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I am a bit confused about interpreting simple linear regression coefficients. My understanding is that for a linear regression equation, increasing x by one unit corresponds to a change in the mean of y.

What I don’t understand is what is exactly meant by “mean of y.” It can’t be the arithmetic mean of the entire y dataset. Same for the x, I’ve heard the regression line passes through the mean of x and mean of y. How is this the case when different values of x will yield larger values of y (assuming positive slope)? How can it be a static number, or is it?

I am trying to get a better grasp of understanding what a one unit change in x corresponding to a k unit change in y really means because it has lost me at this point.

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Your problems of interpretation comes from some sloppy formulations! When you say

mean of y

it has different meanings in different places! Sometimes it is the marginal mean of y, that is, its mean (or in the model, expectation) over all its values, not taking into consideration (that is, conditioning upon) x. But othertimes it is the conditional mean, given the corresponding value of x.

Hopefully this is enough of a hint that you can untangle the mistery yourself now.

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    $\begingroup$ I just think it’s a bit weird to call it the “conditional mean” considering a situation where we have, say, only 3 coordinate pairs. How would the conditional mean of y given x be interpreted then? $\endgroup$
    – user3138766
    Jun 9, 2021 at 17:03

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