Regression Toward the Mean (3 of 6)
There are just not many people who can afford to be unlucky and still
score as high as 750. A person scoring 750 was, more likely than not,
luckier than average. Since, by definition, luck does not hold from one
administration of the test to another, a person scoring 750 on one test
is expected to score below 750 on a second test. This does not mean that
they necessarily will score less than 750, just that it is likely. The
same logic can be applied to someone scoring 250. Since there are more
people with "true" scores between 250 and 300 than between
200 and 250, a person scoring 250 is more likely to have a "true"
score above 250 and be unlucky than a "true" score below 250
and be lucky. This means that a person scoring 250 would be expected
to score higher on the second test. For both the person scoring 750
and the person scoring 250, their expected score on the second test
is between the score they received on the first test and the mean.
This
is the phenomenon called "regression toward the mean." Regression
toward the mean occurs any time people are chosen based on observed
scores that are determined in part or entirely by chance.
On any task that contains both luck and skill, people who score above the mean
are likely to have been luckier than people who score below the mean.
Since luck does not hold from trial to trial, people who score above the mean
can be expected to do worse on a subsequent trial. This counterintuitive phenomenon
is illustrated concretely by a simulation found
here.
In regression with standardized variables, the regression
equation is:
Zy' = (r)Zx
where Zy' is the predicted standardized score, Zx
is the standardized score on the predictor, and r is Pearson's
correlation. This means that
the predicted standardized score will be closer to the mean of zero whenever
the correlation is not perfect (not -1 or 1).
For example, if the SAT had a mean of
500 and a standard deviation of 100, then a score of 750 would have a standard
score equivalent of 2.5 since 750 is two and a half standard deviations above
the mean. If the test-retest correlation were 0.90, then the predicted standard
score for someone with a standard score of 2.5 would be (0.90)(2.5) = 2.25.
Therefore, they would be predicted to be 2.25 standard deviations above the
mean on the retest which is 500 + (2.25)(100) = 725.