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Coefficient of Determination/r^2

Coefficient of Determination/r^2

Author: Katherine Williams

Relate the correlation coefficient to the coefficient of determination in a given situation.

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This tutorial talks about the coefficient of determination. The coefficient of determination is also called r squared. And it's used with a lower case r and the exponent 2, so r squared.

Now, the coefficient of determination tells us something very specific. It tells us how much of the variation in the response variable is due to the variation in the explanatory variable. So for example, if r squared was 0.75, then we could say that 75% of the variation in the response variable, how y is changing, is due to the variation in the explanatory variable, how x is changing.

One key thing to remember about the coefficient of determination is if this 75% of the variation in the response is due to the variation in the explanatory, then what's the deal with that other 25%? And actually, that other 25% can be due to a couple of different things. It can come from the existence of other variables. Or it could just be the statistical randomness of measurements. In any case, that remaining 25% that we have not accounted for with our response variable can be accounted for by one of those two things-- by other variables or statistical randomness of measurement.

Sometimes your either statistical software or calculator or any kind of information sheet might only give you r squared. It might only give you the coefficient determination. But if you want r, you can easily get r. So you take the square root of r squared.

Now, when you take the square root, no matter what, you're going to get a positive number. So in order to decide what the sign for r should be, you have to look at the graph. So in this example here, we have a scatterplot And we have our r squared that's giving us our coefficient of determination.

So it said r squared is 0.04054. So we can then say that 40.54% of the variation in glucose level can be attributed to the variation in h. But if we want r, we need to take the square root of 0.04054. When we do that, we get 0.6367. So that is our r.

Now, in order to determine what the sign is, I need to look at the data points. Do they have a positive correlation? Or do they have a negative correlation? And in fact, they have a positive correlation, so a plus sign. This has been your tutorial on the coefficient of determination r squared.

Terms to Know
Coefficient of Determination (r2)

A value that explains the percent of variation in the response variable that can be explained by a linear association with the explanatory variable. It is the square of the correlation coefficient.