<s>
In	O
probability	O
theory	O
and	O
statistics	O
,	O
skewness	B-General_Concept
is	O
a	O
measure	O
of	O
the	O
asymmetry	O
of	O
the	O
probability	O
distribution	O
of	O
a	O
real-valued	O
random	O
variable	O
about	O
its	O
mean	O
.	O
</s>
<s>
The	O
skewness	B-General_Concept
value	O
can	O
be	O
positive	O
,	O
zero	O
,	O
negative	O
,	O
or	O
undefined	O
.	O
</s>
<s>
For	O
a	O
unimodal	B-Algorithm
distribution	O
,	O
negative	O
skew	O
commonly	O
indicates	O
that	O
the	O
tail	O
is	O
on	O
the	O
left	O
side	O
of	O
the	O
distribution	O
,	O
and	O
positive	O
skew	O
indicates	O
that	O
the	O
tail	O
is	O
on	O
the	O
right	O
.	O
</s>
<s>
In	O
cases	O
where	O
one	O
tail	O
is	O
long	O
but	O
the	O
other	O
tail	O
is	O
fat	O
,	O
skewness	B-General_Concept
does	O
not	O
obey	O
a	O
simple	O
rule	O
.	O
</s>
<s>
These	O
tapering	O
sides	O
are	O
called	O
tails	O
,	O
and	O
they	O
provide	O
a	O
visual	O
means	O
to	O
determine	O
which	O
of	O
the	O
two	O
kinds	O
of	O
skewness	B-General_Concept
a	O
distribution	O
has	O
:	O
</s>
<s>
The	O
distribution	O
is	O
said	O
to	O
be	O
left-skewed	O
,	O
left-tailed	O
,	O
or	O
skewed	B-General_Concept
to	O
the	O
left	O
,	O
despite	O
the	O
fact	O
that	O
the	O
curve	O
itself	O
appears	O
to	O
be	O
skewed	B-General_Concept
or	O
leaning	O
to	O
the	O
right	O
;	O
left	O
instead	O
refers	O
to	O
the	O
left	O
tail	O
being	O
drawn	O
out	O
and	O
,	O
often	O
,	O
the	O
mean	O
being	O
skewed	B-General_Concept
to	O
the	O
left	O
of	O
a	O
typical	O
center	O
of	O
the	O
data	O
.	O
</s>
<s>
The	O
distribution	O
is	O
said	O
to	O
be	O
right-skewed	B-General_Concept
,	O
right-tailed	O
,	O
or	O
skewed	B-General_Concept
to	O
the	O
right	O
,	O
despite	O
the	O
fact	O
that	O
the	O
curve	O
itself	O
appears	O
to	O
be	O
skewed	B-General_Concept
or	O
leaning	O
to	O
the	O
left	O
;	O
right	O
instead	O
refers	O
to	O
the	O
right	O
tail	O
being	O
drawn	O
out	O
and	O
,	O
often	O
,	O
the	O
mean	O
being	O
skewed	B-General_Concept
to	O
the	O
right	O
of	O
a	O
typical	O
center	O
of	O
the	O
data	O
.	O
</s>
<s>
A	O
right-skewed	B-General_Concept
distribution	I-General_Concept
usually	O
appears	O
as	O
a	O
left-leaning	O
curve	O
.	O
</s>
<s>
Skewness	B-General_Concept
in	O
a	O
data	O
series	O
may	O
sometimes	O
be	O
observed	O
not	O
only	O
graphically	O
but	O
by	O
simple	O
inspection	O
of	O
the	O
values	O
.	O
</s>
<s>
We	O
can	O
transform	O
this	O
sequence	O
into	O
a	O
negatively	O
skewed	B-General_Concept
distribution	I-General_Concept
by	O
adding	O
a	O
value	O
far	O
below	O
the	O
mean	O
,	O
which	O
is	O
probably	O
a	O
negative	O
outlier	O
,	O
e.g.	O
</s>
<s>
Similarly	O
,	O
we	O
can	O
make	O
the	O
sequence	O
positively	B-General_Concept
skewed	I-General_Concept
by	O
adding	O
