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 $$x$$ and $$y$$, points that are 1 standard deviation away from the mean of $$x$$ and also 1 standard deviation away from the mean of $$y$$ are on the SD line. The SD line is a useful visual tool since points in a scatter diagram tend to cluster around it, more or less tightly depending on their correlation.

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

Typical distance of points to SD line
The root mean square vertical distance of points from the SD line is $$\sqrt{2(1 - |r|)} \times\sigma_y$$. This gives an idea of the spread of points around the SD line.