Regression to the Mean

Dr. R. Düsing · Osnabrück University

Help — Regression to the Mean

What does this tool show?

People who score extremely high or extremely low on a first measurement tend, on average, to land closer to the mean on a second measurement — even with no intervention at all. The tool simulates this effect using a fictitious blood-pressure study.

The running example

A GP practice measures systolic blood pressure in n patients (baseline T1). Everyone with blood pressure ≥ the threshold is enrolled in a hypertension program and measured again four weeks later (T2) — with no medication. The question: where does the improvement come from?

The math

For bivariate normally distributed measurement occasions X (T1) and Y (T2) with correlation r:

E[Y | X = x] = μ + r · (x − μ)

The regression effect equals (1 − r) · (x − μ) — proportional to how extreme the score is and to (1−r). At r = 1 (perfect measurement) there is no effect. The larger the measurement uncertainty, the stronger the regression to the mean.

Why does this matter?

Regression to the mean can masquerade as a genuine treatment effect. In pre-post studies without a control group, it's unclear whether improvements stem from the intervention or from this statistical artifact. Randomized control groups are the only remedy — they show the same RTM effect, so the difference between groups estimates the adjusted intervention effect.

Controls

Scenarios A–D: predefined examples for comparison. r: reliability / correlation between T1 and T2 — the central parameter. Threshold: the value above which people are selected. Red points: high-selected (≥ threshold), blue points: low-selected (mirror-image). Orange line: E[Y|X] — the theoretical regression line. Dashed: Y = X (no change).

References

Galton, F. (1886). Regression towards mediocrity in hereditary stature. Journal of the Anthropological Institute, 15, 246–263.

📋 Example — Blood-pressure study
A GP practice measures systolic blood pressure in 400 patients (baseline T1, μ = 130, σ = 20 mmHg). Every patient with blood pressure ≥ 160 mmHg is enrolled in a hypertension program and measured again four weeks later (T2) — with no medication. On average, the group improves by — mmHg. Real effect, or statistical artifact?
Basic principle
E[Y | X = x] = μ + r · (x − μ) (regression line)
Regression effect = (1 − r ) · (x − μ) (regression to the mean)
r = 0.70  ·  threshold = 160 mmHg  ·  exp. RTM at threshold: −9.0 mmHg
Scatterplot T1 vs. T2
Selected group
— people
X ≥ — mmHg
Mean T1 (selected)
—
Baseline, mmHg
Observed regression
—
T2 − T1 (mmHg)
Expected from r
—
(r−1)·(T1̄−μ), mmHg
Conclusion: The selected group improves on average by — mmHg — with no treatment at all. This regression effect is exactly predictable from r: — mmHg expected. Without a control group, this artifact would falsely be attributed to the intervention. Also note the blue group (low T1 values): it shows regression upward — the same effect, opposite direction.
Concepts
What is regression to the mean?
Anyone who scores extremely high or low on one measurement tends to land closer to the mean on the next — even if nothing has actually changed. Cause: an extreme result contains a large random component that "disappears" at the next measurement occasion. The less reliable the measurement, the stronger the effect.
Galton's discovery (1886)
Francis Galton observed that children of very tall parents were taller than average — but on average shorter than their parents. Children of very short parents were shorter than average, but taller than their parents. He called it "regression toward mediocrity." The term "regression line" is a direct legacy of this observation.
Reliability r as the key factor
The regression effect equals (1 − r) · (x − μ). At r = 1.0: no effect. At r = 0.7: 30% of the distance to the mean vanishes. At r = 0.4: 60% vanishes. Reliability is the only controllable factor — more precise measurement reduces the RTM effect.
Control group as the solution
The only way to separate a real intervention effect from RTM is a randomized control group. Both groups show the same RTM effect. The difference between them estimates the adjusted intervention effect. Without a control group, a pre-post result is simply not interpretable.
Bidirectionality
High scorers regress downward, low scorers upward. The "sophomore slump" in sports (last season's best player does worse the following year), spontaneous recovery from extreme states, the apparent effect of remedial programs for weak performers — all of these contain an RTM component.
Clinical relevance
Patients with very high blood pressure, severe pain, or poor well-being are often at a temporary "high point" at study enrollment. Even without treatment, many improve — because the next measurement occasion is almost bound to be less extreme. Pre-post studies without a control group almost inevitably produce positive but artifactual results.