a	O
value	O
far	O
above	O
the	O
mean	O
,	O
which	O
is	O
probably	O
a	O
positive	O
outlier	O
,	O
e.g.	O
</s>
<s>
As	O
mentioned	O
earlier	O
,	O
a	O
unimodal	B-Algorithm
distribution	O
with	O
zero	O
value	O
of	O
skewness	B-General_Concept
does	O
not	O
imply	O
that	O
this	O
distribution	O
is	O
symmetric	O
necessarily	O
.	O
</s>
<s>
However	O
,	O
a	O
symmetric	O
unimodal	B-Algorithm
or	O
multimodal	O
distribution	O
always	O
has	O
zero	O
skewness	B-General_Concept
.	O
</s>
<s>
The	O
skewness	B-General_Concept
is	O
not	O
directly	O
related	O
to	O
the	O
relationship	O
between	O
the	O
mean	O
and	O
median	O
:	O
a	O
distribution	O
with	O
negative	O
skew	O
can	O
have	O
its	O
mean	O
greater	O
than	O
or	O
less	O
than	O
the	O
median	O
,	O
and	O
likewise	O
for	O
positive	O
skew	O
.	O
</s>
<s>
In	O
the	O
older	O
notion	O
of	O
nonparametric	O
skew	O
,	O
defined	O
as	O
where	O
is	O
the	O
mean	O
,	O
is	O
the	O
median	O
,	O
and	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
the	O
skewness	B-General_Concept
is	O
defined	O
in	O
terms	O
of	O
this	O
relationship	O
:	O
positive/right	O
nonparametric	O
skew	O
means	O
the	O
mean	O
is	O
greater	O
than	O
(	O
to	O
the	O
right	O
of	O
)	O
the	O
median	O
,	O
while	O
negative/left	O
nonparametric	O
skew	O
means	O
the	O
mean	O
is	O
less	O
than	O
(	O
to	O
the	O
left	O
of	O
)	O
the	O
median	O
.	O
</s>
<s>
However	O
,	O
the	O
modern	O
definition	O
of	O
skewness	B-General_Concept
and	O
the	O
traditional	O
nonparametric	O
definition	O
do	O
not	O
always	O
have	O
the	O
same	O
sign	O
:	O
while	O
they	O
agree	O
for	O
some	O
families	O
of	O
distributions	O
,	O
they	O
differ	O
in	O
some	O
of	O
the	O
cases	O
,	O
and	O
conflating	O
them	O
is	O
misleading	O
.	O
</s>
<s>
If	O
the	O
distribution	O
is	O
symmetric	O
,	O
then	O
the	O
mean	O
is	O
equal	O
to	O
the	O
median	O
,	O
and	O
the	O
distribution	O
has	O
zero	O
skewness	B-General_Concept
.	O
</s>
<s>
If	O
the	O
distribution	O
is	O
both	O
symmetric	O
and	O
unimodal	B-Algorithm
,	O
then	O
the	O
mean	O
=	O
median	O
=	O
mode	O
.	O
</s>
<s>
zero	O
skewness	B-General_Concept
(	O
defined	O
below	O
)	O
does	O
not	O
imply	O
that	O
the	O
mean	O
is	O
equal	O
to	O
the	O
median	O
.	O
</s>
<s>
The	O
skewness	B-General_Concept
of	O
a	O
random	O
variable	O
X	O
is	O
the	O
third	O
standardized	O
moment	O
,	O
defined	O
as	O
:	O
</s>
<s>
where	O
μ	O
is	O
the	O
mean	O
,	O
σ	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
E	O
is	O
the	O
expectation	O
operator	O
,	O
μ3	O
is	O
the	O
third	O
central	B-General_Concept
moment	I-General_Concept
,	O
and	O
κt	O
are	O
the	O
t-th	O
cumulants	O
.	O
</s>
<s>
It	O
is	O
sometimes	O
referred	O
to	O
as	O
Pearson	O
's	O
moment	O
coefficient	O
of	O
