The top test score, next time around

Question

A student scored 3 standard deviations above the class average on an exam. Assume a student's scores on two comparable exams (in standardized units) are bivariate normal with test-retest correlation \rho = 0.6. What is the student's expected score on the next exam? Explain the mechanism, and why this is not evidence the student "got worse."

Accepted Answer

The computation. For a standardized bivariate normal pair, the conditional mean is linear in the observed value: \mathbb{E}[Y \mid X = x] = ho x. With x = 3 and ho = 0.6: the expected next score is 0.6 	imes 3 = 1.8 standard deviations above average. Still excellent, but the student gives back 40\% of the outperformance. (The full conditional distribution is \mathcal{N}(ho x,\, 1 - ho^2) = \mathcal{N}(1.8,\, 0.64), so there is real spread around that 1.8.) The mechanism: signal plus noise. Decompose an extreme score as X = 	ext{ability} + 	ext{luck} (a good day, favorable questions…

The top test score, next time around

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