General_Concept
variance	I-General_Concept
or	O
standard	B-General_Concept
deviation	I-General_Concept
as	O
a	O
measure	O
of	O
scale	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
MAD	O
of	O
a	O
sample	O
from	O
a	O
standard	O
Cauchy	O
distribution	O
is	O
an	O
estimator	O
of	O
the	O
population	O
MAD	O
,	O
which	O
in	O
this	O
case	O
is	O
1	O
,	O
whereas	O
the	O
population	B-General_Concept
variance	I-General_Concept
does	O
not	O
exist	O
.	O
</s>
<s>
These	O
robust	O
estimators	O
typically	O
have	O
inferior	O
statistical	O
efficiency	O
compared	O
to	O
conventional	O
estimators	O
for	O
data	O
drawn	O
from	O
a	O
distribution	O
without	O
outliers	O
(	O
such	O
as	O
a	O
normal	O
distribution	O
)	O
,	O
but	O
have	O
superior	O
efficiency	O
for	O
data	O
drawn	O
from	O
a	O
mixture	O
distribution	O
or	O
from	O
a	O
heavy-tailed	O
distribution	O
,	O
for	O
which	O
non-robust	O
measures	O
such	O
as	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
should	O
not	O
be	O
used	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
data	O
drawn	O
from	O
the	O
normal	O
distribution	O
,	O
the	O
MAD	O
is	O
37%	O
as	O
efficient	O
as	O
the	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
while	O
the	O
Rousseeuw	O
–	O
Croux	O
estimator	O
Qn	O
is	O
88%	O
as	O
efficient	O
as	O
the	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
it	O
computes	O
a	O
symmetric	O
statistic	O
about	O
a	O
location	O
estimate	O
,	O
thus	O
not	O
dealing	O
with	O
skewness	B-General_Concept
.	O
</s>
<s>
For	O
a	O
sample	O
from	O
a	O
normal	O
distribution	O
,	O
Sn	O
is	O
approximately	O
unbiased	O
for	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
even	O
down	O
to	O
very	O
modest	O
sample	O
sizes	O
(	O
<	O
1%	O
bias	O
for	O
n	O
=	O
10	O
)	O
.	O
</s>
<s>
For	O
a	O
large	O
sample	O
from	O
a	O
normal	O
distribution	O
,	O
2.219144465985075864722Qn	O
is	O
approximately	O
unbiased	O
for	O
the	O
population	B-General_Concept
standard	I-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
Then	O
the	O
operator	O
can	O
calculate	O
a	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
for	O
each	O
object	O
,	O
and	O
look	O
for	O
outliers	O
.	O
</s>
<s>
Any	O
object	O
with	O
an	O
unusually	O
large	O
standard	B-General_Concept
deviation	I-General_Concept
probably	O
has	O
an	O
outlier	O
in	O
its	O
data	O
.	O
</s>
<s>
A	O
bootstrap	B-Application
calculation	O
could	O
be	O
used	O
to	O
determine	O
a	O
confidence	O
interval	O
narrower	O
than	O
that	O
calculated	O
from	O
σ	O
,	O
and	O
so	O
obtain	O
some	O
benefit	O
from	O
a	O
large	O
amount	O
of	O
extra	O
work	O
.	O
</s>
<s>
These	O
procedures	O
are	O
robust	O
against	O
procedural	O
errors	O
which	O
are	O
not	O
modeled	O
by	O
the	O
assumption	O
that	O
the	O
balance	O
has	O
a	O
fixed	O
known	O
standard	B-General_Concept
deviation	I-General_Concept
σ	O
.	O
</s>
<s>
Before	O
trusting	O
the	O
results	O
of	O
100	O
objects	O
weighed	O
just	O
three	O
times	O
each	O
to	O
have	O
confidence	O
intervals	O
calculated	O
from	O
σ	O
,	O
it	O
is	O
necessary	O
to	O
test	O
for	O
and	O
remove	O
a	O
reasonable	O
number	O
of	O
outliers	O
(	O
testing	O
the	O
assumption	O
that	O
the	O
operator	O
is	O
careful	O
and	O
correcting	O
for	O
the	O
fact	O
that	O
he	O
is	O
not	O
perfect	O
)	O
,	O
and	O
to	O
test	O
the	O
assumption	O
that	O
the	O
data	O
really	O
have	O
a	O
normal	O
distribution	O
with	O
standard	B-General_Concept
deviation	I-General_Concept
σ	O
.	O
</s>
<s>
The	O
theoretical	O
analysis	O
of	O
such	O
an	O
experiment	O
is	O
complicated	O
,	O
but	O
it	O
is	O
easy	O
to	O
set	O
up	O
a	O
spreadsheet	B-Application
which	O
draws	O
random	O
numbers	O
from	O
a	O
normal	O
distribution	O
with	O
standard	B-General_Concept
deviation	I-General_Concept
σ	O
to	O
simulate	O
the	O
situation	O
;	O
this	O
can	O
be	O
done	O
in	O
Microsoft	B-Application
Excel	I-Application
using	O
=	O
NORMINV(RAND( )	O
,	O
0	O
,	O
σ	O
)	O
)	O
,	O
as	O
discussed	O
in	O
and	O
the	O
same	O
techniques	O
can	O
be	O
used	O
in	O
other	O
spreadsheet	B-Application
programs	I-Application
such	O
as	O
in	O
OpenOffice.org	B-Application
Calc	I-Application
and	O
gnumeric	B-Language
.	O
</s>
<s>
It	O
should	O
be	O
normal	O
with	O
mean	O
near	O
zero	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
a	O
little	O
larger	O
than	O
σ	O
.	O
</s>
<s>
A	O
simple	O
Monte	B-Algorithm
Carlo	I-Algorithm
spreadsheet	B-Application
calculation	O
would	O
reveal	O
typical	O
values	O
for	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
(	O
around	O
105	O
to	O
115%	O
of	O
σ	O
)	O
.	O
</s>
<s>
The	O
mean	O
is	O
identically	O
zero	O
,	O
but	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
should	O
be	O
somewhat	O
smaller	O
(	O
around	O
75	O
to	O
85%	O
of	O
σ	O
)	O
.	O
</s>
