34.3 R-squared (R2)
While using r tells us about the strength and direction of the linear relationship, knowing exactly what the value means is tricky. Interpretation is easier using R2, or ‘R-squared’: the square of the value of r.
The animation below shows some values of R2.
The value of R2 is never negative. However, you need to be careful when using your calculator!
With most calculators,
if you enter -0.5^2
, you will get -0.25
in return.
This is correct, because the calculator interprets your input as meaning
-(0.25^2)
.
(-0,5)^2
.
This will give you the expected answer of 0.25
.
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The value of R2 is the percentage reduction in the unknown variation of y because the value of x is known. In other words, it is the percentage of the variation in y explained by using the linear relationship, rather than just the mean value of y.
Example 34.4 (Values of R2) For the red deer data (Fig. 33.2), the value of R2 from the software output (Fig. 34.5; Fig. 34.6) is R2=(−0.584)2=0.341, usually written as a percentage: 34.1%.
The value of R2 is positive, even though the value of r is negative.For the red deer data, R2 means that about 34.1% of the variation in molar weights can be explained by variation in the age of the deer. The rest of the variation in molar weights is due to extraneous variables, such as weight, diet, amount of exercise, genetics, etc.
Think 34.3 (Interpreting R2) From Example 34.3, the correlation coefficient between the systolic blood pressure and age in the NHANES data is r=0.532.
What is the value of R2? What does it mean?