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The coefficient of determination (denoted by
R^{2}) is a key output of
regression
analysis.
It is interpreted as the proportion of the variance in the
dependent variable that is predictable from the independent variable.

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

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

An R^{2} of 0 means that the dependent variable cannot be
predicted from the independent variable.

An R^{2} of 1 means the dependent variable can be
predicted without error from the independent variable.

An R^{2} between 0 and 1 indicates the extent to which
the dependent variable is predictable. An R^{2} of
0.10 means that 10 percent of the variance in Y is
predictable from X; an R^{2} of 0.20 means
that 20 percent is predictable; and so on.

The formula for computing the coefficient of determination for a
linear regression model with one independent variable is given below.

Coefficient of determination.
The coefficient of determination (R^{2}) for a linear regression model with
one independent variable is:

where N is the number of
observations used to fit the model, Σ is the summation symbol,
x_{i} is the x value for observation i,
x is the mean x value,
y_{i} is the y value for observation i,
y is the mean y value,
σ_{x} is the standard deviation of x, and
σ_{y} is the standard deviation of y.