# Numeracy #9: Regression Toward the Mean

In the late 19th century, Sir Francis Galton was studying how extreme characteristics (such as height) were passed on from parents to offspring. Galton observed that tall parents, on average, produced offspring that were moderately shorter, and short parents, on average, produced offspring that were moderately taller. Galton coined the term regression to describe the fact that parents who lie at the tail end of a distribution produce offspring that tend towards the middle of the distribution.

Today, this concept has wide-ranging applications from sports to finance, but human psychology hasn’t developed in a way to take advantage of it. Humans naturally think in terms of trends rather than statistical models. If something is working, we should do more of it, and going against that flow is counter intuitive. For example, the tech bubble of the late 1990s had a lot of very smart people believing that the mean itself had changed. This ended badly.

### Definition

In nontechnical terms, regression to the mean is the tendency of a random variable to return to “normal.” It describes an evening out of the world. In technical terms, suppose that X and Y are normally distributed random variables that have a correlation of P. Given that we know X, we can expect Y will be closer to its mean than X is to its own. The conditional mean of Y will be equal to PX.

To put it another way, in a random environment, outliers won’t be self-propagating. Consider the height example that we opened with. If regression to the mean didn’t exist, after a few generations, we would have a bunch of 1 foot tall and a bunch of 9 foot tall people in our distribution.

### Regression fallacy

Perhaps you have heard of the “Sports Illustrated cover jinx” or the “Madden Curse.” Both imply a causal relationship between being featured in the given media and a subsequent decline in performance. A better description of this phenomena would simply be regression towards the mean.

The regression fallacy is the tendency to ascribe causation to natural fluctuations. Daniel Kahneman explained it best with the following story:

I had the most satisfying Eureka experience of my career while attempting to teach flight instructors that praise is more effective than punishment for promoting skill-learning. When I had finished my enthusiastic speech, one of the most seasoned instructors in the audience raised his hand and made his own short speech, which began by conceding that positive reinforcement might be good for the birds, but went on to deny that it was optimal for flight cadets.

He said, “On many occasions I have praised flight cadets for clean execution of some aerobatic maneuver, and in general when they try it again, they do worse. On the other hand, I have often screamed at cadets for bad execution, and in general they do better the next time. So please don’t tell us that reinforcement works and punishment does not, because the opposite is the case.”

This was a joyous moment, in which I understood an important truth about the world: because we tend to reward others when they do well and punish them when they do badly, and because there is regression to the mean, it is part of the human condition that we are statistically punished for rewarding others and rewarded for punishing them.

When we get sick, we tend to get better. If we went to the doctor in between, we might credit that with the reason we got better. If we tried some new age homeopathy in between, we might credit that with the reason we got better. In both cases, regression to the mean might deserve the most credit.

### As it relates to investing

When things are going well or bad, we tend to emotionally overreact. We see trends in the market and we predict that they will continue into the future. A company that is growing earnings at an astounding rate, will continue to grow at an astounding rate. A stock that is rallying, will continue to rally.

This is the opposite of a regression to the mean mindset and countless studies have shown it is not the way the world works. Michael Mauboussin once found that:

Various studies conducted over multiple decades document this reversion-to-the-mean pattern. We have reproduced the results here, using data from over 1000 non-financial companies from 1997 to 2006.

We start by ranking companies into quintiles based on their 1997 ROIC. We then follow the median ROIC for the five cohorts through 2006. While all of the returns do not settle at the cost of capital (roughly eight percent) in 2006, they clearly migrate toward that level”

In investing, the only certainty is that whoever is on top today won’t be on top tomorrow. To be an effective investor, we need to apply that statistical model to the world. This helps us avoid the emotional pitfalls that market excess breeds.

### Conclusion

Current conditions rarely last, whether they be the best of times or the worse of times. Regression towards the mean makes sure of that. As humans, we erroneously latch onto trends and apply causation where non exist. We would do better to remember that the only truth in life is: “this too shall pass.”