A key output of the regression analysis is the coefficient determination (indicated by R2). This is interpreted as the proportions of the variance in the dependent variable which can be predicted by the independent variable.

This coefficient of determination results from the square of the correlation (r) between the expected and actual y scores; thus, it ranges from 0 to 1.

With the linear regression, the coefficient of determination is also equal to the square of the correlation between the x and y scores.

An R2 of 0 indicates that the dependent variable is not predictable by the independent variable.

An R2 of 1 indicates that the dependent variable may be predicted without error by the independent variable.

An R2 of 0 to 1 indicates the extent to which the dependent variable is predictable. An R2 of 0.10 means that 10% of the variance in Y is predictable from X; an R2 of 0.20 means that 20% is predictable; and so on.

A formula for calculating the coefficient of determination of a linear regression model using an independent variable is given below.

Determination coefficient. The coefficient of determination (R2) for a linear regression model with an independent variable is:

R2 = { ( 1 / N ) * Σ [ (xi – x) * (yi – y) ] / (σx * σy ) }2

where N is the number of observations used to fit the model, Σ is the sum symbol, xi is the x value for observation i, x is the mean value x, yi is the y value for observation i, y is the mean value y, σx is the standard deviation of x, and σy is the standard deviation of y.