Regression Toward the Mean: Why Big Wins and Bad Days Don't Last

Regression toward the mean is one of the most common — and most misunderstood — phenomena in everyday life. People often mix it up with the gambler’s fallacy or with causal stories (e.g., “they must have gotten lucky” or “they must be slacking now”). In plain terms, regression toward the mean is a statement about probabilities: after an extreme observation, a less extreme observation is usually much more likely than an even more extreme one.

A short history

The phrase and idea trace back to Sir Francis Galton in the late 19th century, who noticed that tall parents tend to have children who are tall but closer to average, and short parents tend to have children who are short but closer to average. Over the next century, statisticians formalized the idea in terms of sampling distributions and expectation, and it became central to understanding measurement variability, reliability, and prediction. The historical takeaway is that what looked like a biological “pull” toward the average was really a probabilistic pattern arising from variability and imperfect correlations between measurements.

Regression toward the mean as a probability comparison

At its core, regression toward the mean is not a mysterious force; it’s a simple comparison of probabilities — and because most measurements in daily life are roughly bell-shaped (approximately normal), those probability comparisons overwhelmingly favor outcomes closer to the mean. And for a normal-like distribution, the bulk of outcomes cluster near the mean, giving the familiar 68–95–99.7 rule: about 68% of outcomes fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ.

The 68-95-99.7 rule: a standard normal curve showing ~68% within ±1σ, ~95% within ±2σ, and ~99.7% within ±3σ.

Suppose a measurement X has a true long-run distribution with mean μ and some variability (standard deviation σ). If you observe an extreme value x far from μ, the distribution of a fresh, independent draw from the same process still centers on μ. That second draw is simply more likely to fall closer to μ than to be even farther out.

Inside vs Outside d from the mean of Normal Distribution, also shows the ratio of these two probabilities.

Concrete classroom example: a student truly masters 80% of a large problem pool. If an exam picks 100 problems at random, the student’s score follows a Binomial distribution with parameters N=100 and p=0.8. Extreme scores (very high or very low) are possible, but unlikely. If the student scored 90 on one test (an extreme positive outcome), the next independent test still follows the same Binomial distribution: scores closer to 80 are far more likely than scores even farther from 80 (like 95 or 65).

Whatever the previous score was, the next independent score follows the same Binomial (or its normal approximation); extreme previous scores are rare, so next scores are much more likely to fall closer to the true mean.

For most everyday examples and many practical purposes, the Binomial can be well approximated by a normal distribution when the rule-of-thumb N*p≥5 and N*(1-p)≥5 holds, as shown in the following image. 

Binomial (N = 100, p = 0.8) with its normal approximation (μ = 80, σ = 4); the discrete distribution and smooth curve show how the Binomial closely matches a normal shape for the test score case.

The intuition: being 3σ above the mean is rare; the probability mass even farther out is tiny. So after seeing an extreme, the odds strongly favor a next score that’s closer to the mean rather than still more extreme. This is a ratio/ comparison of probabilities, not a statement that the process “wants” the mean more than before.

When regression toward the mean can fail or reverse

Regression toward the mean depends on the shape of the underlying distribution and on independence between measurements.

If the distribution is multi-modal or heavily skewed, an extreme value in one mode might be followed by another value in the same distant mode, making “regression” less evident or even producing apparent divergence for some ranges. Like the following example, the distribution is a mixture of two normal distributions and one beta distribution. Luckily, most everyday measurements follow an approximately normal, bell-shaped pattern, so the usual regression‑toward‑the‑mean intuition holds in most practical cases.

A multimodal distribution: multiple peaks produce less mass near the center (≈45% within ±1σ) and more weight in distant modes, so values can remain far from the overall mean rather than regressing toward it at least for one σ from the mean.

Dependence matters: if measurements are not independent — true skill changes, learning effects, carelessness after success, carefulness after failure, fatigue, selection effects — then future outcomes aren’t just resampled from the same underlying distribution. In those cases, you must separate regression from real change. Regression toward the mean can sometimes reflect genuine causes (for example, learning or skill decay), but it also follows purely from chance — even when nothing real has changed, extreme observations are statistically likely to be followed by values closer to the average.

Always ask: Are you comparing independent draws from the same process, or is the process itself changing after the extreme observation?

Everyday examples

Sports: A player who has an unusually great season often posts a less extreme season afterwards. That’s frequently a mixture of regression toward the mean (performance variability) and some real effects (injury, aging).  

Investing: a stock or fund that dramatically outperforms in one year is statistically unlikely to match that same outperformance the next year because returns are noisy and short-term winners often reflect luck or transient conditions. Over long periods, this tendency helps explain why an equal-weighted index fund can outperform market-cap-weighted ones: cap-weighting concentrates on the recent large winners, but many of those winners shrink relative to the market later, while equally weighting treats past performance as less decisive. This is a statistical tendency, not a rule—some firms do sustain long-term outperformance.

Testing and measurement: an exceptionally high or low test score often reverts toward a student’s long-term average on repeat testing unless there was a real change in knowledge.

Conclusion

Regression toward the mean is simple and powerful: after seeing an extreme observation, a less extreme observation is usually more probable than an even more extreme one. It is properly understood as a probability comparison about repeated measurements or independent draws from the same distribution. Confusing this fact with causal explanations (or with the gambler’s fallacy, like I discussed in this blog post) leads to bad reasoning. When evaluating extremes, look for independence and for changes in the underlying process before inferring causes — and remember that sometimes the simplest explanation is just chance pulling outcomes closer to the mean.