## October 22, 2004

### Inference of between-individual differences from between-group differences

I have often witnessed a particular logical error, namely the inference of between-individual differences based on knowledge about between-group differences.

Imagine for example two groups, A and B, and two numerical traits X and Y.

Group A has an average value of X(A)=10 and an average value of Y(A)=20.

Group B has an average value of X(B)=15 and an average value of Y(B)=30.

You can add more groups, maintaining the property that a higher group value for X is associated with a higher group value for Y. Indeed, you might regress X(.) on Y(.) and find a strong positive correlation between the two: high X(.) value corresponds to high Y(.) value.

The logical error is the following: to infer that this means that individuals with a high X value also tend to have a high Y value.

To see why this is the case, suppose that group A has three individuals a, b, c with values X(a)=5, X(b)=10, X(c)=15 and Y(a)=30, Y(b)=20, Y(c)=10. It is easy to see that these values do in fact correspond to a group average of X(A)=10 and Y(A)=20, but it is obvious that within the group low X values are associated with high Y values and vice versa.

The same situation might also be the case for group B, and it would thus turn out that when it comes to individuals low X values are associated with high Y values, that is the opposite correlation to that of the group averages.

The lesson from this example is the following: whenever you hear that e.g., group A is "greener" than group B and group A is "smarter" than group B, therefore "greener" people are "smarter", you should ask to see the correlation between "green" and "smart" individuals and question the assumption that group-based differences have anything to do with individual difference.

PS: I had meant to write something on the topic for a while, but I was reminded to do so by Steve Sailer's new article on John Kerry's IQ. Steve brings up a recent hoax which he uncovered regarding supposedly "smarter" Democratic-voting versus "dumber" Republic-voting states. Liberals were eager to treat this as evidence that liberals are smarter than conservatives. As the analysis above shows, this would emphatically not be the case, even if those results were correct. But, note also, that the fact that Democratic-leaning states vs. Republican-leaning states tend to have similar average IQs does not prove that Democrats and Republicans tend to have similar IQs, for exactly the same reason. It is quite possible that individual-based comparisons within each state and consequently within the total US population might indicate that political orientation and intelligence are correlated.