Pythagorean Expectation Part I: The Luck Factor

Let's start with the most basic of basics. How does a team win? Score more runs (or goals, points) than the opponent, and allow fewer runs than the opponent. That is true in any given game.  It is also the same over the course of a season. The team with the better run differential has a better tendency to win because they score more than they allow. With this concept, the following formula known as the Pythagorean Expectation was created:

PW% = PS^x / (PS^x + PA^x)

where   PW% = Pythag win percentage

            PS = Runs/Goals/Points scored

            PA = Runs/Goals/Points allowed

            x = 1.82 for MLB, 2.0 for NHL, 2.37 for NFL and 13.91 for NBA. More of this later.

This formula approximates a team's win percentage based on their scoring capability. Using this approximation, we can compare it to the team's actual win percentage and determine whether a team has been "lucky" or "unlucky" and whether they have been under performing or over performing. Let's take a look at an example on how Pythag is calculated:

As of 3/12/12, the Boston Bruins (40-28) scored 222 goals and allowed 164 goals. Plugging in the numbers, this gives the Bruins a PW% of 0.647 as opposed to their actual win percentage of 0.588. This implies that the Bruins were either under performing or unlucky. Have they been getting a lot of bad breaks? Have they been lackadaisical in their games? Have injuries become a factor? Eventually, this "luck" will even itself out and regress to the Pythagorean Expectation. A PW% of 0.647 suggests that Bruins are really a 44 win team with 24 losses. PW% = 0.5, when PS = PA.

Using this theory, when two teams are matched together, we can calculate their Pythagorean difference to find value in matchups. It can be calculated as follows:

P_win = PW% * (GP) 
P_diff = P_win - W

where P_win = Pythag wins
          GP = Games played
          W = Actual wins
          P_diff = Difference in P_win and actual wins

The Boston Bruins have a P_win of 44 and a P_diff of 4.0. Tomorrow 3/13/12, the Bruins play the Tampa Bay Lightning, who hold a P_diffof -3.53. Put the two together, the Bruins have a 7.53 game advantage in wins. Looking at their actual records, the Bruins already look like the better team (Lightning are 30-37).  Take into account the "luck" and performance factor, the Bruins should be an even more favorable team. By looking at the P_diffalone, Bruins should win the matchup. However, we must keep in mind that this is the first filter and other statistics, such as goaltending and offensive power must be taken into consideration.

More to come soon.... and for those of you who don't believe in the "luck" factor. Watch the following videos and think about it. You watch enough sports, and you'll start seeing some crazy, unbelievable and WTF moments.

Rangers vs. Giants: Game 2 of 2010 World Series

Sharks vs Avalanche: Game 3 of 2010 Western Conference Quarterfinals