Statistics and Probability Dictionary
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Coefficient of Determination
The coefficient of determination (denoted by
r2) 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.
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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.
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With linear regression (the type of regression we are using
in this tutorial), the coefficient of determination is also
equal to the square of the correlation between x and
y scores.
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An r2 of 0 means that the dependent variable cannot be
predicted from the independent variable.
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An r2 of 1 means the dependent variable can be
predicted without error from the independent variable.
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An r2 between 0 and 1 indicates the extent to which
the dependent variable is predictable. An r2 of
0.10 means that 10 percent of the variance in Y is
predictable from X; an r2 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:
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 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.