<s>
The	O
Shapiro	B-General_Concept
–	I-General_Concept
Francia	I-General_Concept
test	I-General_Concept
is	O
a	O
statistical	B-General_Concept
test	I-General_Concept
for	I-General_Concept
the	I-General_Concept
normality	I-General_Concept
of	O
a	O
population	O
,	O
based	O
on	O
sample	O
data	O
.	O
</s>
<s>
It	O
was	O
introduced	O
by	O
S	O
.	O
S	O
.	O
Shapiro	O
and	O
R	O
.	O
S	O
.	O
Francia	O
in	O
1972	O
as	O
a	O
simplification	O
of	O
the	O
Shapiro	B-General_Concept
–	I-General_Concept
Wilk	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Let	O
be	O
the	O
mean	O
of	O
the	O
th	O
order	B-General_Concept
statistic	I-General_Concept
when	O
making	O
independent	O
draws	O
from	O
a	O
normal	O
distribution	O
.	O
</s>
<s>
Under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
the	O
data	O
is	O
drawn	O
from	O
a	O
normal	O
distribution	O
,	O
this	O
correlation	O
will	O
be	O
strong	O
,	O
so	O
values	O
will	O
cluster	O
just	O
under	O
1	O
,	O
with	O
the	O
peak	O
becoming	O
narrower	O
and	O
closer	O
to	O
1	O
as	O
increases	O
.	O
</s>
<s>
This	O
test	O
is	O
a	O
formalization	O
of	O
the	O
older	O
practice	O
of	O
forming	O
a	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
to	O
compare	O
two	O
distributions	O
,	O
with	O
the	O
playing	O
the	O
role	O
of	O
the	O
quantile	O
points	O
of	O
the	O
sample	O
distribution	O
and	O
the	O
playing	O
the	O
role	O
of	O
the	O
corresponding	O
quantile	O
points	O
of	O
a	O
normal	O
distribution	O
.	O
</s>
<s>
Compared	O
to	O
the	O
Shapiro	B-General_Concept
–	I-General_Concept
Wilk	I-General_Concept
test	I-General_Concept
statistic	O
,	O
the	O
Shapiro	B-General_Concept
–	I-General_Concept
Francia	I-General_Concept
test	I-General_Concept
statistic	O
is	O
easier	O
to	O
compute	O
,	O
because	O
it	O
does	O
not	O
require	O
that	O
we	O
form	O
and	O
invert	O
the	O
matrix	O
of	O
covariances	O
between	O
order	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
Monte-Carlo	B-Algorithm
simulations	I-Algorithm
have	O
shown	O
that	O
the	O
transformed	O
statistic	O
is	O
nearly	O
normally	O
distributed	O
,	O
with	O
values	O
of	O
the	O
mean	O
and	O
standard	O
deviation	O
that	O
vary	O
slowly	O
with	O
in	O
an	O
easily	O
parameterized	O
form	O
.	O
</s>
<s>
Comparison	O
studies	O
have	O
concluded	O
that	O
order	B-General_Concept
statistic	I-General_Concept
correlation	O
tests	O
such	O
as	O
Shapiro	O
–	O
Francia	O
and	O
Shapiro	B-General_Concept
–	I-General_Concept
Wilk	I-General_Concept
are	O
among	O
the	O
most	O
powerful	B-General_Concept
of	O
the	O
established	O
statistical	B-General_Concept
tests	I-General_Concept
for	I-General_Concept
normality	I-General_Concept
.	O
</s>
<s>
One	O
might	O
assume	O
that	O
the	O
covariance-adjusted	O
weighting	O
of	O
different	O
order	B-General_Concept
statistics	I-General_Concept
used	O
by	O
the	O
Shapiro	B-General_Concept
–	I-General_Concept
Wilk	I-General_Concept
test	I-General_Concept
should	O
make	O
it	O
slightly	O
better	O
,	O
but	O
in	O
practice	O
the	O
Shapiro	B-General_Concept
–	I-General_Concept
Wilk	I-General_Concept
and	O
Shapiro	O
–	O
Francia	O
variants	O
are	O
about	O
equally	O
good	O
.	O
</s>
