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.

  • 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 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.
  • An R2 of 0 means that the dependent variable cannot be predicted from the independent variable.
  • An R2 of 1 means the dependent variable can be predicted without error from the independent variable.
  • 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 (R2) 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, xi is the x value for observation i, x is the mean x value, yi is the y value for observation i, y is the mean y value, σx is the standard deviation of x, and σy is the standard deviation of y.


See also:   AP Statistics Tutorial: Least Squares Linear Regression | AP Statistics Tutorial: A Simple Regression Example