<s>
In	O
statistics	O
,	O
a	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
(	O
quantile-quantile	B-Application
plot	I-Application
)	O
is	O
a	O
probability	O
plot	O
,	O
a	O
graphical	B-Application
method	I-Application
for	O
comparing	O
two	O
probability	O
distributions	O
by	O
plotting	O
their	O
quantiles	O
against	O
each	O
other	O
.	O
</s>
<s>
If	O
the	O
two	O
distributions	O
being	O
compared	O
are	O
similar	O
,	O
the	O
points	O
in	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
will	O
approximately	O
lie	O
on	O
the	O
identity	O
line	O
.	O
</s>
<s>
If	O
the	O
distributions	O
are	O
linearly	O
related	O
,	O
the	O
points	O
in	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
will	O
approximately	O
lie	O
on	O
a	O
line	O
,	O
but	O
not	O
necessarily	O
on	O
the	O
line	O
.	O
</s>
<s>
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
can	O
also	O
be	O
used	O
as	O
a	O
graphical	O
means	O
of	O
estimating	O
parameters	O
in	O
a	O
location-scale	O
family	O
of	O
distributions	O
.	O
</s>
<s>
A	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
used	O
to	O
compare	O
the	O
shapes	O
of	O
distributions	O
,	O
providing	O
a	O
graphical	O
view	O
of	O
how	O
properties	O
such	O
as	O
location	O
,	O
scale	O
,	O
and	O
skewness	B-General_Concept
are	O
similar	O
or	O
different	O
in	O
the	O
two	O
distributions	O
.	O
</s>
<s>
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
can	O
be	O
used	O
to	O
compare	O
collections	O
of	O
data	O
,	O
or	O
theoretical	O
distributions	O
.	O
</s>
<s>
The	O
use	O
of	O
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
to	O
compare	O
two	O
samples	O
of	O
data	O
can	O
be	O
viewed	O
as	O
a	O
non-parametric	B-General_Concept
approach	O
to	O
comparing	O
their	O
underlying	O
distributions	O
.	O
</s>
<s>
A	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
generally	O
more	O
diagnostic	O
than	O
comparing	O
the	O
samples	O
 '	O
histograms	B-Algorithm
,	O
but	O
is	O
less	O
widely	O
known	O
.	O
</s>
<s>
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
are	O
commonly	O
used	O
to	O
compare	O
a	O
data	O
set	O
to	O
a	O
theoretical	O
model	O
.	O
</s>
<s>
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
are	O
also	O
used	O
to	O
compare	O
two	O
theoretical	O
distributions	O
to	O
each	O
other	O
.	O
</s>
<s>
Since	O
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
compare	O
distributions	O
,	O
there	O
is	O
no	O
need	O
for	O
the	O
values	O
to	O
be	O
observed	O
as	O
pairs	O
,	O
as	O
in	O
a	O
scatter	B-Application
plot	I-Application
,	O
or	O
even	O
for	O
the	O
numbers	O
of	O
values	O
in	O
the	O
two	O
groups	O
being	O
compared	O
to	O
be	O
equal	O
.	O
</s>
<s>
The	O
term	O
"	O
probability	O
plot	O
"	O
sometimes	O
refers	O
specifically	O
to	O
a	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
,	O
sometimes	O
to	O
a	O
more	O
general	O
class	O
of	O
plots	O
,	O
and	O
sometimes	O
to	O
the	O
less	O
commonly	O
used	O
P	B-Application
–	I-Application
P	I-Application
plot	I-Application
.	O
</s>
<s>
The	O
probability	B-Application
plot	I-Application
correlation	I-Application
coefficient	I-Application
plot	I-Application
(	O
PPCC	B-Application
plot	I-Application
)	O
is	O
a	O
quantity	O
derived	O
from	O
the	O
idea	O
of	O