skewness	B-General_Concept
,	O
or	O
simply	O
the	O
moment	O
coefficient	O
of	O
skewness	B-General_Concept
,	O
but	O
should	O
not	O
be	O
confused	O
with	O
Pearson	O
's	O
other	O
skewness	B-General_Concept
statistics	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
The	O
last	O
equality	O
expresses	O
skewness	B-General_Concept
in	O
terms	O
of	O
the	O
ratio	O
of	O
the	O
third	O
cumulant	O
κ3	O
to	O
the	O
1.5th	O
power	O
of	O
the	O
second	O
cumulant	O
κ2	O
.	O
</s>
<s>
This	O
is	O
analogous	O
to	O
the	O
definition	O
of	O
kurtosis	B-Error_Name
as	O
the	O
fourth	O
cumulant	O
normalized	O
by	O
the	O
square	O
of	O
the	O
second	O
cumulant	O
.	O
</s>
<s>
The	O
skewness	B-General_Concept
is	O
also	O
sometimes	O
denoted	O
Skew[X]	O
.	O
</s>
<s>
If	O
σ	O
is	O
finite	O
,	O
μ	O
is	O
finite	O
too	O
and	O
skewness	B-General_Concept
can	O
be	O
expressed	O
in	O
terms	O
of	O
the	O
non-central	O
moment	O
E[X3]	O
by	O
expanding	O
the	O
previous	O
formula	O
,	O
</s>
<s>
Examples	O
of	O
distributions	O
with	O
finite	O
skewness	B-General_Concept
include	O
the	O
following	O
.	O
</s>
<s>
where	O
is	O
the	O
sample	O
mean	O
,	O
s	O
is	O
the	O
sample	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
m2	O
is	O
the	O
(	O
biased	O
)	O
sample	O
second	O
central	B-General_Concept
moment	I-General_Concept
,	O
and	O
m3	O
is	O
the	O
sample	O
third	O
central	B-General_Concept
moment	I-General_Concept
.	O
</s>
<s>
This	O
adjusted	O
Fisher	O
–	O
Pearson	O
standardized	O
moment	O
coefficient	O
is	O
the	O
version	O
found	O
in	O
Excel	B-Application
and	O
several	O
statistical	O
packages	O
including	O
Minitab	B-Application
,	O
SAS	B-Language
and	O
SPSS	B-Algorithm
.	O
</s>
<s>
Under	O
the	O
assumption	O
that	O
the	O
underlying	O
random	O
variable	O
is	O
normally	O
distributed	O
,	O
it	O
can	O
be	O
shown	O
that	O
all	O
three	O
ratios	O
,	O
and	O
are	O
unbiased	O
and	O
consistent	O
estimators	O
of	O
the	O
population	O
skewness	B-General_Concept
,	O
with	O
,	O
i.e.	O
,	O
their	O
distributions	O
converge	O
to	O
a	O
normal	O
distribution	O
with	O
mean	O
0	O
and	O
variance	O
6	O
(	O
Fisher	O
,	O
1930	O
)	O
.	O
</s>
<s>
The	O
variance	O
of	O
the	O
sample	O
skewness	B-General_Concept
is	O
thus	O
approximately	O
for	O
sufficiently	O
large	O
samples	O
.	O
</s>
<s>
For	O
non-normal	O
distributions	O
,	O
,	O
and	O
are	O
generally	O
biased	O
estimators	O
of	O
the	O
population	O
skewness	B-General_Concept
;	O
their	O
expected	O
values	O
can	O
even	O
have	O
the	O
opposite	O
sign	O
from	O
the	O
true	O
skewness	B-General_Concept
.	O
</s>
<s>
For	O
instance	O
,	O
a	O
mixed	O
distribution	O
consisting	O
of	O
very	O
thin	O
Gaussians	O
centred	O
at	O
−99	O
,	O
0.5	O
,	O
and	O
2	O
with	O
weights	O
0.01	O
