Co-authored by Lisa Ashworth
On Friday night in Oklahoma City, Russell Westbrook took a 3-pointer with 0.1 seconds left in overtime. The shot sank through the net giving Oklahoma City a one point victory over the Golden State Warriors. That shot will be replayed on highlight reels. On otherwise empty courts, young players will create their own versions taking shots that lead to last second victories in fictional games. Enacting such dramas enlivens hours and hours of practice. A generation ago, young players across the country were shooting layups with their tongues waggling in the air to simply "Be like Mike."
In the mathematical classroom, similar role-playing can enhance learning. Rather than being like Mike, let's see how to be like Bill -- James, that is. The greats of sports transform the game. Bill James is undoubtedly one of the greats of baseball. His greatest asset wasn't a great arm or bat. It was mathematics. He is now on the staff for the Red Sox using mathematics to enhance the team that won the World Series again this year, further burying the curse of the Bambino.
Keep in mind: Even the giants of sport were younger, less experienced versions of themselves. Michael Jordan still had to learn to shoot, pass and dribble. Keep in mind, he was always a future NBA star, but that's something we know now. As a sophomore in high school, he didn't make the basketball team in Wilmington, North Carolina. He did his junior year and had several 40-point games.
In a similar way, it is unlikely that Bill James, when he was first creating the formulas that served as a foundation for what is known now as Moneyball, was thinking about being named one of the most influential people in the world by Time magazine as he was in 2006. Many of his baseball articles were penned while he worked nightshifts as a security guard at the Stokely-Van Camp's pork and beans cannery. When Billy Beane applied James' ideas to Major League Baseball and the 2002 A's, he probably was thinking more about keeping his job than being played some day by Brad Pitt.
Michael Jordan loved basketball and spent hours on the court. Billy Beane loved baseball and wanted to compete with big league teams with even bigger, much bigger team salaries. Bill James also loved baseball and delved into data and analysis to better understand and predict the game.
Like a youth imagining a swish on a playground court wins a big game, students in classrooms can act like big leaguers mathematically. Let's see how to develop Bill James' Pythagorean Expectation with students. This idea developed out of a Math and Sports seminar in the Charlotte Teachers Institute.
The formula estimates how many games a baseball team should win in a season based on the number of runs that they score and allow. In particular:
winning percentage = (runs scored)^2/((runs scored)^2 +(runs allowed)^2)
Bill James' formula has been applied to other sports like basketball.
Let's see how to find a suitable exponent for basketball, which can allow students to flex their mathematical muscle and "Be like Bill." Note, advanced statistical methods can be applied for this purpose. Here we present an approach suitable for many secondary classrooms.
In moving from baseball to basketball, we change terminology. Points rather than runs are scored. Now, we are searching for the exponent in the Pythagorean Expectation so it will be treated as our unknown. So our equation is:
winning percentage = (points scored)^x/((points scored)^x + (points allowed)^x)
How do we determine a suitable exponent? As we saw for Major League Baseball, Bill James found x to equal 2.
One way to approach this is to find the exact exponent for each team. Let's do this for 2012-13 NBA season. For example, consider the Miami Heat. They scored 8436 points, allowed 7791 points and won 66 of their 82 games. So, the Pythagorean Expectation for them becomes:
8436^x/(8436^x + 7791^x) = 66/82 = 0.8048.
There are two ways to find x.
First, we can apply algebra techniques and solve. Warning, this can become pretty involved! Just look at the exact answer:
x = (-3log(2) + log(3) + log(11))/(2log(2) - 2log(7) + log(19) + log(37) - log(53))
Alternatively, one can make a guess. Here is how this works. Let's lean on baseball and take x = 2. Now: 8436^2/(8436^2 + 7791^2) = 0.5397
...which can be found using the Google calculator. This estimate is too low. We will see in a moment that we also want an exponent that produces an estimated winning percentage that's too high. Let's try x = 20. We find:
8436^20/(8436^20 + 7791^20) = 0.8307
...which is indeed too high. So, the value for x lives somewhere between 2 and 20. We can use a method known as the bisection method and simply choose the value halfway between 2 and 20, which equals 11. We try that value now:
8436^11/(8436^11 + 7791^11) = 0.7057
...which is too low. So, we now know that value for x lives between 11 and 20. We again pick the value in the middle and test it. In this way, we keep finding two values for x, one that produces an estimate that's too high and one too low. When we get a value that produces a value for the corresponding Pythagorean Expectation that's close to 0.8048, we have found our desired value for x, which for the Miami Heat equals 17.81.
So, the value for the NBA is, after rounding, 18! Actually, no. This is the value that works for the Heat. A value 5.87 works for the Houston Rockets and 33.35 for the Golden State Warriors. We need to know the value for x for every team.
For classroom use, each student can be assigned a team and find the corresponding value of x. So, a student assigned the Heat would find x = 17.81 and another would find x = 5.87 for the Houston Rockets.
With so much disagreement, how do we see anything? A few approaches could be taken. We can simply take an average of the computed x values for all the teams. If we do this, we find x is about 12. We could also find the median which equals 15.
Generally, the exponent in the Pythagorean Expectation for basketball is taken to equal about 14. Our average was screwed a bit low due to outliers like the Utah Jazz. They scored 8038 points and allowed 8045. The Pythagorean Expectation estimates that they should have lost more games than they won. Think about it. They scored less points than they allowed. But, they won 43 and lost 41. Teams that win more games than the Pythagorean Expectation indicates are referred to as lucky.
Note, you can try this for other sports, like baseball. Just like our computations with NBA data, we can run the numbers on the 2013 regular season of Major League Baseball. When we take the average of the computed exponents for all the teams, we find x = 2. Want to see the numbers? Break out pencil, paper, and a calculator and be like Bill James and start crunching the numbers!