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
,	O
which	O
measures	O
the	O
agreement	O
of	O
a	O
fitted	O
distribution	O
with	O
observed	O
data	O
and	O
which	O
is	O
sometimes	O
used	O
as	O
a	O
means	O
of	O
fitting	O
a	O
distribution	O
to	O
data	O
.	O
</s>
<s>
A	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
a	O
plot	O
of	O
the	O
quantiles	O
of	O
two	O
distributions	O
against	O
each	O
other	O
,	O
or	O
a	O
plot	O
based	O
on	O
estimates	O
of	O
the	O
quantiles	O
.	O
</s>
<s>
The	O
main	O
step	O
in	O
constructing	O
a	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
calculating	O
or	O
estimating	O
the	O
quantiles	O
to	O
be	O
plotted	O
.	O
</s>
<s>
If	O
one	O
or	O
both	O
of	O
the	O
axes	O
in	O
a	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
based	O
on	O
a	O
theoretical	O
distribution	O
with	O
a	O
continuous	O
cumulative	O
distribution	O
function	O
(	O
CDF	O
)	O
,	O
all	O
quantiles	O
are	O
uniquely	O
defined	O
and	O
can	O
be	O
obtained	O
by	O
inverting	O
the	O
CDF	O
.	O
</s>
<s>
If	O
a	O
theoretical	O
probability	O
distribution	O
with	O
a	O
discontinuous	O
CDF	O
is	O
one	O
of	O
the	O
two	O
distributions	O
being	O
compared	O
,	O
some	O
of	O
the	O
quantiles	O
may	O
not	O
be	O
defined	O
,	O
so	O
an	O
interpolated	B-Algorithm
quantile	O
may	O
be	O
plotted	O
.	O
</s>
<s>
If	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
based	O
on	O
data	O
,	O
there	O
are	O
multiple	O
quantile	O
estimators	O
in	O
use	O
.	O
</s>
<s>
Rules	O
for	O
forming	O
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
when	O
quantiles	O
must	O
be	O
estimated	O
or	O
interpolated	B-Algorithm
are	O
called	O
plotting	O
positions	O
.	O
</s>
<s>
In	O
that	O
case	O
,	O
to	O
make	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
,	O
one	O
orders	O
each	O
set	O
in	O
increasing	O
order	O
,	O
then	O
pairs	O
off	O
and	O
plots	O
the	O
corresponding	O
values	O
.	O
</s>
<s>
To	O
construct	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
in	O
this	O
case	O
,	O
it	O
is	O
necessary	O
to	O
use	O
an	O
interpolated	B-Algorithm
quantile	O
estimate	O
so	O
that	O
quantiles	O
corresponding	O
to	O
the	O
same	O
underlying	O
probability	O
can	O
be	O
constructed	O
.	O
</s>
<s>
More	O
abstractly	O
,	O
given	O
two	O
cumulative	O
probability	O
distribution	O
functions	O
and	O
,	O
with	O
associated	O
quantile	O
functions	O
and	O
(	O
the	O
inverse	O
function	O
of	O
the	O
CDF	O
is	O
the	O
quantile	O
function	O
)	O
,	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
draws	O
the	O
-th	O
quantile	O
of	O
against	O
the	O
-th	O
quantile	O
of	O
for	O
a	O
range	O
of	O
values	O
of	O
.	O
</s>
<s>
Thus	O
,	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
a	O
parametric	O
curve	O
indexed	O
over	O
 [ 0 , 1 ] 	O
with	O
values	O
in	O
the	O
real	O
plane	O
.	O
</s>
<s>
The	O
points	O
plotted	O
in	O
a	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
are	O
always	O
non-decreasing	O
when	O
viewed	O
from	O
left	O
to	O
right	O
.	O
</s>
<s>
If	O
the	O
two	O
distributions	O
being	O
compared	O
are	O
identical	O
,	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
follows	O
the	O
45°	O
line	O
.	O
</s>
<s>
If	O
the	O
two	O
distributions	O
agree	O
after	O
linearly	O
transforming	O
the	O
values	O
in	O
one	O
of	O
the	O
