Editing Fisher transformation (section)

==Definition==

Given a set of ''N'' bivariate sample pairs (''X''<sub>''i''</sub>,&nbsp;''Y''<sub>''i''</sub>), ''i''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''N'',  the [[Pearson product-moment correlation coefficient|sample correlation coefficient]] ''r'' is given by

:<math>r = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} = \frac{\sum ^N _{i=1}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum ^N _{i=1}(X_i - \bar{X})^2} \sqrt{\sum ^N _{i=1}(Y_i - \bar{Y})^2}}.</math>

Here <math>\operatorname{cov}(X,Y)</math> stands for the [[covariance]] between the variables <math>X</math> and <math>Y</math> and <math>\sigma</math> stands for the [[standard deviation]] of the respective variable.  Fisher's z-transformation of ''r'' is defined as
:<math>z = {1 \over 2}\ln\left({1+r \over 1-r}\right) = \operatorname{artanh}(r),</math>
where "ln" is the [[natural logarithm]] function and "artanh" is the [[inverse hyperbolic function|inverse hyperbolic tangent function]].

If (''X'',&nbsp;''Y'') has a [[bivariate normal distribution]] with correlation ρ and the pairs (''X''<sub>''i''</sub>,&nbsp;''Y''<sub>''i''</sub>) are [[Independent and identically distributed random variables|independent and identically distributed]], then ''z'' is approximately [[normal distribution|normally distributed]] with mean
:<math>{1 \over 2}\ln\left({{1+\rho} \over {1-\rho}}\right),</math>
and a standard deviation which does not depend on the value of the correlation rho (i.e., a [[Variance-stabilizing transformation]])
:<math>{1 \over \sqrt{N-3}},</math>
where ''N'' is the sample size, and ρ is the true correlation coefficient.

This transformation, and its inverse
:<math>r = \frac{\exp(2z)-1}{\exp(2z)+1} = \operatorname{tanh}(z),</math>
can be used to construct a large-sample [[confidence interval]] for&nbsp;''r'' using standard normal theory and derivations. See also application to [[partial correlation]].