<s>
The	O
average	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
(	O
AAD	O
)	O
of	O
a	O
data	O
set	O
is	O
the	O
average	O
of	O
the	O
absolute	O
deviations	B-General_Concept
from	O
a	O
central	O
point	O
.	O
</s>
<s>
AAD	O
includes	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
and	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
(	O
both	O
abbreviated	O
as	O
MAD	O
)	O
.	O
</s>
<s>
Several	O
measures	O
of	O
statistical	O
dispersion	O
are	O
defined	O
in	O
terms	O
of	O
the	O
absolute	O
deviation	B-General_Concept
.	O
</s>
<s>
The	O
term	O
"	O
average	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
"	O
does	O
not	O
uniquely	O
identify	O
a	O
measure	O
of	O
statistical	O
dispersion	O
,	O
as	O
there	O
are	O
several	O
measures	O
that	O
can	O
be	O
used	O
to	O
measure	O
absolute	O
deviations	B-General_Concept
,	O
and	O
there	O
are	O
several	O
measures	O
of	O
central	O
tendency	O
that	O
can	O
be	O
used	O
as	O
well	O
.	O
</s>
<s>
Thus	O
,	O
to	O
uniquely	O
identify	O
the	O
absolute	O
deviation	B-General_Concept
it	O
is	O
necessary	O
to	O
specify	O
both	O
the	O
measure	O
of	O
deviation	B-General_Concept
and	O
the	O
measure	O
of	O
central	O
tendency	O
.	O
</s>
<s>
Unfortunately	O
,	O
the	O
statistical	O
literature	O
has	O
not	O
yet	O
adopted	O
a	O
standard	O
notation	O
,	O
as	O
both	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
around	O
the	O
mean	O
and	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
around	O
the	O
median	O
have	O
been	O
denoted	O
by	O
their	O
initials	O
"	O
MAD	O
"	O
in	O
the	O
literature	O
,	O
which	O
may	O
lead	O
to	O
confusion	O
,	O
since	O
in	O
general	O
,	O
they	O
may	O
have	O
values	O
considerably	O
different	O
from	O
each	O
other	O
.	O
</s>
<s>
The	O
choice	O
of	O
measure	O
of	O
central	O
tendency	O
,	O
,	O
has	O
a	O
marked	O
effect	O
on	O
the	O
value	O
of	O
the	O
mean	O
deviation	B-General_Concept
.	O
</s>
<s>
The	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
(	O
MAD	O
)	O
,	O
also	O
referred	O
to	O
as	O
the	O
"	O
mean	O
deviation	B-General_Concept
"	O
or	O
sometimes	O
"	O
average	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
"	O
,	O
is	O
the	O
mean	O
of	O
the	O
data	O
's	O
absolute	O
deviations	B-General_Concept
around	O
the	O
data	O
's	O
mean	O
:	O
the	O
average	O
(	O
absolute	O
)	O
distance	O
from	O
the	O
mean	O
.	O
</s>
<s>
"	O
Average	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
"	O
can	O
refer	O
to	O
either	O
this	O
usage	O
,	O
or	O
to	O
the	O
general	O
form	O
with	O
respect	O
to	O
a	O
specified	O
central	O
point	O
(	O
see	O
above	O
)	O
.	O
</s>
<s>
MAD	O
has	O
been	O
proposed	O
to	O
be	O
used	O
in	O
place	O
of	O
standard	B-General_Concept
deviation	I-General_Concept
since	O
it	O
corresponds	O
better	O
to	O
real	O
life	O
.	O
</s>
<s>
Because	O
the	O
MAD	O
is	O
a	O
simpler	O
measure	O
of	O
variability	O
than	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
it	O
can	O
be	O
useful	O
in	O
school	O
teaching	O
.	O
</s>
<s>
This	O
method	O
's	O
forecast	O
accuracy	O
is	O
very	O
closely	O
related	O
to	O
the	O
mean	B-Algorithm
squared	I-Algorithm
error	I-Algorithm
(	O
MSE	O
)	O
method	O
which	O
is	O
just	O
the	O
average	O
squared	O
error	O
of	O
the	O
forecasts	O
.	O
</s>
<s>
For	O
the	O
normal	O
distribution	O
,	O
the	O
ratio	O
of	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
from	O
the	O
mean	O
to	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
for	O
a	O
normal	O
distribution	O
,	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
is	O
about	O
0.8	O
times	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
However	O
,	O
in-sample	O
measurements	O
deliver	O
values	O
of	O
the	O
ratio	O
of	O
mean	O
average	O
deviation	B-General_Concept
/	O
standard	B-General_Concept
deviation	I-General_Concept
for	O
a	O
given	O
Gaussian	O
sample	O
n	O
with	O
the	O
following	O
bounds	O
:	O
,	O
with	O
a	O
bias	O
for	O
small	O
n	O
.	O
</s>
<s>
The	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
from	O
the	O
mean	O
is	O
less	O
than	O
or	O
equal	O
to	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
;	O
one	O
way	O
of	O
proving	O
this	O
relies	O
on	O
Jensen	O
's	O
inequality	O
.	O
</s>
<s>
The	O
median	O
is	O
the	O
point	O
about	O
which	O
the	O
mean	O
deviation	B-General_Concept
is	O
minimized	O
.	O
</s>
<s>