distributions	O
,	O
then	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
follows	O
some	O
line	O
,	O
but	O
not	O
necessarily	O
the	O
line	O
.	O
</s>
<s>
If	O
the	O
general	O
trend	O
of	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
flatter	O
than	O
the	O
line	O
,	O
the	O
distribution	O
plotted	O
on	O
the	O
horizontal	O
axis	O
is	O
more	O
dispersed	O
than	O
the	O
distribution	O
plotted	O
on	O
the	O
vertical	O
axis	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
the	O
general	O
trend	O
of	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
steeper	O
than	O
the	O
line	O
,	O
the	O
distribution	O
plotted	O
on	O
the	O
vertical	O
axis	O
is	O
more	O
dispersed	O
than	O
the	O
distribution	O
plotted	O
on	O
the	O
horizontal	O
axis	O
.	O
</s>
<s>
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
are	O
often	O
arced	O
,	O
or	O
"	O
S	O
"	O
shaped	O
,	O
indicating	O
that	O
one	O
of	O
the	O
distributions	O
is	O
more	O
skewed	B-General_Concept
than	O
the	O
other	O
,	O
or	O
that	O
one	O
of	O
the	O
distributions	O
has	O
heavier	O
tails	O
than	O
the	O
other	O
.	O
</s>
<s>
Although	O
a	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
is	O
based	O
on	O
quantiles	O
,	O
in	O
a	O
standard	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
it	O
is	O
not	O
possible	O
to	O
determine	O
which	O
point	O
in	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
determines	O
a	O
given	O
quantile	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
is	O
not	O
possible	O
to	O
determine	O
the	O
median	O
of	O
either	O
of	O
the	O
two	O
distributions	O
being	O
compared	O
by	O
inspecting	O
the	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
.	O
</s>
<s>
Some	O
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
indicate	O
the	O
deciles	O
to	O
make	O
determinations	O
such	O
as	O
this	O
possible	O
.	O
</s>
<s>
The	O
intercept	B-Algorithm
and	O
slope	O
of	O
a	O
linear	O
regression	O
between	O
the	O
quantiles	O
gives	O
a	O
measure	O
of	O
the	O
relative	O
location	O
and	O
relative	O
scale	O
of	O
the	O
samples	O
.	O
</s>
<s>
If	O
the	O
median	O
of	O
the	O
distribution	O
plotted	O
on	O
the	O
horizontal	O
axis	O
is	O
0	O
,	O
the	O
intercept	B-Algorithm
of	O
a	O
regression	O
line	O
is	O
a	O
measure	O
of	O
location	O
,	O
and	O
the	O
slope	O
is	O
a	O
measure	O
of	O
scale	O
.	O
</s>
<s>
The	O
distance	O
between	O
medians	O
is	O
another	O
measure	O
of	O
relative	O
location	O
reflected	O
in	O
a	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
.	O
</s>
<s>
The	O
"	O
probability	B-Application
plot	I-Application
correlation	I-Application
coefficient	I-Application
"	O
(	O
PPCC	B-Application
plot	I-Application
)	O
is	O
the	O
correlation	O
coefficient	O
between	O
the	O
paired	O
sample	O
quantiles	O
.	O
</s>
<s>
For	O
distributions	O
with	O
a	O
single	O
shape	O
parameter	O
,	O
the	O
probability	B-Application
plot	I-Application
correlation	I-Application
coefficient	I-Application
plot	I-Application
provides	O
a	O
method	O
for	O
estimating	O
the	O
shape	O
parameter	O
–	O
one	O
simply	O
computes	O
the	O
correlation	O
coefficient	O
for	O
different	O
values	O
of	O
the	O
shape	O
parameter	O
,	O
and	O
uses	O
the	O
one	O
with	O
the	O
best	O
fit	O
,	O
just	O
as	O
if	O
one	O
were	O
comparing	O
