Pearson Correlation (r, R^2, significance)
Pearson r and R^2 for paired x / y series, with the t-test (t = r sqrt(n-2) / sqrt(1-r^2), n-2 df) and the two-tailed p-value from the Student-t CDF. Per OpenIntro Statistics Ch. 8.
Formula and source
Pearson r = sum((x - xbar)(y - ybar)) / sqrt(sum(x - xbar)^2 * sum(y - ybar)^2). R^2 = r^2. Significance test for rho = 0: t = r * sqrt(n - 2) / sqrt(1 - r^2) on n - 2 degrees of freedom; two-tailed p = 2 * (1 - tcdf(|t|, n - 2)).
OpenIntro Statistics 4th ed. Chapter 8 (introduction to linear regression) by name; the Student-t CDF via the regularized incomplete beta function per Numerical Recipes in C 2nd ed. ยง6.4.
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