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Pearson's correlation is among the mostly widely reported measures ofassociation. The strength of the statistical evidence for linear association isdetermined by the p-value of a hypothesis test. If the true distribution of adataset is bivariate normal, then under specific data transformations at-statistic returns the exact p-value, otherwise it is an approximation.Alternatively, the p-value can be estimated by analyzing the distribution ofthe sample correlation under permutations of the data. Moment approximations ofthis distribution are not as widely used since estimation of the momentsthemselves are numerically intensive with greater uncertainties. In this paperwe derive an inductive formula allowing for the analytic expression of thesample moments of the sample correlation under permutations of the data interms of the central moments of the data. These formulas placed in a properstatistical framework could open up the possibility of new estimation methodsfor computing the p-value.

Derivation of Analytic Formulas for the Sample Moments of the Sample Correlation over Permutations of Data