distributions	O
of	O
different	O
types	O
.	O
</s>
<s>
Another	O
common	O
use	O
of	O
Q	B-Application
–	I-Application
Q	I-Application
plots	I-Application
is	O
to	O
compare	O
the	O
distribution	O
of	O
a	O
sample	O
to	O
a	O
theoretical	O
distribution	O
,	O
such	O
as	O
the	O
standard	O
normal	O
distribution	O
,	O
as	O
in	O
a	O
normal	B-Application
probability	I-Application
plot	I-Application
.	O
</s>
<s>
As	O
in	O
the	O
case	O
when	O
comparing	O
two	O
samples	O
of	O
data	O
,	O
one	O
orders	O
the	O
data	O
(	O
formally	O
,	O
computes	O
the	O
order	B-General_Concept
statistics	I-General_Concept
)	O
,	O
then	O
plots	O
them	O
against	O
certain	O
quantiles	O
of	O
the	O
theoretical	O
distribution	O
.	O
</s>
<s>
In	O
using	O
a	O
normal	B-Application
probability	I-Application
plot	I-Application
,	O
the	O
quantiles	O
one	O
uses	O
are	O
the	O
rankits	B-Application
,	O
the	O
quantile	O
of	O
the	O
expected	O
value	O
of	O
the	O
order	B-General_Concept
statistic	I-General_Concept
of	O
a	O
standard	O
normal	O
distribution	O
.	O
</s>
<s>
More	O
generally	O
,	O
Shapiro	B-General_Concept
–	I-General_Concept
Wilk	I-General_Concept
test	I-General_Concept
uses	O
the	O
expected	O
values	O
of	O
the	O
order	B-General_Concept
statistics	I-General_Concept
of	O
the	O
given	O
distribution	O
;	O
the	O
resulting	O
plot	O
and	O
line	O
yields	O
the	O
generalized	O
least	O
squares	O
estimate	O
for	O
location	O
and	O
scale	O
(	O
from	O
the	O
intercept	B-Algorithm
and	O
slope	O
of	O
the	O
fitted	O
line	O
)	O
.	O
</s>
<s>
However	O
,	O
this	O
requires	O
calculating	O
the	O
expected	O
values	O
of	O
the	O
order	B-General_Concept
statistic	I-General_Concept
,	O
which	O
may	O
be	O
difficult	O
if	O
the	O
distribution	O
is	O
not	O
normal	O
.	O
</s>
<s>
Alternatively	O
,	O
one	O
may	O
use	O
estimates	O
of	O
the	O
median	O
of	O
the	O
order	B-General_Concept
statistics	I-General_Concept
,	O
which	O
one	O
can	O
compute	O
based	O
on	O
estimates	O
of	O
the	O
median	O
of	O
the	O
order	B-General_Concept
statistics	I-General_Concept
of	O
a	O
uniform	O
distribution	O
and	O
the	O
quantile	O
function	O
of	O
the	O
distribution	O
;	O
this	O
was	O
suggested	O
by	O
.	O
</s>
<s>
Several	O
different	O
formulas	O
have	O
been	O
used	O
or	O
proposed	O
as	O
affine	B-Algorithm
symmetrical	O
plotting	O
positions	O
.	O
</s>
<s>
The	O
order	B-General_Concept
statistic	I-General_Concept
medians	O
are	O
the	O
medians	O
of	O
the	O
order	B-General_Concept
statistics	I-General_Concept
of	O
the	O
distribution	O
.	O
</s>
<s>
These	O
can	O
be	O
expressed	O
in	O
terms	O
of	O
the	O
quantile	O
function	O
and	O
the	O
order	B-General_Concept
statistic	I-General_Concept
medians	O
for	O
the	O
continuous	O
uniform	O
distribution	O
by	O
:	O
</s>
<s>
where	O
are	O
the	O
uniform	O
order	B-General_Concept
statistic	I-General_Concept
medians	O
and	O
is	O
the	O
quantile	O
function	O
for	O
the	O
desired	O
distribution	O
.	O
</s>
<s>
James	O
J	O
.	O
Filliben	O
uses	O
the	O
following	O
estimates	O
for	O
the	O
uniform	O
order	B-General_Concept
statistic	I-General_Concept
medians	O
:	O
</s>
<s>
The	O
reason	O
for	O
this	O
estimate	O
is	O
that	O
the	O
order	B-General_Concept
statistic	I-General_Concept
medians	O
do	O
not	O
have	O
a	O
simple	O
form	O
.	O
</s>
