Standard deviation line

Plot of the standard deviation line (SD line), dashed, and the regression line, solid, for a scatter diagram of 20 points.

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.

Properties

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.