Uncertainty can be a Good Thing

Consistency is a word that is often used in sports to describe teams and players. An analyst on ESPN might say that he doesn’t have a lot of confidence in a pitcher because he’s just not that consistent. Another may argue that the Mariners just don’t have it in them to make the playoffs because they’re too inconsistent. These analysts generally argue that consistency is good, and that a lack thereof is bad.

The opposite of consistency might be considered uncertainty, or variance, and I’m going to argue that it can actually be a good thing in some situations—situations like that the Mariners find themselves in this season. It’s optimal to be consistently good, but if you’re bad, you want to be inconsistently bad.

Understanding how to measure variance, and then what to do with those measurements is important in sports analysis, but let’s start with another example of when variance comes into play: casino games. Any introductory probability class learns to calculate how the casino makes money. If you make a bet on red at the roulette table, you have 18-in-38 chances of winning your bet, or 47.4%. That means the casino has a 52.6% to 47.4% edge over you, and when it comes to winnings, we would say that the casino has a higher expectation than you do. Ok, so if we expect the casino to win, then why did your friends come back from Vegas that one time with hundreds of dollars? The answer to that question is variance.

Variance helps describe the range of possibilities. When you play casino games, there are many, many possibilities in even ten roulette spins or blackjack deals. If we assigned each of those possibilities with a mathematical probability, and then divided out those probabilities between you and the casino, the casino would have more. It would have more probability, but it would not have all the probability. See chart.

# Bets Win %

1

47.4%

5

45.1%

10

31.4%

20

32.2%

50

30.3%

100

26.5%

200

20.7%

500

11.1%

1000

4.5%

Here, the Win% column indicates your chances of leaving Vegas ahead at the roulette table after the given number of bets. The casino starts to develop a pretty weighty advantage as more and more bets are made, but you still do have a chance to make money even after 1000 bets. Vegas wouldn’t be in existence if casino games did not have variance. Think about a game of blackjack where the dealer consistently gets blackjack every time without fail. That’s a stupid game, and no one would play. Basically, variance allows for unexpected things to happen. The more the variability, the crazier the results. Even though the expectation is in favor of the casino, you could still win money over the course of a weekend, and that’s what draws people to casinos: the hope and possibility of winning.

Speaking of hope and things that draw us back to the Mariners every spring… Even though the Mariners are expected to win only 75-80 games, there is a lot of variance in even a 162-game season. 162 roulette spins wasn’t a large enough sample to squelch out your chances in Vegas, and it isn’t a big enough sample to squelch the Mariners in 2013. So while math nerds expect the M’s to win about 79 games and miss out on the playoffs, variance allows for the possibility of a post-season. And for a lower-end team like Seattle, more variance is a good thing. Observe.

Season Variance ComparisonProjecting a team’s win-loss record is not an exact science. We take what we know about a team’s players, derive a best expectation for the season, and then admit there are factors we can’t predict. Those factors we can’t predict lead to variance, or uncertainty. The two teams depicted over there by the red and green distributions articulate why variance is a good thing for bad teams. Think of the height of each curve as a probability for the various win totals along the x-axis.

The low-variance team in red has less spread to its prediction. In other words, the outcome is more certain for red than for the green team. I can say almost surely that the red team will not make the playoffs since its probability curve hits zero around 88 wins. But the green team actually has a legit shot at the playoffs. There is some positive probability associated with the green team in the upper 80s and lower 90s, indicating some chance of that many wins.

Of course, the green team is also more likely to challenge the Astros for the cellar. We might say that the green team has a high ceiling but also a low floor, while the red team has a high floor but low ceiling projection. In this case, it’s probably better to be the green team—despite both teams having the same expectation of 79 wins—because the green team might actually make the playoffs.

Rye bread, mustard and salami sound good this year, Justin.

I would suggest that outcomes in baseball have higher variances than, say, the roulette table, and that the Mariners are an especially variable baseball team. Youth and uncertainty surround this Seattle squadron, and that makes them hard to predict. Or, better put, that makes the variance around their prediction higher like the green team.

Despite Baseball Prospectus’ 79-win expectation for the M’s, you’ll also notice our playoffs chances are estimated at 14.1%. That all comes from variance and uncertainty, so here’s to Dustin Ackley returning to 2011 form, and Justin Smoak to last September’s form.