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I'm not sure if I'm asking this correctly, but I'll give it a shot:

If I have two independent variables (A and B), both varying from 1 to 5, and have the same correlation with a dependent variable C, but A has a greater standard deviation than B. Would I expect A or B to have a larger coefficient if I do two regression analyses: one for C as a function of A, and a second for C as a function of B?

Basically, if all else is the same, would a different standard deviation change how much of an impact a coefficient would have on an outcome?

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  • $\begingroup$ If indeed all else is the same, you are asking what happens to the coefficient of $A$ when you rescale $A.$ The answer follows upon considering that the units of measurement of its coefficient is the units of the response divided by the units of $A.$ $\endgroup$
    – whuber
    Commented Mar 9, 2022 at 18:38

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In a bivariate analysis, the correlation coefficient is related to the slope by:

$$ \beta = \rho \frac{\sigma_y}{\sigma_x} $$

In other words, if I have a random, bivariate sample $(X_1, Y_1), \ldots, (X_n, Y_n)$, and the correlation between $X$ and $Y$ is known to be a fixed value $\rho$, then two cases where $\sigma_x^{(1)} > \sigma_x^{(2)}$ we expect the slope to be smaller when the variability of the $X$ is greater.

You can visualize this as a point-cloud relating $X$ to $Y$ with a line of best fit. If the plot is "stretched" horizontally, the slope of the line of best fit is attenuated toward 0.

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