<s>
In	O
statistics	O
,	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
(	O
or	O
Fisher	B-Algorithm
z-transformation	I-Algorithm
)	O
of	O
a	O
Pearson	O
correlation	O
coefficient	O
is	O
its	O
inverse	O
hyperbolic	O
tangent	O
(	O
artanh	O
)	O
.	O
</s>
<s>
When	O
the	O
sample	O
correlation	O
coefficient	O
r	B-Language
is	O
near	O
1	O
or	O
-1	O
,	O
its	O
distribution	O
is	O
highly	O
skewed	B-General_Concept
,	O
which	O
makes	O
it	O
difficult	O
to	O
estimate	O
confidence	O
intervals	O
and	O
apply	O
tests	O
of	O
significance	O
for	O
the	O
population	O
correlation	O
coefficient	O
ρ	O
.	O
</s>
<s>
The	O
Fisher	B-Algorithm
transformation	I-Algorithm
solves	O
this	O
problem	O
by	O
yielding	O
a	O
variable	O
whose	O
distribution	O
is	O
approximately	O
normally	O
distributed	O
,	O
with	O
a	O
variance	O
that	O
is	O
stable	O
over	O
different	O
values	O
of	O
r	B-Language
.	O
</s>
<s>
Here	O
stands	O
for	O
the	O
covariance	O
between	O
the	O
variables	O
and	O
and	O
stands	O
for	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
respective	O
variable	O
.	O
</s>
<s>
See	O
also	O
application	O
to	O
partial	B-Language
correlation	I-Language
.	O
</s>
<s>
Hotelling	O
gives	O
a	O
concise	O
derivation	O
of	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
To	O
derive	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
,	O
one	O
starts	O
by	O
considering	O
an	O
arbitrary	O
increasing	O
,	O
twice-differentiable	O
function	O
of	O
,	O
say	O
.	O
</s>
<s>
The	O
extra	O
terms	O
are	O
not	O
part	O
of	O
the	O
usual	O
Fisher	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
The	O
near-constant	O
variance	O
of	O
the	O
transformation	O
is	O
the	O
result	O
of	O
removing	O
its	O
skewness	B-General_Concept
–	O
the	O
actual	O
improvement	O
is	O
achieved	O
by	O
the	O
latter	O
,	O
not	O
by	O
the	O
extra	O
terms	O
.	O
</s>
<s>
The	O
application	O
of	O
Fisher	B-Algorithm
's	I-Algorithm
transformation	I-Algorithm
can	O
be	O
enhanced	O
using	O
a	O
software	O
calculator	O
as	O
shown	O
in	O
the	O
figure	O
.	O
</s>
<s>
Assuming	O
that	O
the	O
r-squared	O
value	O
found	O
is	O
0.80	O
,	O
that	O
there	O
are	O
30	O
data	O
,	O
and	O
accepting	O
a	O
90%	O
confidence	O
interval	O
,	O
the	O
r-squared	O
value	O
in	O
another	O
random	O
sample	O
from	O
the	O
same	O
population	O
may	O
range	O
from	O
0.588	O
to	O
0.921	O
.	O
</s>
<s>
When	O
r-squared	O
is	O
outside	O
this	O
range	O
,	O
the	O
population	O
is	O
considered	O
to	O
be	O
different	O
.	O
</s>
<s>
However	O
,	O
if	O
a	O
certain	O
data	O
set	O
is	O
analysed	O
with	O
two	O
different	O
regression	O
models	O
while	O
the	O
first	O
model	O
yields	O
r-squared	O
=	O
0.80	O
and	O
the	O
second	O
r-squared	O
is	O
0.49	O
,	O
one	O
may	O
conclude	O
that	O
the	O
second	O
model	O
is	O
insignificant	O
as	O
the	O
value	O
0.49	O
is	O
below	O
the	O
critical	O
value	O
0.588	O
.	O
</s>
<s>
The	O
Fisher	B-Algorithm
transformation	I-Algorithm
is	O
an	O
approximate	O
variance-stabilizing	B-General_Concept
transformation	I-General_Concept
for	O
r	B-Language
when	O
X	O
and	O
Y	O
follow	O
a	O
bivariate	O
normal	O
distribution	O
.	O
</s>
<s>
Without	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
,	O
the	O
variance	O
of	O
r	B-Language
grows	O
smaller	O
as	O
|ρ|	O
gets	O
closer	O
to	O
1	O
.	O
</s>
<s>
Since	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
is	O
approximately	O
the	O
identity	O
function	O
when	O
|r|	O
<	O
1/2	O
,	O
it	O
is	O
sometimes	O
useful	O
to	O
remember	O
that	O
the	O
variance	O
of	O
r	B-Language
is	O
well	O
approximated	O
by	O
1/N	O
as	O
long	O
as	O
|ρ|	O
is	O
not	O
too	O
large	O
and	O
N	O
is	O
not	O
too	O
small	O
.	O
</s>
<s>
This	O
is	O
related	O
to	O
the	O
fact	O
that	O
the	O
asymptotic	O
variance	O
of	O
r	B-Language
is	O
1	O
for	O
bivariate	O
normal	O
data	O
.	O
</s>
<s>
While	O
the	O
Fisher	B-Algorithm
transformation	I-Algorithm
is	O
mainly	O
associated	O
with	O
the	O
Pearson	O
product-moment	O
correlation	O
coefficient	O
for	O
bivariate	O
normal	O
observations	O
,	O
it	O
can	O
also	O
be	O
applied	O
to	O
Spearman	B-General_Concept
's	I-General_Concept
rank	I-General_Concept
correlation	I-General_Concept
coefficient	I-General_Concept
in	O
more	O
general	O
cases	O
.	O
</s>