,	O
0.66	O
,	O
and	O
0.33	O
has	O
a	O
skewness	B-General_Concept
of	O
about	O
−	O
9.77	O
,	O
but	O
in	O
a	O
sample	O
of	O
3	O
has	O
an	O
expected	O
value	O
of	O
about	O
0.32	O
,	O
since	O
usually	O
all	O
three	O
samples	O
are	O
in	O
the	O
positive-valued	O
part	O
of	O
the	O
distribution	O
,	O
which	O
is	O
skewed	B-General_Concept
the	O
other	O
way	O
.	O
</s>
<s>
Skewness	B-General_Concept
is	O
a	O
descriptive	O
statistic	O
that	O
can	O
be	O
used	O
in	O
conjunction	O
with	O
the	O
histogram	B-Algorithm
and	O
the	O
normal	B-Application
quantile	I-Application
plot	I-Application
to	O
characterize	O
the	O
data	O
or	O
distribution	O
.	O
</s>
<s>
Skewness	B-General_Concept
indicates	O
the	O
direction	O
and	O
relative	O
magnitude	O
of	O
a	O
distribution	O
's	O
deviation	O
from	O
the	O
normal	O
distribution	O
.	O
</s>
<s>
With	O
pronounced	O
skewness	B-General_Concept
,	O
standard	O
statistical	O
inference	O
procedures	O
such	O
as	O
a	O
confidence	O
interval	O
for	O
a	O
mean	O
will	O
be	O
not	O
only	O
incorrect	O
,	O
in	O
the	O
sense	O
that	O
the	O
true	O
coverage	O
level	O
will	O
differ	O
from	O
the	O
nominal	O
(	O
e.g.	O
,	O
95%	O
)	O
level	O
,	O
but	O
they	O
will	O
also	O
result	O
in	O
unequal	O
error	O
probabilities	O
on	O
each	O
side	O
.	O
</s>
<s>
Skewness	B-General_Concept
can	O
be	O
used	O
to	O
obtain	O
approximate	O
probabilities	O
and	O
quantiles	O
of	O
distributions	O
(	O
such	O
as	O
value	O
at	O
risk	O
in	O
finance	O
)	O
via	O
the	O
Cornish-Fisher	B-General_Concept
expansion	I-General_Concept
.	O
</s>
<s>
The	O
normal	O
distribution	O
has	O
a	O
skewness	B-General_Concept
of	O
zero	O
.	O
</s>
<s>
So	O
,	O
an	O
understanding	O
of	O
the	O
skewness	B-General_Concept
of	O
the	O
dataset	O
indicates	O
whether	O
deviations	O
from	O
the	O
mean	O
are	O
going	O
to	O
be	O
positive	O
or	O
negative	O
.	O
</s>
<s>
Agostino	B-General_Concept
's	I-General_Concept
K-squared	I-General_Concept
test	I-General_Concept
is	O
a	O
goodness-of-fit	O
normality	B-General_Concept
test	I-General_Concept
based	O
on	O
sample	O
skewness	B-General_Concept
and	O
sample	O
kurtosis	B-Error_Name
.	O
</s>
<s>
Other	O
measures	O
of	O
skewness	B-General_Concept
have	O
been	O
used	O
,	O
including	O
simpler	O
calculations	O
suggested	O
by	O
Karl	O
Pearson	O
(	O
not	O
to	O
be	O
confused	O
with	O
Pearson	O
's	O
moment	O
coefficient	O
of	O
skewness	B-General_Concept
,	O
see	O
above	O
)	O
.	O
</s>
<s>
Bowley	O
's	O
measure	O
of	O
skewness	B-General_Concept
(	O
from	O
1901	O
)	O
,	O
also	O
called	O
Yule	O
's	O
coefficient	O
(	O
from	O
1912	O
)	O
is	O
defined	O
as	O
:	O
</s>
<s>
The	O
numerator	O
is	O
difference	O
between	O
the	O
average	O
of	O
the	O
upper	O
and	O
lower	O
quartiles	O
