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The covariance (and correlation) are described as the strength of a linear association between variables. Is there a way to see where the 'linear' part comes from in the formula?

Steve
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  • The comments at the end of https://stats.stackexchange.com/a/18200/919 might help, for a start. A completely different set of insights is available by examining some posts showing how [the correlation is a linear regression coefficient](https://stats.stackexchange.com/search?q=correlation+linear+regression). – whuber Mar 11 '21 at 21:24
  • Perhaps, and I am overthinking things, but even from that I guess I am not entirely sure what about the formula makes it linear- perhaps if there is something much more evident from the equation directly that makes it inherently about linear relationships – Steve Mar 11 '21 at 21:32
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    @Steve the covariance is proportional to the OLS regression slope. Neither tell you the relationship is strictly linear. If the trend is curvilinear, the OLS slope is interpretable as the average first-order trend in the relationship: *on average* by how much does the response trend upward for a unit difference in the predictor. That is the most that can be said. Does that answer your question? – AdamO Mar 11 '21 at 21:53
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    Many thanks, @firebug. I remembered such a thread existed but was interrupted before I could search for it. Upon re-reading it, it looks spot on. – whuber Mar 11 '21 at 22:11
  • @whuber what theread? – Charlie Parker Oct 13 '21 at 20:22
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    @Charlie Apologies: the comment to which I was referring has since been deleted. It linked to https://stats.stackexchange.com/questions/381149 . – whuber Oct 13 '21 at 21:01

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