# Standard deviation impact on regression coefficients

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?

• 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 $A.$
– whuber
Commented Mar 9, 2022 at 18:38

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