statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, the Standard deviation line (or SD line) marks points on a

scatter diagram A scatter plot (also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram) is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. ...

that are an equal number of

standard deviations In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...

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 In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...

Properties

Relation to regression line

The SD line goes through the point of averages and has a slope of

\frac

when the

between

x

and

y

is positive, and

-\frac

when the correlation is negative. Unlike the

regression line In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is call ...

, 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

where

a

is the slope of the regression line,

r

is the

correlation coefficient A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...

, and

\frac

is the magnitude of the slope of the SD line.

Typical distance of points to SD line

The

root mean square In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...

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. Statistics Mathematics