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I understand statisticians call the predicted y in a simple linear regression the “mean of y.” Let’s assume we only have 3 pairs of x and y values: (1,1), (2,3), (3,1). Whatever the regression line ends up being, for a value of x equal to 4 the regression line would output something representing the “mean of y given x = 4.” In what sense is this predicted y from the regression line at x = 4 the “mean of y?” It must be a guess of the mean of y at x = 4? It can’t be an “actual” mean calculated from our data. Do you see my confusion? I don’t understand why every y value on the regression line is the mean of y given whatever x is chosen.

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In what sense is this predicted y from the regression line at x = 4 the “mean of y?” It must be a guess of the mean of y at x = 4?

Correct. It is an estimate of $E(Y \vert X = x)$.

It can’t be an “actual” mean calculated from our data

Also correct. It is an estimate of the actual mean.

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  • $\begingroup$ Thank you! No one has clearly said this to me when asking this question, so thanks a ton. One more question if you’re willing: I am seeing depictions of “conditional distributions” of y given x for every value of x on a scatter plot with the regression line. Each y value has a normal distribution with mean equal to the predicted value given by the regression line. How can we say the regression outputs at each x represent the mean of y at that value? Isn’t that a reach given we know, in my example, only of three coordinate pairs? $\endgroup$ – user3138766 Jun 9 at 15:35
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    $\begingroup$ @user3138766 The quality of the estimate is something the analyst has to consider. If all you have is 3 data points, then yea I would be dubious of all estimates. $\endgroup$ – Demetri Pananos Jun 9 at 15:37
  • $\begingroup$ Thank you Demetri. $\endgroup$ – user3138766 Jun 9 at 15:43

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