Friday, June 04, 2004

Galton's Fallacy

Here's a post to file in the "must carefully re-read later" category.

Imagine you want to conduct a study on the effectiveness of punishment and reward on flight training. You praise trainees for good landings and reprimand them for near crashes, and observe whether this has any effects.

After spending some time on the airfield your data indicate trainees did worse on their next flight if they had just been praised, and fly better just after being yelled at.

Then you have a conversation with the flight instructor who does the written examinations. He tells you that experience shows students who do well on midterms slack off on their final exam, while students who do badly generally get better.

What does this mean? Do students’ test scores tend to revert to some mean?

If you answered yes, you’ve just committed the famous “Galton’s fallacy”.

What did the 19th century statistician and geneticist Sir Francis Galton, Charles Darwin’s cousin, do wrong?

Well, he plotted the height of fathers against the height of their sons and discovered sons of tall fathers tended to be tall, but on average not as tall as their fathers. Similarly, sons of short fathers tended to be short, but on average not as short as their fathers. He—and this was his mistake—was immediately concerned that the sons of tall fathers are regressing into a pool of mediocrity along with the sons of everybody else.

So what’s wrong with that? Well, height tends to be normally distributed—i.e., the distribution takes a bell-shaped form—and sons’ heights are, due to heredity, correlated with the father’s height. So there is a linear dependence between fathers’ and the sons’ heights.

The link to the paper by Danny Quah is also well worth archiving.

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