<s>
In	O
statistics	O
,	O
normality	B-General_Concept
tests	I-General_Concept
are	O
used	O
to	O
determine	O
if	O
a	O
data	B-General_Concept
set	I-General_Concept
is	O
well-modeled	O
by	O
a	O
normal	O
distribution	O
and	O
to	O
compute	O
how	O
likely	O
it	O
is	O
for	O
a	O
random	O
variable	O
underlying	O
the	O
data	B-General_Concept
set	I-General_Concept
to	O
be	O
normally	O
distributed	O
.	O
</s>
<s>
In	O
descriptive	B-General_Concept
statistics	I-General_Concept
terms	O
,	O
one	O
measures	O
a	O
goodness	O
of	O
fit	O
of	O
a	O
normal	O
model	O
to	O
the	O
data	O
–	O
if	O
the	O
fit	O
is	O
poor	O
then	O
the	O
data	O
are	O
not	O
well	O
modeled	O
in	O
that	O
respect	O
by	O
a	O
normal	O
distribution	O
,	O
without	O
making	O
a	O
judgment	O
on	O
any	O
underlying	O
variable	O
.	O
</s>
<s>
In	O
frequentist	B-General_Concept
statistics	I-General_Concept
statistical	O
hypothesis	O
testing	O
,	O
data	O
are	O
tested	O
against	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
it	O
is	O
normally	O
distributed	O
.	O
</s>
<s>
In	O
Bayesian	O
statistics	O
,	O
one	O
does	O
not	O
"	O
test	O
normality	O
"	O
per	O
se	O
,	O
but	O
rather	O
computes	O
the	O
likelihood	O
that	O
the	O
data	O
come	O
from	O
a	O
normal	O
distribution	O
with	O
given	O
parameters	O
μ	O
,	O
σ	O
(	O
for	O
all	O
μ	O
,	O
σ	O
)	O
,	O
and	O
compares	O
that	O
with	O
the	O
likelihood	O
that	O
the	O
data	O
come	O
from	O
other	O
distributions	O
under	O
consideration	O
,	O
most	O
simply	O
using	O
a	O
Bayes	B-General_Concept
factor	I-General_Concept
(	O
giving	O
the	O
relative	O
likelihood	O
of	O
seeing	O
the	O
data	O
given	O
different	O
models	O
)	O
,	O
or	O
more	O
finely	O
taking	O
a	O
prior	O
distribution	O
on	O
possible	O
models	O
and	O
parameters	O
and	O
computing	O
a	O
posterior	O
distribution	O
given	O
the	O
computed	O
likelihoods	O
.	O
</s>
<s>
A	O
normality	B-General_Concept
test	I-General_Concept
is	O
used	O
to	O
determine	O
whether	O
sample	O
data	O
has	O
been	O
drawn	O
from	O
a	O
normally	O
distributed	O
population	O
(	O
within	O
some	O
tolerance	O
)	O
.	O
</s>
<s>
A	O
number	O
of	O
statistical	O
tests	O
,	O
such	O
as	O
the	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
and	O
the	O
one-way	O
and	O
two-way	O
ANOVA	O
,	O
require	O
a	O
normally	O
distributed	O
sample	O
population	O
.	O
</s>
<s>
An	O
informal	O
approach	O
to	O
testing	O
normality	O
is	O
to	O
compare	O
a	O
histogram	B-Algorithm
of	O
the	O
sample	O
data	O
to	O
a	O
normal	O
probability	O
curve	O
.	O
</s>
<s>
The	O
empirical	O
distribution	O
of	O
the	O
data	O
(	O
the	O
histogram	B-Algorithm
)	O
should	O
be	O
bell-shaped	O
and	O
resemble	O
the	O
normal	O
distribution	O
.	O
</s>
<s>
Lack	O
of	O
fit	O
to	O
the	O
regression	B-General_Concept
line	I-General_Concept
suggests	O
a	O
departure	O
from	O
normality	O
(	O
see	O
Anderson	O
Darling	O
coefficient	O
and	O
minitab	O
)	O
.	O
</s>
<s>
A	O
graphical	O
tool	O
for	O
assessing	O
normality	O
is	O
the	O
normal	B-Application
probability	I-Application
plot	I-Application
,	O
a	O
quantile-quantile	B-Application
plot	I-Application
(	O
QQ	B-Application
plot	I-Application
)	O
of	O
the	O
standardized	O
data	O
against	O
the	O
standard	O
normal	O
distribution	O
.	O
</s>
<s>
For	O
normal	O
data	O
the	O
points	O
plotted	O
in	O
the	O
QQ	B-Application
plot	I-Application
should	O
fall	O
approximately	O
on	O
a	O
straight	O
line	O
,	O
indicating	O
high	O
positive	O
correlation	O
.	O
</s>
<s>
This	O
test	O
is	O
useful	O
in	O
cases	O
where	O
one	O
faces	O
kurtosis	B-Error_Name
risk	O
–	O
where	O
large	O
deviations	O
matter	O
–	O
and	O
has	O
the	O
benefits	O
that	O
it	O
is	O
very	O
easy	O
to	O
compute	O
and	O
to	O
communicate	O
:	O
non-statisticians	O
can	O
easily	O
grasp	O
that	O
"	O
6σ	O
events	O
are	O
very	O
rare	O
in	O
normal	O
distributions	O
