Statistical distribution
Not to be confused with Fisher z-transformation .
"z-distribution" redirects here. For the distribution related to z-scores, see Normal distribution § Standard normal distribution .
Fisher's z
Probability density function Parameters
d
1
>
0
,
d
2
>
0
{\displaystyle d_{1}>0,\ d_{2}>0}
Support
x
∈
(
−
∞
;
+
∞
)
{\displaystyle x\in (-\infty ;+\infty )\!}
PDF
2
d
1
d
1
/
2
d
2
d
2
/
2
B
(
d
1
/
2
,
d
2
/
2
)
e
d
1
x
(
d
1
e
2
x
+
d
2
)
(
d
1
+
d
2
)
/
2
{\displaystyle {\frac {2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}}{\frac {e^{d_{1}x}}{\left(d_{1}e^{2x}+d_{2}\right)^{\left(d_{1}+d_{2}\right)/2}}}\!}
Mode
0
{\displaystyle 0}
Ronald Fisher Fisher's z -distribution is the statistical distribution of half the logarithm of an F -distributionvariate :
z
=
1
2
log
F
{\displaystyle z={\frac {1}{2}}\log F}
It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto . Nowadays one usually uses the F -distribution instead.
The probability density function and cumulative distribution function can be found by using the F -distribution at the value of
x
′
=
e
2
x
{\displaystyle x'=e^{2x}\,}
The probability density function is
f
(
x
;
d
1
,
d
2
)
=
2
d
1
d
1
/
2
d
2
d
2
/
2
B
(
d
1
/
2
,
d
2
/
2
)
e
d
1
x
(
d
1
e
2
x
+
d
2
)
(
d
1
+
d
2
)
/
2
,
{\displaystyle f(x;d_{1},d_{2})={\frac {2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}}{\frac {e^{d_{1}x}}{\left(d_{1}e^{2x}+d_{2}\right)^{(d_{1}+d_{2})/2}}},}
where B is the beta function .
When the degrees of freedom becomes large (
d
1
,
d
2
→
∞
{\displaystyle d_{1},d_{2}\rightarrow \infty }
normality with mean
x
¯
=
1
2
(
1
d
2
−
1
d
1
)
{\displaystyle {\bar {x}}={\frac {1}{2}}\left({\frac {1}{d_{2}}}-{\frac {1}{d_{1}}}\right)}
and variance
σ
x
2
=
1
2
(
1
d
1
+
1
d
2
)
.
{\displaystyle \sigma _{x}^{2}={\frac {1}{2}}\left({\frac {1}{d_{1}}}+{\frac {1}{d_{2}}}\right).}
Related distribution
If
X
∼
FisherZ
(
n
,
m
)
{\displaystyle X\sim \operatorname {FisherZ} (n,m)}
e
2
X
∼
F
(
n
,
m
)
{\displaystyle e^{2X}\sim \operatorname {F} (n,m)\,}
F -distribution
If
X
∼
F
(
n
,
m
)
{\displaystyle X\sim \operatorname {F} (n,m)}
log
X
2
∼
FisherZ
(
n
,
m
)
{\displaystyle {\tfrac {\log X}{2}}\sim \operatorname {FisherZ} (n,m)}
References
Fisher, R. A. (1924). "On a Distribution Yielding the Error Functions of Several Well Known Statistics" (PDF). Proceedings of the International Congress of Mathematics, Toronto . 2 : 805–813. Archived from the original (PDF) on April 12, 2011.
^ Leo A. Aroian (December 1941). "A study of R. A. Fisher's z distribution and the related F distribution" . The Annals of Mathematical Statistics . 12 (4): 429–448. doi :10.1214/aoms/1177731681 . JSTOR 2235955 .
Charles Ernest Weatherburn (1961). A first course in mathematical statistics
External links
Categories :
Fisher's
z -distribution
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↑
deg. of freedom
. However, the mean and variance do not follow the same transformation.
), the distribution approaches 
then 
(
then 
