Everyday Math # 1
For me, baseball turned out to be a gateway drug to mathematics. I realize that by talking about both baseball and math I’ve probably already alienated enough people to ensure no one is still reading this sentence. But for those brave souls who are still reading, I think you will find something interesting about how bits of our lives are interconnected to many other things, sometimes even connected to mathematics.
I’m sure even casual sports fans wouldn’t find the relationship between math and baseball to be a surprise. Most of the numbers used in baseball are counts or rates. Batting average is a rate. Runs batted in is a count. These are very simple for everyone to understand, that’s part of the reason why we use them. But it’s still math. As a young baseball fan, I understood the existence of math, but I refused to think of it in any greater detail. Why should I? It’s all pretty simple. What could be gained?
Well, to understand the answer, first you have to understand what kind of baseball fan I am. Those of you who have been long time fans of the Boston Red Sox might understand where I’m coming from when I tell you I’m a lifelong fan of the Seattle Mariners. It’s not my fault. That’s where I fell in love with the sport and it’s the team I was able to see on a regular basis. The Mariners are bad. They haven’t always been bad. In the middle of the 1990’s they were great, and in 2001 they actually set a record for the number of wins in a season. But that’s less than a decade they were good, and before and after that time span, they’ve been bad. Horrible at times. Bad enough to make any fan wonder if the time and energy devoted to being a fan of a particular team could be worth the frustration.
But a true fan has no option. As a way to cope with the disappointment inherent in being a Mariner fan, I started looking for possible signs of life, little things that could indicate bigger and better things down the road. A reason for hope. I was tired of seeing the Mariners trade for a player with previous success only to see that success never duplicated while in Seattle. The trouble, I found, was in the math. It turns out there are a number of reasons to look closer at the math involved with baseball.
Traditional baseball mathematics are pretty simple, and it gives us a relatively easy way to evaluate players. That math, however, is flawed. Batting average doesn’t include walks, and cannot account for the quality of hits (doubles are worth more than singles, for example). Runs batted in requires your teammates to be setting the field for you, so this cannot account for individual ability. Even the batting order is important with that counting stat. Even earned run average (ERA), the go to staple for evaluating pitchers, is too muddled with other factors that it’s impossible to understand the pitcher’s true talent from that statistic. If the infielders can’t field well, a pitcher’s ERA is going to skyrocket. The converse is true as well, great fielders and a large stadium can really drive a pitcher’s ERA down. That math is limited. And the more you understand how the math is limited, the more you want better math to help understand the game and what a player brings to the table. The Oakland A’s are famous for starting a trend of using advanced stats to improve their ball club. The Red Sox have probably benefited the most from the new stats as they’ve also had more money to spread around.
There are a lot of different statistics that I could talk about, but this isn’t really a post about baseball statistics. This is more about the relationship between baseball and math. So I’m going to talk about one single statistical measure that I’ve found to revolutionize how I appreciate baseball. I’m talking about Win Probability.
At the start of each game, before a pitch has been thrown, each team has the same chance to win that game. They also have the same chance to lose that game. Win Probability takes into account the current game state (which is to say, how many runs have been scored, what inning are we in, how many outs, and so on) to compare to every game that’s already happened before. So let’s say the game is in the 9th inning and the home team is winning by 2 runs. No outs, no one on base. In all of baseball (since around 1970), there have been 8,240 other times this situation has occurred and the win percentage of those games for the home team has been 1. That means the number of times the home team has won in those 8,240 games is so close to 100%, we would expect the home team to win. That does not mean the home team will win. Statistics is funny that way.
Why does this matter? Well, it helps us to understand how important an at bat is or an out can be. If we let the situation play out just a little bit and the away team gets the bases loaded with no outs, the win percentage for the away team becomes .4 (40% chance the away team will win). Without scoring any runs, the away team has improved their odds of winning to 40%. That’s a huge gain.
So the value in Win Expectancy ranges between 1 and 0. 1 is a win (100%) and 0 is a loss (0%). The game always starts at .5 (50%) and each change in the game state (out, inning, runners on base, score) will move the Win Expectancy positive or negative. This means we can track each player and know how much they have attributed to the Win Expectancy.
The Mariners did something strange last night against the Texas Rangers. They won. Here’s the Win Expectancy chart for that game, courtesy of Fangraphs.com:
Notice the sharp decline into Mariner’s territory right at the end. That’s due to a dramatic three run 9th inning. Before the Mariners scored a single run in the 9th, the chance of them winning, based on all other games that have been in that same situation, was about 5%. They pulled a fast one. The biggest contributor towards a win in this game was John Jaso, with .333 WPA (Win Probability Added). WPA is the amount of change in the Win Expectancy between the start and end of every play John Jaso was involved in. This doesn’t mean he has the best skill on the team (No disrespect towards John Jaso, but that would be bad even for the Mariners), but it can tell us, especially over the course of a full year, how much a player contributed towards his team’s win and loss record. It can show the value of a stolen base, or a sacrifice bunt.
There are problems with evaluating players this way. I won’t go into detail except to say that a person who hits a home run in the first inning and another player who wins the game by doing the same thing in the 9th inning are using the same talent. But the 9th inning is highly leveraged, and so that’s where most of the credit goes. Over the course of a full season, it might make sense to use WPA because of regression to the mean, but that exceeds the point of this post.
What is perfectly in scope with this post is this; math has a place in everything and is a tool that can be used to better understand and sometimes better enjoy something. I love baseball, but the newer statistics being used to evaluate baseball has helped me better appreciate the game. It helps me to fully understand a player’s contribution, and when a team wins in dramatic fashion, like the Mariners did last night, it helps me to fully appreciate how anything can happen, even when the math says you don’t have a chance. When the Mariners started the 9th inning, down by two runs, they had a 5% chance of winning. 8,928 games have entered the 9th inning just like that since 1970, and only 503 times have the trailing team won. Baseball is amazing simply because of the things math cannot capture, but the math is amazing in how it frames the value of amazing things happening in baseball.
I am not a mathematician. I was an English major who ran away from math most of my life. I’m still not very good at math, I use my fingers to count sometimes, and always feel more comfortable with a calculator than not. But mathematics has enhanced my enjoyment of baseball, and has helped me learn to think mathematically and become a better problem solver. If you struggle with math I would encourage you to find ways to get comfortable with math through things you’re already interested in. Although you’ve heard it countess times before, take it from someone who’s been embarrassed and scared of math for a long time; math has a huge world of application and really is accessible. For some of us that just means finding the right angle.
If you’re still with me, and wow, I’m proud of you for sticking with it (unless you’re my mother), and you want to know more about some of the baseball statistics I was talking about and more, you can try fangraphs.com (start with the glossary). Where else have you found fun with mathematics?
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