The	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
from	O
the	O
median	O
is	O
less	O
than	O
or	O
equal	O
to	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
from	O
the	O
mean	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
from	O
the	O
median	O
is	O
always	O
less	O
than	O
or	O
equal	O
to	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
from	O
any	O
other	O
fixed	O
number	O
.	O
</s>
<s>
While	O
in	O
principle	O
the	O
mean	O
or	O
any	O
other	O
central	O
point	O
could	O
be	O
taken	O
as	O
the	O
central	O
point	O
for	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
,	O
most	O
often	O
the	O
median	O
value	O
is	O
taken	O
instead	O
.	O
</s>
<s>
The	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
(	O
also	O
MAD	O
)	O
is	O
the	O
median	O
of	O
the	O
absolute	O
deviation	B-General_Concept
from	O
the	O
median	O
.	O
</s>
<s>
It	O
is	O
a	O
robust	B-General_Concept
estimator	I-General_Concept
of	I-General_Concept
dispersion	I-General_Concept
.	O
</s>
<s>
For	O
the	O
example	O
{	O
2	O
,	O
2	O
,	O
3	O
,	O
4	O
,	O
14}	O
:	O
3	O
is	O
the	O
median	O
,	O
so	O
the	O
absolute	O
deviations	B-General_Concept
from	O
the	O
median	O
are	O
{	O
1	O
,	O
1	O
,	O
0	O
,	O
1	O
,	O
11}	O
(	O
reordered	O
as	O
{	O
0	O
,	O
1	O
,	O
1	O
,	O
1	O
,	O
11}	O
)	O
with	O
a	O
median	O
of	O
1	O
,	O
in	O
this	O
case	O
unaffected	O
by	O
the	O
value	O
of	O
the	O
outlier	O
14	O
,	O
so	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
is	O
1	O
.	O
</s>
<s>
For	O
a	O
symmetric	O
distribution	O
,	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
is	O
equal	O
to	O
half	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
.	O
</s>
<s>
The	O
maximum	O
absolute	O
deviation	B-General_Concept
around	O
an	O
arbitrary	O
point	O
is	O
the	O
maximum	O
of	O
the	O
absolute	O
deviations	B-General_Concept
of	O
a	O
sample	O
from	O
that	O
point	O
.	O
</s>
<s>
While	O
not	O
strictly	O
a	O
measure	O
of	O
central	O
tendency	O
,	O
the	O
maximum	O
absolute	O
deviation	B-General_Concept
can	O
be	O
found	O
using	O
the	O
formula	O
for	O
the	O
average	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
as	O
above	O
with	O
,	O
where	O
is	O
the	O
sample	O
maximum	O
.	O
</s>
<s>
The	O
measures	O
of	O
statistical	O
dispersion	O
derived	O
from	O
absolute	O
deviation	B-General_Concept
characterize	O
various	O
measures	O
of	O
central	O
tendency	O
as	O
minimizing	O
dispersion	O
:	O
</s>
<s>
The	O
median	O
is	O
the	O
measure	O
of	O
central	O
tendency	O
most	O
associated	O
with	O
the	O
absolute	O
deviation	B-General_Concept
.	O
</s>
<s>
L1	O
norm	O
statistics	O
:	O
the	O
median	O
minimizes	O
average	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
,	O
</s>
<s>
trimmed	O
L∞	O
norm	O
statistics	O
:	O
for	O
example	O
,	O
the	O
midhinge	B-General_Concept
(	O
average	O
of	O
first	O
and	O
third	O
quartiles	O
)	O
which	O
minimizes	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
of	O
the	O
whole	O
distribution	O
,	O
also	O
minimizes	O
the	O
maximum	O
absolute	O
deviation	B-General_Concept
of	O
the	O
distribution	O
after	O
the	O
top	O
and	O
bottom	O
25%	O
have	O
been	O
trimmed	O
off	O
.	O
</s>
<s>
The	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
of	O
a	O
sample	O
is	O
a	O
biased	O
estimator	O
of	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
of	O
the	O
population	O
.	O
</s>
<s>
In	O
order	O
for	O
the	O
absolute	O
deviation	B-General_Concept
to	O
be	O
an	O
unbiased	O
estimator	O
,	O
the	O
expected	O
value	O
(	O
average	O
)	O
of	O
all	O
the	O
sample	O
absolute	O
deviations	B-General_Concept
must	O
equal	O
the	O
population	O
absolute	O
deviation	B-General_Concept
.	O
</s>
<s>
For	O
the	O
population	O
1	O
,	O
2	O
,	O
3	O
both	O
the	O
population	O
absolute	O
deviation	B-General_Concept
about	O
the	O
median	O
and	O
the	O
population	O
absolute	O
deviation	B-General_Concept
about	O
the	O
mean	O
are	O
2/3	O
.	O
</s>
<s>
The	O
average	O
of	O
all	O
the	O
sample	O
absolute	O
deviations	B-General_Concept
about	O
the	O
mean	O
of	O
size	O
3	O
that	O
can	O
be	O
drawn	O
from	O
the	O
population	O
is	O
44/81	O
,	O
while	O
the	O
average	O
of	O
all	O
the	O
sample	O
absolute	O
deviations	B-General_Concept
about	O
the	O
median	O
is	O
4/9	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
absolute	O
deviation	B-General_Concept
is	O
a	O
biased	O
estimator	O
.	O
</s>
