What is the relationship between the standard deviation of y and x when calculating the slope (m) in a regression line?

The relationship between the standard deviation of y and x when calculating the slope of a regression line is described by the formula for the slope (m) in simple linear regression. The slope can be expressed as the product of the correlation coefficient (r) and the ratio of the standard deviations of y and x. This encapsulates how changes in the independent variable (x) influence changes in the dependent variable (y).

In simple linear regression, the slope of the line quantifies the expected change in y for a one-unit increase in x. The correlation coefficient (r) measures the strength and direction of the linear relationship between the two variables. By multiplying r with the quotient of the standard deviation of y (the variability of the dependent variable) and the standard deviation of x (the variability of the independent variable), you arrive at the slope.

Thus, the correct formula captures the intrinsic relationship between the variability of both variables and the strength of their correlation, indicating how steep the regression line will be. This relationship reflects that when the variability of y is high relative to x, the slope will also be greater (if the correlation is positive), showing a stronger association between the two variables.