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Given that that correlation coefficient can only be between -1 and 1, it can never increase the steepness of the slope when calculating the steepness of a slope in a regression line.

Say correlation coefficient r = 0.946, standard deviation along the x-axis Sx = 0.816, and standard deviation along y-axis is Yx = 2.160 (values same as in reference Khan Academy video to follow along with example) then slope m = r(Sx/Sy). Here Sx/Sy will be the slope if the correlation coefficient is 1 and all the data plots match perfectly to the line. When we multiply it by r=0.946 however, the steepness of the slope decreases. My question is, how do we know that the only possibility here is where the slope decreases and not increases? The correlation coefficient only indicates how accurate the data plot is compared with the regression line, it says nothing about in which direction (more steep or less steep) the inaccuracy points to. So why are we multiply r with Sx/Sy when we know a r with a absolute value of less than 1 can only decrease the steepness of the slope but never increase it?

Edit: This is how Sal Khan explains the concept in his lesson video, that the slope decreases after multiplying by the correlation coefficient.

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  • $\begingroup$ Trying to self-study statistics. Sorry guys if my question is a bit confusing. $\endgroup$
    – Simon Suh
    May 29, 2023 at 11:28
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    $\begingroup$ There is a story to tell here about regression to the mean... $\endgroup$
    – BigBendRegion
    May 30, 2023 at 10:29
  • $\begingroup$ If you can provide another answer, that would be great! I did checkmark the answer but I still don't completely understand this. $\endgroup$
    – Simon Suh
    May 30, 2023 at 22:57
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    $\begingroup$ See p. 36-40 of books.google.com/books/about/… $\endgroup$
    – BigBendRegion
    May 31, 2023 at 14:35
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    $\begingroup$ the example explaining the correlation between the heights of fathers and sons was a great example from that textbook link you gave me. So basically, when r correlation coefficient is low, it means there is a less of a correlation between the regression line and the sample data, so a better prediction would be the mean, therefore the slope of the regression line will be closer to 0 (flat line) the lower the r value is. $\endgroup$
    – Simon Suh
    Jun 6, 2023 at 12:50

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In the formula for the unstandardized bivariate linear regression slope coefficient

b = r*(SD_Y/SD_X)

it may help to think of the standard deviations SD_Y and SD_X as mere scaling parameters (they "adjust" r for the different--and often arbitrary--units of measurement of Y and X). If you consider the standardized regression weight b_S (where SD_Y and SD_X = 1 due to standardization and thus the regression weight does not depend on the scaling/units of measurement of Y and X), the correlation r does reflect the steepness of the regression line directly:

b_S = r

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