Editing Pearson correlation coefficient (section)

===Testing using Student's ''t''-distribution===
[[File:Critical correlation vs. sample size.svg|thumb|324x324px|Critical values of Pearson's correlation coefficient that must be exceeded to be considered significantly nonzero at the 0.05 level]]For pairs from an uncorrelated [[bivariate normal distribution]], the [[sampling distribution]] of the [[studentized]] Pearson's correlation coefficient follows [[Student's t-distribution|Student's ''t''-distribution]] with degrees of freedom ''n''&nbsp;−&nbsp;2.  Specifically, if the underlying variables have a bivariate normal distribution, the variable

:<math>t = \frac{r}{\sigma_r} = r\sqrt{\frac{n-2}{1 - r^2}}</math>

has a student's ''t''-distribution in the null case (zero correlation).<ref>Rahman, N. A. (1968) ''A Course in Theoretical Statistics'', Charles Griffin and Company, 1968</ref> This holds approximately in case of non-normal observed values if sample sizes are large enough.<ref>Kendall, M. G., Stuart, A. (1973) ''The Advanced Theory of Statistics, Volume 2: Inference and Relationship'', Griffin. {{isbn|0-85264-215-6}} (Section 31.19)</ref> For determining the critical values for ''r'' the inverse function is needed:

:<math>r = \frac{t}{\sqrt{n - 2 + t^2}}.</math>

Alternatively, large sample, asymptotic approaches can be used.

Another early paper<ref>{{cite journal |last1=Soper |first1=H.E. |author-link=H. E. Soper |last2=Young |first2=A.W. |last3=Cave |first3=B.M. |last4=Lee |first4=A. |last5=Pearson |first5=K. |year=1917 |title=On the distribution of the correlation coefficient in small samples. Appendix II to the papers of "Student" and R.A. Fisher. A co-operative study |url=https://zenodo.org/record/1431587 |journal=[[Biometrika]] |volume=11 |issue=4 |pages=328–413 |doi=10.1093/biomet/11.4.328}}</ref> provides graphs and tables for general values of ''ρ'', for small sample sizes, and discusses computational approaches.

In the case where the underlying variables are not normal, the sampling distribution of Pearson's correlation coefficient follows a Student's ''t''-distribution, but the degrees of freedom are reduced.<ref>{{cite journal |last1=Davey |first1=Catherine E. |last2=Grayden |first2=David B. |last3=Egan |first3=Gary F. |last4=Johnston |first4=Leigh A. |title=Filtering induces correlation in fMRI resting state data |journal=NeuroImage |date=January 2013 |volume=64 |pages=728–740 |doi=10.1016/j.neuroimage.2012.08.022 |pmid=22939874 |hdl=11343/44035 |s2cid=207184701 |hdl-access=free }}</ref>