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gedymin comments on The "best" mathematically-informed topics? - Less Wrong Discussion
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Example of a mathematical fact: a formula for calculating correlation coefficient. Example of a statistical intuition: knowing when to conclude that close-to-zero correlation implies independence. (To see the problem, see this picture for some datasets in which variables are uncorrelated, but not independent.)
Not sure why are you calling this "intuition". Understanding that Pearson correlation attempts to measure a linear relationship and many relationships are not linear is just statistical knowledge, only a bit higher level than knowing the formula.