Editing Pearson correlation coefficient (section)

===Using the Fisher transformation===
{{main|Fisher transformation}}

In practice, [[confidence intervals]] and [[hypothesis test]]s relating to ''ρ'' are usually carried out using the, [[Variance-stabilizing transformation]], [[Fisher transformation]], <math>F</math>'':

:<math>F(r) \equiv \tfrac{1}{2} \, \ln \left(\frac{1 + r}{1 - r}\right) = \operatorname{artanh}(r)</math>

''F''(''r'') approximately follows a [[normal distribution]] with

:<math>\text{mean} = F(\rho) = \operatorname{artanh}(\rho)</math>{{spaces|4}}and [[standard error]] <math>=\text{SE} = \frac{1}{\sqrt{n - 3}},</math>

where ''n'' is the sample size. The approximation error is lowest for a large sample size <math>n</math> and small <math>r</math> and <math>\rho_0</math> and increases otherwise.

Using the approximation, a [[standard score|z-score]] is

:<math>z = \frac{x - \text{mean}}{\text{SE}} = [F(r) - F(\rho_0)]\sqrt{n - 3}</math>

under the [[null hypothesis]] that <math>\rho = \rho_0</math>, given the assumption that the sample pairs are [[independent and identically distributed]] and follow a [[bivariate normal distribution]].  Thus an approximate [[p-value]] can be obtained from a normal probability table.  For example, if ''z''&nbsp;=&nbsp;2.2 is observed and a two-sided p-value is desired to test the null hypothesis that <math>\rho = 0</math>, the p-value is {{nowrap|1=2Φ(−2.2) = 0.028}}, where Φ is the standard normal [[cumulative distribution function]].

To obtain a confidence interval for ρ, we first compute a confidence interval for ''F''(''<math>\rho</math>''):

:<math>100(1 - \alpha)\%\text{CI}: \operatorname{artanh}(\rho) \in [\operatorname{artanh}(r) \pm z_{\alpha/2}\text{SE}]</math>

The inverse Fisher transformation brings the interval back to the correlation scale.

:<math>100(1 - \alpha)\%\text{CI}: \rho \in [\tanh(\operatorname{artanh}(r) - z_{\alpha/2}\text{SE}), \tanh(\operatorname{artanh}(r) + z_{\alpha/2}\text{SE})]</math>

For example, suppose we observe ''r''&nbsp;=&nbsp;0.7 with a sample size of ''n''=50, and we wish to obtain a 95% confidence interval for&nbsp;''ρ''. The transformed value is <math display="inline">\operatorname{arctanh} \left ( r \right ) = 0.8673</math>, so the confidence interval on the transformed scale is <math>0.8673 \pm \frac{1.96}{\sqrt{47}} </math>, or (0.5814,&nbsp;1.1532). Converting back to the correlation scale yields (0.5237,&nbsp;0.8188).