Standard deviation line

The SD line goes through the point of averages and has a slope of ${\displaystyle {\frac {\sigma _{y}}{\sigma _{x}}}}$ when the correlation between ${\displaystyle x}$ and ${\displaystyle y}$ is positive, and ${\displaystyle -{\frac {\sigma _{y}}{\sigma _{x}}}}$ when the correlation is negative.[1][2] Unlike the regression line, the SD line does not take into account the relationship between ${\displaystyle x}$ and ${\displaystyle y}$.[3] The slope of the SD line is related to that of the regression line by ${\displaystyle a=r{\frac {\sigma _{y}}{\sigma _{x}}}}$ where ${\displaystyle a}$ is the slope of the regression line, ${\displaystyle r}$ is the correlation coefficient, and ${\displaystyle {\frac {\sigma _{y}}{\sigma _{x}}}}$ is the magnitude of the slope of the SD line.[2]

The root mean square vertical distance of points from the SD line is ${\displaystyle {\sqrt {2(1-|r|)}}\times \sigma _{y}}$.[1] This gives an idea of the spread of points around the SD line.