<s>
In	O
statistics	O
,	O
Agostino	O
's	O
K2	O
test	O
,	O
named	O
for	O
Ralph	O
D'Agostino	O
,	O
is	O
a	O
goodness-of-fit	O
measure	O
of	O
departure	O
from	O
normality	O
,	O
that	O
is	O
the	O
test	O
aims	O
to	O
gauge	O
the	O
compatibility	O
of	O
given	O
data	O
with	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
the	O
data	O
is	O
a	O
realization	O
of	O
independent	O
,	O
identically	O
distributed	O
Gaussian	O
random	O
variables	O
.	O
</s>
<s>
The	O
test	O
is	O
based	O
on	O
transformations	O
of	O
the	O
sample	O
kurtosis	B-Error_Name
and	O
skewness	B-General_Concept
,	O
and	O
has	O
power	O
only	O
against	O
the	O
alternatives	O
that	O
the	O
distribution	O
is	O
skewed	B-General_Concept
and/or	O
kurtic	O
.	O
</s>
<s>
In	O
the	O
following	O
,	O
{	O
xi	O
}	O
denotes	O
a	O
sample	O
of	O
n	O
observations	O
,	O
g1	O
and	O
g2	O
are	O
the	O
sample	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
,	O
mj	O
’s	O
are	O
the	O
j-th	O
sample	O
central	B-General_Concept
moments	I-General_Concept
,	O
and	O
is	O
the	O
sample	O
mean	O
.	O
</s>
<s>
Frequently	O
in	O
the	O
literature	O
related	O
to	O
normality	B-General_Concept
testing	I-General_Concept
,	O
the	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
are	O
denoted	O
as	O
and	O
β2	O
respectively	O
.	O
</s>
<s>
These	O
quantities	O
consistently	O
estimate	O
the	O
theoretical	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
of	O
the	O
distribution	O
,	O
respectively	O
.	O
</s>
<s>
Moreover	O
,	O
if	O
the	O
sample	O
indeed	O
comes	O
from	O
a	O
normal	O
population	O
,	O
then	O
the	O
exact	O
finite	O
sample	O
distributions	O
of	O
the	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
can	O
themselves	O
be	O
analysed	O
in	O
terms	O
of	O
their	O
means	O
μ1	O
,	O
variances	O
μ2	O
,	O
skewnesses	O
γ1	O
,	O
and	O
kurtosis	B-Error_Name
γ2	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
sample	O
with	O
size	O
drawn	O
from	O
a	O
normally	O
distributed	O
population	O
can	O
be	O
expected	O
to	O
have	O
a	O
skewness	B-General_Concept
of	O
and	O
a	O
kurtosis	B-Error_Name
of	O
,	O
where	O
SD	O
indicates	O
the	O
standard	O
deviation	O
.	O
</s>
<s>
The	O
sample	O
skewness	B-General_Concept
g1	O
and	O
kurtosis	B-Error_Name
g2	O
are	O
both	O
asymptotically	O
normal	O
.	O
</s>
<s>
For	O
example	O
even	O
with	O
observations	O
the	O
sample	O
kurtosis	B-Error_Name
g2	O
has	O
both	O
the	O
skewness	B-General_Concept
and	O
the	O
kurtosis	B-Error_Name
of	O
approximately	O
0.3	O
,	O
which	O
is	O
not	O
negligible	O
.	O
</s>
<s>
In	O
particular	O
,	O
suggested	O
the	O
following	O
transformation	O
for	O
sample	O
skewness	B-General_Concept
:	O
</s>
<s>
and	O
where	O
μ2	O
=	O
μ2(g1 )	O
is	O
the	O
variance	O
of	O
g1	O
,	O
and	O
γ2	O
=	O
γ2(g1 )	O
is	O
the	O
kurtosis	B-Error_Name
—	O
the	O
expressions	O
given	O
in	O
the	O
previous	O
section	O
.	O
</s>
<s>
Statistics	O
Z1	O
and	O
Z2	O
can	O
be	O
combined	O
to	O
produce	O
an	O
omnibus	O
test	O
,	O
able	O
to	O
detect	O
deviations	O
from	O
normality	O
due	O
to	O
either	O
skewness	B-General_Concept
or	O
kurtosis	B-Error_Name
:	O
</s>
<s>
If	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
normality	O
is	O
true	O
,	O
then	O
K2	O
is	O
approximately	O
χ2-distributed	O
with	O
2	O
degrees	O
of	O
freedom	O
.	O
</s>
