# 3.9: Regression to the Mean Fallacy

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Humans are prone to see causes even when no such cause is present. For example, if I have just committed some wrong and then immediately after the thunder cracks, I may think that my wrong action caused the lightning (e.g., because the gods were angry with me). The term “snake oil” refers to a product that promises certain (e.g., health) benefits but is actually fraudulent and has no benefits whatsoever. For example, consider a product that is supposed to help you recover from a common cold. You take the medicine and then within a few days, you are all better! No cold! It must have been the medicine. Or maybe you just regressed to the mean. Regression to the mean describes the tendency of things to go back to normal or to return to something close to the relevant statistical average. In the case of a cold, when you have a cold, you are outside of the average in terms of health. But you will naturally return to the state of health, with or without the “medicine.” If anyone were to try to convince you to buy such a medicine, you shouldn’t. Because the fact that you got better from your cold more likely has to do with the fact that you will naturally regress to the mean (return to normal) than it has to do with the special medicine.

Another example.

Suppose you live in Lansing and it has been over 100 degrees for two weeks straight. Someone says that if you pay tribute and do a special dance to Baal, the temperature will drop. Suppose you do this and the temperature does drop. Was it Baal or just regression to the mean? Probably regression to the mean, unless we have some special reason for thinking it is Baal. The point is, extreme situations tend to regress towards less extreme, more average situations. Since it is very rare for it to ever be over 100 degrees in Lansing, the fact that the temperature drops is to be expected, regardless of one’s prayers to Baal.

Suppose that a professional golfer has been on a hot streak. She has been winning every tournament she enters by ten strokes—she’s beating the competition like they were middle school golfers. She is just playing so much better than them. Then something happens. The golfer all of a sudden starts playing like average. What explains her fall from greatness? The sports commentators speculate: could it be that she switched her caddy, or that it is warmer now than is was when she was on her streak, or perhaps it was fame that went to her head once she had started winning all those tournaments? Chances are, none of these are the right explanation because no such explanation is needed. Most likely she just regressed to the mean and is now playing like everyone else—still like a pro, just not like a golfer who is out of this world good. Even those who are skilled can get lucky (or unlucky) and when they do, we should expect that eventually that luck will end and they will regress to the mean.

As these examples illustrate, one commits the regression to the mean fallacy when one tries to give a causal explanation of a phenomenon that is merely statistical or probabilistic in nature. The best way to rule out that something is not to be explained as regression to the mean is by doing a study where one compares two groups. For example, suppose we could get our snake oil salesman to agree to a study in which a group of people who had colds took the medicine (experimental group) and another group of people didn’t take the medicine or took a placebo (control group). In this situation, if we found that the experimental group got better and the control group didn’t, or if the experimental group got better more quickly than the control group, then perhaps we’d have to say that maybe there is something to this snake oil medicine. But without the evidence of a control for comparison, even if lots of people took the snake oil medicine and got better from their colds, it wouldn’t prove anything about the efficacy of the medicine.

This page titled 3.9: Regression to the Mean Fallacy is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Matthew Van Cleave.