Editing Pearson correlation coefficient (section)

===Adjusted correlation coefficient===
The sample correlation coefficient {{mvar|r}} is not an unbiased estimate of {{mvar|ρ}}. For data that follows a [[bivariate normal distribution]], the expectation {{math|E[''r'']}} for the sample correlation coefficient  {{mvar|r}} of a normal bivariate is<ref>{{Cite journal | first = H. | last = Hotelling | year = 1953 | title = New Light on the Correlation Coefficient and its Transforms | journal = Journal of the Royal Statistical Society. Series B (Methodological) | volume = 15 | issue = 2 | pages = 193–232 | jstor = 2983768| doi = 10.1111/j.2517-6161.1953.tb00135.x }}</ref>

:<math>\operatorname\mathbb{E}\left[r\right] = \rho - \frac{\rho \left(1 - \rho^2\right)}{2n} + \cdots, \quad</math> therefore {{mvar|r}} is a biased estimator of <math>\rho.</math>

The unique minimum variance unbiased estimator {{math|''r''<sub>adj</sub>}} is given by<ref>{{Cite journal | first=Ingram | last=Olkin |author2=Pratt, John W.  |date=March 1958  | title=Unbiased Estimation of Certain Correlation Coefficients | journal=The Annals of Mathematical Statistics | volume=29| issue=1 | pages=201–211 | jstor=2237306 | doi=10.1214/aoms/1177706717| doi-access=free }}.</ref>

{{NumBlk|:|<math> r_\text{adj} = r \, \mathbf{_2F_1}\left(\frac{1}{2}, \frac{1}{2}; \frac{n - 1}{2}; 1 - r^2\right),</math>|{{EquationRef|1}}}}

where:
*<math>r, n</math> are defined as above,
*<math>\mathbf{_2 F_1}(a, b; c; z)</math> is the [[hypergeometric function|Gaussian hypergeometric function]].

An approximately unbiased estimator {{math|''r''<sub>adj</sub>}} can be obtained{{citation needed|date=April 2012}} by truncating {{math|E[''r'']}} and solving this truncated equation:

{{NumBlk|:|<math> r = \operatorname\mathbb{E}[r] \approx r_\text{adj} - \frac{r_\text{adj} \left(1 - r_\text{adj}^2\right)}{2n}.</math>|{{EquationRef|2}}}}

An approximate solution{{citation needed|date=April 2012}} to equation ({{EquationNote|2}}) is

{{NumBlk|:|<math> r_\text{adj} \approx r \left[1 + \frac{1 - r^2}{2n}\right],</math>|{{EquationRef|3}}}}

where in ({{EquationNote|3}})
*<math>r, n</math> are defined as above,
*{{math|''r''<sub>adj</sub>}} is a suboptimal estimator,{{citation needed|date=April 2012}}{{clarify|date=February 2015| reason=suboptimal in what sense?}}
*{{math|''r''<sub>adj</sub>}} can also be obtained by maximizing log(''f''(''r'')),
*{{math|''r''<sub>adj</sub>}} has minimum variance for large values of {{mvar|n}},
*{{math|''r''<sub>adj</sub>}} has a bias of order {{math|{{frac|1|(''n'' − 1)}}}}.

Another proposed<ref name="RealCorBasic"/> adjusted correlation coefficient is{{citation needed|date=February 2015|reason=is this in a published article?}}

:<math>r_\text{adj}=\sqrt{1-\frac{(1-r^2)(n-1)}{(n-2)}}.</math>

{{math|''r''<sub>adj</sub> ≈ ''r''}} for large values of&nbsp;{{mvar|n}}.