What regression to the mean is
When a measurement is partly noise, extreme scores tend to be followed by less extreme ones. Nothing has caused the change: an unusually high score usually included some good luck, and the luck does not repeat. See regression to the mean explained.
The formula
Expected retest score = mean + r × (score − mean), where r is the correlation between the two measurements. The weaker the correlation, the further the expected score falls back towards the mean.
Worked example
A test has mean 100 and SD 15, and r = 0.5 between two sittings. Someone scoring 130 is two SDs above the mean, but is expected to score 115 next time, only one SD above. Half the gap disappears with no change in ability. Likewise, someone scoring 70 is expected to rise to 85. The 95% range uses a residual SD of 15 × √(1 − r²) = 13.0 and assumes the scores are jointly normal.
Why it matters
Selecting people because of extreme scores, such as the weakest students or the patients with the highest blood pressure, and measuring them again guarantees apparent change. A tutoring scheme or a new drug can look effective when nothing has happened. A randomised control group solves the problem, because it regresses by the same amount. To analyse a before-and-after design, use the t-test calculator; to estimate r from your data, use the correlation calculator.
Frequently asked questions
What is the formula for regression to the mean?
Expected second score = mean + r x (first score - mean), where r is the correlation between the two measurements.
Does regression to the mean mean everyone becomes average?
No. The overall spread of scores stays the same. Different people take the extreme positions next time, while those who were extreme tend to move back.
How do I avoid being fooled by it?
Use a randomised control group. Both groups regress equally, so any extra change in the treated group can be attributed to the treatment.
Where do I get the correlation r?
From a test-retest reliability study or from your own repeated measurements. Published tests often report their reliability.