The first result contains some luck.
In the experiment, each person has an unchanged underlying ability. Their two scores add fresh, independent noise to that same ability. Selecting the highest first scores also selects people whose first measurement was unusually favorable.
The next measurement usually removes some of that advantage at the group level. Nobody had to become less capable. Selecting on an unusually bad first result creates the opposite tendency.
Inside this model
Ability is normally distributed with mean 50 and standard deviation 10. Both scores add normally distributed noise with mean zero and the standard deviation you choose.
The model’s correlation between repeated scores is 100 / (100 + noise²). With no noise the correlation is 1 and scores match exactly. With noise, the conditional expected second score is 50 + correlation × (first score − 50). Scores are unbounded units.
Why it matters
A change following an unusually good or bad result does not by itself identify a cause. A comparison group can help distinguish a real intervention from selection and noise.
A useful lens. Not a universal law.
- The model has stable ability and independent errors. Learning, fatigue, shared conditions, and real trends can change the pattern.
- Regression is a conditional average tendency, not a force that guarantees the next individual score will be closer to 50.
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
For a broader treatment of regression, measurement, and inference, see the authors’ resources for Regression and Other Stories.
Gelman, Hill & Vehtari — Regression and Other Stories ↗