Why does SAT score correlate weakly with college GPA when measured only among admitted students?

## The setup A university admits students partly on a test score, then asks whether the score predicts first-year grade point average. The correlation computed among enrolled students comes back modest — often far lower than expected — and someone concludes the test is close to worthless. The conclusion does not follow, because of how the sample was built. ## Why a narrow predictor range shrinks r Start from the decomposition. If the relationship is `y = a + b * x + e`, with `e` the part of y that x does not explain, then in a sample `r^2 = (b^2 * var(x)) / (b^2 * var(x) + var(e))` Read that expression as a competition between two quantities: how much the predictor moves things around, versus how much everything else does. Selection on the score truncates `var(x)` — the admitted band might span a fraction of the applicant range — while `var(e)`, the influence of motivation, health, course choice, teaching quality and luck, is essentially untouched by admission. The numerator falls, the denominator falls by less, and `r` drops. Nothing about the underlying relationship changed; only the sample's composition did. This is why range restriction is a *sample* property rather than a property of the relationship. Two analysts can compute honest correlations from the same underlying process and report very different numbers because their samples were selected differently. ## The slope survives, the correlation does not The most useful thing to say in an interview is the distinction between the two. Under selection on the predictor alone — everyone above a score cutoff is admitted, nobody below — the conditional relationship of y given x within the retained band is the same relationship it always was, so the regression slope `b` is essentially preserved. Correlation is `b * sd(x) / sd(y)`: it carries `sd(x)` inside it, so it moves with the sample's spread. That is the whole asymmetry. A slope is expressed in real units (GPA points per score point) and answers 'how much does the outcome change'; a correlation is unitless and answers 'how much of this sample's variation is accounted for'. Restriction of range attacks the second question, not the first. The caveat worth stating: this clean result holds for selection on the predictor. If selection depends on the outcome too, or on unmeasured things related to the outcome, more can go wrong than range restriction, and the slope is no longer safe. ## Corrections and their price There are classical corrections that take the restricted correlation and the ratio of restricted to unrestricted predictor spread and estimate what the correlation would have been in the full population. They are standard in personnel-selection work. They require knowing the unrestricted variance of the selection variable — usually available for a test score, since applicants who were rejected still took the test — and they assume the relationship and the residual spread are the same outside the observed band, which is an extrapolation nobody can check. Use them to argue that a modest observed correlation is consistent with a useful predictor, not to manufacture a precise number. ## The same trap, other clothes Restriction of range appears whenever the analysis population is filtered on something related to the predictor: - A hiring test evaluated only on people who were hired because they scored well on it. - A credit model scored only on applicants who were approved, so the risky end of the score distribution has no outcomes at all. - An engagement metric correlated with retention among users who already stayed thirty days. - Salary versus experience within a single job level, where the level itself was assigned by experience. In each case, the correlation is computed in a window carved out by the very variable under study. ## The mirror image The opposite error inflates rather than deflates. Pool two groups that differ in both variables — two schools, two product tiers, two time periods — and the combined scatter stretches along a line joining the group centres, which pushes the correlation up. Neither within-group relationship has to be strong for the pooled number to look impressive. So the discipline runs both ways: before quoting a correlation, ask what defined the sample, and whether that definition narrowed or widened the spread of the variable being credited. ## What to say in an interview Name the mechanism (selection truncated the predictor's variance), state the consequence in the right direction (r understates, slope roughly survives), give the remedy (report the slope in units, use unrestricted variance where available, be explicit that the number describes the selected sample), and note the mirror-image inflation. That sequence shows the judgment rather than the vocabulary.