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I think I've found the original paper from Pearson on correlation coefficients (equation on p 279, also cited here), but I'm not sure it's the right one. Pearson mentions normality multiple times in the paper, but I'm not sure it actually applies to the correlation equation.

Two questions -- is this (the first paper, rsta) the right paper (as in the original paper for Pearson's correlation coefficient), and what are the original assumptions in the formulation of the correlation coefficient? I've seen things ranging from "normality and homoscedasticity" to only "variables are interval".

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    $\begingroup$ Pearson fixated on the correlation named for him as a parameter of a bivariate normal distribution, but few have followed him all the way. Suppose $x$ is uniformly distributed and $y = a + bx$ exactly: here $b > 0$. Is the corresponding $r = 1$ for correlation of $(x, y)$ invalid because neither marginal distributions nor joint distributions show normality? $r$ can be directly interpreted as measuring linearity; inference is for just about 40 years possible with the bootstrap, so (fascinating history of ideas aside) there is no need to adopt Pearson's emphasis. $\endgroup$
    – Nick Cox
    Commented May 28, 2018 at 6:45

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