Standard deviation line

In statistics, the standard deviation line (or SD line) marks points on a scatter plot that are an equal number of standard deviations away from the average in each dimension. For example, in a 2-dimensional scatter diagram with variables ${\displaystyle x}$ and ${\displaystyle y}$, points that are 1 standard deviation away from the mean of ${\displaystyle x}$ and also 1 standard deviation away from the mean of ${\displaystyle y}$ are on the SD line.^[1] The SD line is a useful visual tool since points in a scatter diagram tend to cluster around it,^[1] more or less tightly depending on their correlation.

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.