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Randomness in Your Poker Results? Don't Forget "Regression to the Mean"

  • Robert WoolleyRobert Woolley
Randomness in Your Poker Results? Don't Forget "Regression to the Mean"
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  • If you run especially hot or cold at the poker tables, be ready for a "regression to the mean."

  • Robert Woolley shows how a simple math concept readily explains some of your poker results.

I just finished reading a brand-new book: What the Luck? The Surprising Role of Chance in Our Everyday Lives by Gary Smith.

The title doesn't disclose the actual subject of the book, which is a statistical phenomenon known as "regression to the mean." I first learned of regression to the mean in a stats class in my formal education back in the Stone Age, but I confess I dismissed it as something useful only for passing the final exam, not in real life. Smith's lively book has completely changed my view. It's now clear to me that as Smith argues, the concept is not just important, but comes up practically everywhere.

The fact that I'm writing about regression to the mean for PokerNews probably gives you a clue that I think it yields interesting insights into our favorite game — and you're right. But it's a sufficiently tricky concept that I'll first need to spend this week's column explaining how it works generally, then do a follow-up piece next week dedicated to the poker implications.

"Regression to the Mean" and Assessing Results

Here's the basic idea: Take any field in which (1) there is at least some element of luck to outcomes, and (2) what we can measure is an imperfect yardstick of the characteristic in which we're interested. Given those conditions, exceptionally high or low performances on one occasion will tend not to be repeated on the next occasion. Regression to the mean is, as Smith dubs it, a "mediocrity magnet."

Take baseball hitting percentages as an example. Each player has some inherent batting ability, but we have no perfect measurement of it. Instead, we look at results, which are an imperfect surrogate measure of ability, because they're subject, in part, to the vagaries of chance — things like lucky bounces and wind currents, which pitchers the batter faces, whether he gets favourable calls from the umpires, his health, and many other factors outside the player's control.

Regression to the mean tells us that those who have the highest batting averages in a season often won't do as well the next year. This is because the outstanding performance that we're looking at was partly due to those luck factors breaking the right way more than average. The player's high one-season batting average, then, overestimates his true ability. In other seasons it will be less exceptional, because he won't keep getting those lucky breaks.

It's true at the bottom end, too. The worst seasonal batting averages generally underestimate the players' true abilities, because those players got more than their fair share of bad luck. In future seasons, we would expect to see their batting averages improve toward their true ability, because their bad luck probably won't continue.

For example, Smith discusses the 10 major-league players with the best batting averages for 2014. Of those 10, nine hit higher than their career averages; they were performing above their ability. And for nine out of 10, their batting averages dipped in 2015, just as regression to the mean predicts.

Of course, there is a lot of variability in players' natural batting abilities. Nothing about regression to the mean suggests that they all share the same prowess. Differences in performance are dependent on both intrinsic ability and luck, in a complex mix.

Exceptionally Good or Bad Performances Tend Not to Recur

This leads us to one of the important mistakes that occurs when we don't understand or take into account regression to the mean when assessing results — settling for other, less rational ways to explain why extremely good or bad performances don't keep happening.

Looking again at examples from sports, there are all sorts of labels for failure to repeat an exceptionally strong performance.

There's the "rookie of the year jinx," used to describe a high-performing first-year player whose second-year results are diminished. There's the "Sports Illustrated jinx," named for the fact that sports performances remarkable enough to earn a cover photo will usually be followed by less amazing performances. There's the "John Madden curse," a reference to the same phenomenon occurring to NFL players who appear on the cover of the newest version of the popular video game.

None of these things are "jinxes" or "curses" or anything supernatural — they're simply examples of regression to the mean in action.

Remember the same is true on the bottom end, though there aren't as many titles and recognitions to be had in the basement. Understand, though, that exceptionally poor performances will usually not be repeated, and subsequent endeavors will typically rise back toward the person's natural ability.

But What Does "Regression to the Mean" Really Mean?

This phenomenon pops up everywhere. As Smith demonstrates, regression to the mean influences fluctuations in such disparate fields as...

  • College students who score the highest on the midterm will tend to do well, but not as well, on the final, because luck boosted their performance once, but will probably not do so again.
  • Corporations with the best gains in profits this year will usually not continue so hot next year.
  • New drugs that are most promising in clinical trials will typically have less impressive results when released for general use.
  • The tallest parents will tend to have children who are taller than average, but not as tall as their parents; same with the shortest.
  • College applicants who are the most stellar-looking on paper will usually underperform relative to their super-high expectations.
  • Abnormally high or low blood test results can lead to false diagnoses if they are just random deviations from the patient's true average value, as measured across time.
  • Fund managers who assemble an investment portfolio that performs incredibly well one year tend to attract a lot of new clients, who are then disappointed the next year when the returns aren't as spectacular.

Regression to the mean does not mean that everybody and everything in which the phenomenon can be observed is trending over time to a generalized homogeneity, a uniform mediocrity. This year's outstanding performances mostly won't be repeated, but equally outstanding performances will be put in by other people, teams, companies, and so on. The "mean" to which exceptional performances regress is the individual person or entity's own true level of ability, not the population average.

Of course, true ability can change over time; nothing about the concept suggests otherwise. But for simplicity of illustration here, we're assuming it stays constant.

Conclusion

Here's the key insight to take away today — because we erroneously think exceptional performances accurately reflect ability and therefore should be repeated, we're subject to all manner of wrong theories about what causes the lack of repetition.

For example, if the lowest-scoring students are given special tutoring, and then do better on the next test, we're inclined to think that the intervention was what made the difference, when the truth may be that the change was entirely due to random variance, and the tutor made no difference.

If the best players or teams don't repeat their championship performances, we might think they got cocky or complacent or lazy — or were "jinxed" — when all that really happened is that their good luck didn't continue.

Okay, that's enough about regression to the mean as a general phenomenon. Next time, we'll explore more specifically what the concept can teach us about poker. Spoiler alert — that last point about erroneous attribution of causes will figure prominently, so you might want to read it again and ponder its implications.

Robert Woolley lives in Asheville, NC. He spent several years in Las Vegas and chronicled his life in poker on the "Poker Grump" blog.

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