In his 2011 book entitled “Thinking: Fast and Slow,” Daniel Kahneman stated (on page 181):
“The correlation coefficient between two measures, which varies between 0 and 1, is a measure of the relative weight of the factors they share.”
Evidently, Kahneman truly did think that correlation coefficients must land on a continuum that extends from 0 to 1. That’s because he tried to help his readers “appreciate the meaning of the correlation measure” by presenting these 5 examples:
- The correlation between the size of objects measured with precision in English or in metric is 1.
- The correlation between self-reported height and weight among adult American males is .41.
- The correlation between SAT scores and college GPA is approximately .60.
- The correlation between income and educational level in the United States is approximately .40.
- The correlation between family income and the last four digits of their phone number is 0.
Note that each of these examples contains a correlation coefficient with a value somewhere between 0 and 1, inclusive. Not one example of a negative correlation (e.g., –.80) was provided.
There are 2 ways to correct Kahneman’s inaccurate sentence containing the words: “…varies between 0 and 1.” One obvious option is to change 0 to –1. The second option is to not change the 0, but instead to add these 3 words at the beginning: “The square of….”
If we square a correlation coefficient, we produce something called the “coefficient of determination.” (For example, if the correlation between 2 variables, X and Y, is –.50, the coefficient of determination is .25.) If we now change the coefficient of determination into a percentage, we get the percentage of variability in the X variable that is associated with, or explained by, variability in the Y variable. Note that this works just as well with negative correlations as it does with positive correlations.