Regression to the Mean
Why extreme scores tend to move toward the average on retest — a consequence of imperfect correlation between the two occasions, not real change
Students take the same test twice. The test-retest reliability slider controls how much random error affects scores. At low reliability, extreme Time-1 scorers regress strongly toward the mean at Time-2 — not because they changed, but because their extreme score partly reflected luck. Highlight the top or bottom performers to see the regression effect in action.
Test-retest r: —
Top group: T1 mean = —, T2 mean = —
Bot group: T1 mean = —, T2 mean = —
Expected regression: —
The diagonal dashed line is the identity (T1 = T2). The solid line is the OLS regression of T2 on T1. Because both scores are in SD units, its slope estimates the test-retest correlation ρ — but only estimates it. The value printed on the plot scatters around the slider setting: at ρ = 0.20 with N = 120, a sample slope of 0.10 is entirely ordinary.
When ρ < 1, the regression line is less steep than the identity. This means: if you scored 2 SD above the mean at T1, your expected T2 score is only 2ρ SD above the mean.
Real-world consequences: Selecting people because they scored at an extreme guarantees movement toward the mean on retest, and the direction follows the selection. Recruit the lowest scorers and they appear to improve; recruit the highest scorers and they appear to deteriorate — even if the treatment does nothing at all. So regression to the mean can manufacture a treatment effect in a low-scoring sample and hide a real one in a high-scoring sample. Only a control group selected the same way can separate the two. Always include one.
Citation
Persson, B. N. (2026). Regression to the Mean [Interactive visualization]. https://bjorn-persson.github.io/visualizations/regression-to-the-mean/
@misc{Persson2026regressiontothemean,
author = {Björn N. Persson},
year = {2026},
title = {Regression to the Mean},
note = {Interactive visualization},
url = {https://bjorn-persson.github.io/visualizations/regression-to-the-mean/}}