<s>
In	O
statistics	O
,	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
(	O
MAD	O
)	O
is	O
a	O
robust	O
measure	O
of	O
the	O
variability	O
of	O
a	O
univariate	B-General_Concept
sample	O
of	O
quantitative	O
data	O
.	O
</s>
<s>
For	O
a	O
univariate	B-General_Concept
data	O
set	O
X1	O
,	O
X2	O
,...,	O
Xn	O
,	O
the	O
MAD	O
is	O
defined	O
as	O
the	O
median	O
of	O
the	O
absolute	O
deviations	B-General_Concept
from	O
the	O
data	O
's	O
median	O
:	O
</s>
<s>
that	O
is	O
,	O
starting	O
with	O
the	O
residuals	O
(	O
deviations	B-General_Concept
)	O
from	O
the	O
data	O
's	O
median	O
,	O
the	O
MAD	O
is	O
the	O
median	O
of	O
their	O
absolute	O
values	O
.	O
</s>
<s>
The	O
absolute	O
deviations	B-General_Concept
about	O
2	O
are	O
(	O
1	O
,	O
1	O
,	O
0	O
,	O
0	O
,	O
2	O
,	O
4	O
,	O
7	O
)	O
which	O
in	O
turn	O
have	O
a	O
median	O
value	O
of	O
1	O
(	O
because	O
the	O
sorted	O
absolute	O
deviations	B-General_Concept
are	O
(	O
0	O
,	O
0	O
,	O
1	O
,	O
1	O
,	O
2	O
,	O
4	O
,	O
7	O
)	O
)	O
.	O
</s>
<s>
So	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
for	O
this	O
data	O
is	O
1	O
.	O
</s>
<s>
The	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
is	O
a	O
measure	O
of	O
statistical	O
dispersion	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
MAD	O
is	O
a	O
robust	O
statistic	O
,	O
being	O
more	O
resilient	O
to	O
outliers	O
in	O
a	O
data	O
set	O
than	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
In	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
the	O
distances	O
from	O
the	O
mean	O
are	O
squared	O
,	O
so	O
large	O
deviations	B-General_Concept
are	O
weighted	O
more	O
heavily	O
,	O
and	O
thus	O
outliers	O
can	O
heavily	O
influence	O
it	O
.	O
</s>
<s>
In	O
the	O
MAD	O
,	O
the	O
deviations	B-General_Concept
of	O
a	O
small	O
number	O
of	O
outliers	O
are	O
irrelevant	O
.	O
</s>
<s>
Because	O
the	O
MAD	O
is	O
a	O
more	O
robust	O
estimator	O
of	O
scale	O
than	O
the	O
sample	O
variance	B-General_Concept
or	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
it	O
works	O
better	O
with	O
distributions	O
without	O
a	O
mean	O
or	O
variance	B-General_Concept
,	O
such	O
as	O
the	O
Cauchy	O
distribution	O
.	O
</s>
<s>
The	O
MAD	O
may	O
be	O
used	O
similarly	O
to	O
how	O
one	O
would	O
use	O
the	O
deviation	B-General_Concept
for	O
the	O
average	O
.	O
</s>
<s>
where	O
is	O
a	O
constant	O
scale	B-Algorithm
factor	I-Algorithm
,	O
which	O
depends	O
on	O
the	O
distribution	O
.	O
</s>
<s>
we	O
have	O
that	O
,	O
from	O
which	O
we	O
obtain	O
the	O
scale	B-Algorithm
factor	I-Algorithm
.	O
</s>
<s>
In	O
the	O
case	O
of	O
complex	O
values	O
(	O
X+iY	O
)	O
,	O
the	O
relation	O
of	O
MAD	O
to	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
unchanged	O
for	O
normally	O
distributed	O
data	O
.	O
</s>
<s>
Analogously	O
to	O
how	O
the	O
median	O
generalizes	O
to	O
the	O
geometric	B-General_Concept
median	I-General_Concept
(	O
gm	O
)	O
in	O
multivariate	O
data	O
,	O
MAD	O
can	O
be	O
generalized	O
to	O
MADGM	O
(	O
median	O
of	O
distances	O
to	O
gm	O
)	O
in	O
n	O
dimensions	O
.	O
</s>
<s>
This	O
is	O
done	O
by	O
replacing	O
the	O
absolute	O
differences	O
in	O
one	O
dimension	O
by	O
euclidian	O
distances	O
of	O
the	O
data	O
points	O
to	O
the	O
geometric	B-General_Concept
median	I-General_Concept
in	O
n	O
dimensions	O
.	O
</s>
<s>
This	O
gives	O
the	O
identical	O
result	O
as	O
the	O
univariate	B-General_Concept
MAD	O
in	O
1	O
dimension	O
and	O
generalizes	O
to	O
any	O
number	O
of	O
dimensions	O
.	O
</s>
<s>
MADGM	O
needs	O
the	O
geometric	B-General_Concept
median	I-General_Concept
to	O
be	O
found	O
,	O
which	O
is	O
done	O
by	O
an	O
iterative	O
process	O
.	O
</s>
<s>
Unlike	O
the	O
variance	B-General_Concept
,	O
which	O
may	O
be	O
infinite	O
or	O
undefined	O
,	O
the	O
population	O
MAD	O
is	O
always	O
a	O
finite	O
number	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
standard	O
Cauchy	O
distribution	O
has	O
undefined	O
variance	B-General_Concept
,	O
but	O
its	O
MAD	O
is	O
1	O
.	O
</s>