"	O
.	O
</s>
<s>
Agostino	B-General_Concept
's	I-General_Concept
K-squared	I-General_Concept
test	I-General_Concept
,	O
</s>
<s>
Jarque	B-General_Concept
–	I-General_Concept
Bera	I-General_Concept
test	I-General_Concept
,	O
</s>
<s>
Anderson	B-General_Concept
–	I-General_Concept
Darling	I-General_Concept
test	I-General_Concept
,	O
</s>
<s>
Cramér	B-General_Concept
–	I-General_Concept
von	I-General_Concept
Mises	I-General_Concept
criterion	I-General_Concept
,	O
</s>
<s>
Kolmogorov	B-General_Concept
–	I-General_Concept
Smirnov	I-General_Concept
test	I-General_Concept
(	O
this	O
one	O
only	O
works	O
if	O
the	O
mean	O
and	O
the	O
variance	O
of	O
the	O
normal	O
are	O
assumed	O
known	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
)	O
,	O
</s>
<s>
Lilliefors	B-General_Concept
test	I-General_Concept
(	O
based	O
on	O
the	O
Kolmogorov	B-General_Concept
–	I-General_Concept
Smirnov	I-General_Concept
test	I-General_Concept
,	O
adjusted	O
for	O
when	O
also	O
estimating	O
the	O
mean	O
and	O
variance	O
from	O
the	O
data	O
)	O
,	O
</s>
<s>
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
A	O
2011	O
study	O
concludes	O
that	O
Shapiro	O
–	O
Wilk	O
has	O
the	O
best	O
power	B-General_Concept
for	O
a	O
given	O
significance	O
,	O
followed	O
closely	O
by	O
Anderson	B-General_Concept
–	I-General_Concept
Darling	I-General_Concept
when	O
comparing	O
the	O
Shapiro	O
–	O
Wilk	O
,	O
Kolmogorov	B-General_Concept
–	I-General_Concept
Smirnov	I-General_Concept
,	O
Lilliefors	O
,	O
and	O
Anderson	B-General_Concept
–	I-General_Concept
Darling	I-General_Concept
tests	I-General_Concept
.	O
</s>
<s>
Some	O
published	O
works	O
recommend	O
the	O
Jarque	B-General_Concept
–	I-General_Concept
Bera	I-General_Concept
test	I-General_Concept
,	O
but	O
the	O
test	O
has	O
weakness	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
test	O
has	O
low	O
power	B-General_Concept
for	O
distributions	O
with	O
short	O
tails	O
,	O
especially	O
for	O
bimodal	O
distributions	O
.	O
</s>
<s>
Historically	O
,	O
the	O
third	O
and	O
fourth	O
standardized	O
moments	O
(	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
)	O
were	O
some	O
of	O
the	O
earliest	O
tests	O
for	O
normality	O
.	O
</s>
<s>
The	O
Jarque	B-General_Concept
–	I-General_Concept
Bera	I-General_Concept
test	I-General_Concept
is	O
itself	O
derived	O
from	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
estimates	O
.	O
</s>
<s>
Mardia	O
's	O
multivariate	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
tests	O
generalize	O
the	O
moment	O
tests	O
to	O
the	O
multivariate	O
case	O
.	O
</s>
<s>
Other	O
early	O
test	B-General_Concept
statistics	I-General_Concept
include	O
the	O
ratio	O
of	O
the	O
mean	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
to	O
the	O
standard	O
deviation	O
and	O
of	O
the	O
range	O
to	O
the	O
standard	O
deviation	O
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
normality	B-General_Concept
tests	I-General_Concept
based	O
on	O
this	O
property	O
,	O
the	O
first	O
attributable	O
to	O
Vasicek	O
.	O
</s>
<s>
Spiegelhalter	O
suggests	O
using	O
a	O
Bayes	B-General_Concept
factor	I-General_Concept
to	O
compare	O
normality	O
with	O
a	O
different	O
class	O
of	O
distributional	O
alternatives	O
.	O
</s>
<s>
One	O
application	O
of	O
normality	B-General_Concept
tests	I-General_Concept
is	O
to	O
the	O
residuals	O
from	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
.	O
</s>
<s>
If	O
they	O
are	O
not	O
normally	O
distributed	O
,	O
the	O
residuals	O
should	O
not	O
be	O
used	O
in	O
Z	O
tests	O
or	O
in	O
any	O
other	O
tests	O
derived	O
from	O
the	O
normal	O
distribution	O
,	O
such	O
as	O
t	B-General_Concept
tests	I-General_Concept
,	O
F	B-General_Concept
tests	I-General_Concept
and	O
chi-squared	B-General_Concept
tests	I-General_Concept
.	O
</s>
