The relationship between the correlation coefficient rrr and the slope bbb of the least-squares regression line for the same data set can be summarized as follows:
- The slope bbb of the least-squares regression line is directly related to the correlation coefficient rrr through the formula:
b=r×sysxb=r\times \frac{s_y}{s_x}b=r×sxsy
where sys_ysy and sxs_xsx are the standard deviations of the yyy- and xxx- variables, respectively
- This means the slope bbb reflects both the strength and direction of the linear relationship (captured by rrr) and the relative scales of the variables (captured by the ratio of standard deviations).
- The sign of the slope bbb is always the same as the sign of the correlation coefficient rrr. A positive correlation corresponds to a positive slope, and a negative correlation corresponds to a negative slope
- While the correlation coefficient rrr measures the strength and direction of the linear association and is bounded between -1 and 1, the slope bbb measures the rate of change of yyy with respect to xxx and can take any real value, reflecting how steeply yyy changes as xxx changes
- If both variables are standardized (mean zero, standard deviation one), the slope bbb equals the correlation coefficient rrr
In essence, the slope bbb can be viewed as a scaled version of the correlation coefficient rrr, adjusted by the variability in the variables. Correlation gives a dimensionless measure of association strength and direction, while slope gives the actual rate of change in the units of the data.
Summary: The slope bbb of the least-squares line equals the correlation coefficient rrr multiplied by the ratio of the standard deviations of yyy and xxx, and both share the same sign indicating the direction of the linear relationship