(	O
a	O
measure	O
of	O
location	O
)	O
and	O
the	O
median	O
(	O
another	O
measure	O
of	O
location	O
)	O
,	O
while	O
the	O
denominator	O
is	O
the	O
semi-interquartile	O
range	O
,	O
which	O
for	O
symmetric	O
distributions	O
is	O
the	O
MAD	B-General_Concept
measure	O
of	O
dispersion	O
.	O
</s>
<s>
Other	O
names	O
for	O
this	O
measure	O
are	O
Galton	O
's	O
measure	O
of	O
skewness	B-General_Concept
,	O
the	O
Yule	O
–	O
Kendall	O
index	O
and	O
the	O
quartile	O
skewness	B-General_Concept
,	O
</s>
<s>
Bowley	O
's	O
measure	O
of	O
skewness	B-General_Concept
is	O
γ(u )	O
evaluated	O
at	O
u	O
=	O
3/4	O
while	O
Kelly	O
's	O
measure	O
of	O
skewness	B-General_Concept
is	O
γ(u )	O
evaluated	O
at	O
u	O
=	O
9/10	O
.	O
</s>
<s>
Quantile-based	O
skewness	B-General_Concept
measures	O
are	O
at	O
first	O
glance	O
easy	O
to	O
interpret	O
,	O
but	O
they	O
often	O
show	O
significantly	O
larger	O
sample	O
variations	O
than	O
moment-based	O
methods	O
.	O
</s>
<s>
This	O
means	O
that	O
often	O
samples	O
from	O
a	O
symmetric	O
distribution	O
(	O
like	O
the	O
uniform	O
distribution	O
)	O
have	O
a	O
large	O
quantile-based	O
skewness	B-General_Concept
,	O
just	O
by	O
chance	O
.	O
</s>
<s>
Groeneveld	O
and	O
Meeden	O
have	O
suggested	O
,	O
as	O
an	O
alternative	O
measure	O
of	O
skewness	B-General_Concept
,	O
</s>
<s>
This	O
is	O
closely	O
related	O
in	O
form	O
to	O
Pearson	O
's	O
second	O
skewness	B-General_Concept
coefficient	O
.	O
</s>
<s>
Use	O
of	O
L-moments	O
in	O
place	O
of	O
moments	O
provides	O
a	O
measure	O
of	O
skewness	B-General_Concept
known	O
as	O
the	O
L-skewness	O
.	O
</s>
<s>
A	O
value	O
of	O
skewness	B-General_Concept
equal	O
to	O
zero	O
does	O
not	O
imply	O
that	O
the	O
probability	O
distribution	O
is	O
symmetric	O
.	O
</s>
<s>
It	O
is	O
called	O
distance	O
skewness	B-General_Concept
and	O
denoted	O
by	O
dSkew	O
.	O
</s>
<s>
Distance	O
skewness	B-General_Concept
is	O
always	O
between	O
0	O
and	O
1	O
,	O
equals	O
0	O
if	O
and	O
only	O
if	O
X	O
is	O
diagonally	O
symmetric	O
with	O
respect	O
to	O
θ	O
(	O
X	O
and	O
2θ−X	O
have	O
the	O
same	O
probability	O
distribution	O
)	O
and	O
equals	O
1	O
if	O
and	O
only	O
if	O
X	O
is	O
a	O
constant	O
c''	O
(	O
)	O
with	O
probability	O
one	O
.	O
</s>
<s>
Thus	O
there	O
is	O
a	O
simple	O
consistent	O
statistical	O
test	O
of	O
diagonal	O
symmetry	O
based	O
on	O
the	O
sample	O
distance	O
skewness	B-General_Concept
:	O
</s>
<s>
The	O
medcouple	B-General_Concept
is	O
a	O
scale-invariant	O
robust	O
measure	O
of	O
skewness	B-General_Concept
,	O
with	O
a	O
breakdown	O
point	O
of	O
25%	O
.	O
</s>
<s>
It	O
can	O
be	O
seen	O
as	O
the	O
median	O
of	O
all	O
possible	O
quantile	O
skewness	B-General_Concept
measures	O
.	O
</s>
