Regression to the mean is the tendency for unusually high or unusually low measurements to be followed by measurements that are closer to the average. It matters because many real measurements contain both a stable signal and random variation. When a result is extreme, part of that extremeness is often due to luck, noise, or temporary conditions.
On a second measurement, that random part is less likely to be extreme in the same direction.
In a scatterplot of first measurement versus second measurement, perfect repeatability would put every point on the diagonal line y = x. Real data usually form a cloud around that line, and the most extreme first measurements tend to have second measurements pulled toward the mean. This effect is not a mysterious force and does not mean individuals are destined to become average.
It is a statistical pattern caused by imperfect correlation between repeated measurements.
Key Facts
- Regression to the mean occurs when extreme observations are followed by less extreme observations on average.
- Perfect repeatability would give y = x for first measurement x and second measurement y.
- If the correlation is less than 1, predicted standardized scores move toward 0: z_y = r z_x.
- The mean is the balancing point of a data set: mean = sum of values / number of values.
- The weaker the correlation r between two measurements, the stronger the regression effect.
- Regression to the mean can make random change look like improvement, decline, punishment effects, or treatment effects.
Vocabulary
- Regression to the mean
- A statistical pattern in which extreme measurements tend to be followed by measurements closer to the average.
- Mean
- The arithmetic average found by adding all values and dividing by the number of values.
- Correlation
- A number from -1 to 1 that describes the strength and direction of a linear relationship between two variables.
- Random variation
- Unpredictable fluctuation in measurements caused by chance, noise, or temporary factors.
- Standardized score
- A value expressed in standard deviation units from the mean, often written as a z-score.
Common Mistakes to Avoid
- Mistaking regression to the mean for a real cause, because an extreme score followed by a more average score can happen even when nothing caused a change.
- Judging a treatment only on people with extreme initial scores, because some improvement is expected naturally from regression to the mean.
- Assuming every individual must move toward the average, because regression to the mean describes group averages, not a guaranteed path for each person.
- Ignoring measurement error and random variation, because extreme results often include temporary noise that is unlikely to repeat exactly.
Practice Questions
- 1 A class has a mean test score of 70 with a standard deviation of 10. A student scores 90 on the first test. If the correlation between first and second test scores is r = 0.6, what is the predicted second-test score using z_y = r z_x?
- 2 A basketball player averages 50% on free throws but makes 9 out of 10 in one game. In the next game, the player makes 5 out of 10. Explain numerically how the second result is closer to the long-term average.
- 3 A school gives extra tutoring only to students who scored extremely low on a practice exam. Their scores improve on the next exam. Explain why this improvement is not automatically proof that the tutoring caused all of the gain.