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.^[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.

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.^[1]^[2] Unlike the regression line, the SD line does not take into account the relationship between $x$ and $y$ .^[3] 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.^[2]

Typical distance of points to SD line

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

1 2 3 4 Freedman, David (1998). Statistics. Robert Pisani, Roger Purves (3rd ed.). New York: W.W. Norton. ISBN 0-393-97083-3. OCLC 36922529.
1 2 Stark. "Regression". www.stat.berkeley.edu. Retrieved 2022-11-12.
↑ Cochran. "Regression". www.stat.ucla.edu. Retrieved 2022-11-12.

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[:0-1] 1 2 3 4 Freedman, David (1998). Statistics. Robert Pisani, Roger Purves (3rd ed.). New York: W.W. Norton. ISBN 0-393-97083-3. OCLC 36922529.

[:1-2] 1 2 Stark. "Regression". www.stat.berkeley.edu. Retrieved 2022-11-12.

[3] Cochran. "Regression". www.stat.ucla.edu. Retrieved 2022-11-12.

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