Coefficient of Determination

ScatterplotThe coefficient of determination is often referred to as R^2 . The coefficient of determination, simply put, is the measure of how well a regression models a data set. If you have a data set that has an R^2 value of 0.95, then that means the regression explains 95% of the variation of the data, which is excellent. If you have a different data set that gives you an R^2 value of 0.5, then only 50% of the variation is explained by the regression, and that is not ideal.

There are three main ways of calculating R^2 :

  1. Calculate the square of the correlation coefficient (r). Since -1 \leq r \leq 1 then 0 \leq R^2 \leq 1.
  2. Calculate the ratio of the regression sum of squares and the total sum of squares: R^2=\frac{\sum (\hat{y}-\bar{y})^2}{\sum (y-\bar{y})^2}.
  3. The total variation will always be 1 (or 100%), thus if you are given the unexplained variation you can calculate R^2 from the equation R^2=1-\frac{\sum(\hat{y}-y)^2}{\sum(y-\bar{y})^2}. Where 1 is the total variation and the ratio is the residual sum of squares of the total sum of squares which is the unexplained variation, also known as the coefficient of non-determination.

For a very basic definition, visit this web site for more information on R^2. This web site gives a very good, easy to understand definition and then goes more in depth than the first site; this is the best site for those new to statistics but want to develop a relatively deep understanding. Of course the Wikipedia page is usually a first stop in learning a new topic; however, I found the article to be a bit confusing for first exposure to this idea, though it is very expansive and covers much on the subject.


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