Pythagoras Explained

Pythagoras of Samos, mathematician and philosopher, died about 2500 years ago. Nevertheless, his name is familiar to baseball fans. The "Pythagorean Expectation", invented by Bill James in the 1980s, predicts a team's winning percentage from runs scored and runs allowed. Despite the intimidating name, Pythagorean win expectations can now be found on mainstream sites like ESPN and MLB.com.

The Pythagorean Theorem

The original Pythagorean Theorem states the following for right triangles (triangles with a 90-degree angle):

The square of the longest side equals the sum of the squares of the two shorter sides.

To put it visually, the area of square 'c' always equals the combined areas of squares 'a' and 'b':

Pythagorean Expectations

James' Pythagorean Expectation uses a similar formula to express the fact that a team's winning percentage can be predicted by comparing the square of the team's runs scored to the square of the team's runs allowed.

Specifically, the ratio of wins to losses correlates with the ratio of those two squares:

Wins : Losses   =   Runs Scored ² : Runs Allowed ²

The reason this formula evokes Pythagoras' name can be seen in this diagram. If the lengths of each side are represented by a team's runs scored and allowed, their balance of wins to losses is shown by the area of the squares.

Take the 2004 Red Sox as an example. They scored 949 runs while only allowing 768.

• 949 ² = 900601
• 768 ² = 589824
• Projected Winning Percentage = Projected Wins / (Projected Wins + Projected Losses)
• Projected Winning Percentage = 900601 / (900601 +  589824)
• Projected Winning Percentage = .604 (98 - 64)

A .604 winning percentage in a 162-game season equals a record of 98-64. In real life, the 2004 Red Sox were 98-64. So, as some stat nerds might say, "Pythagoras was right".

Player Value

Predicting team winning percentage is useful, but the Holy Grail for general managers is a formula that determines player value. So, we use the Pythagorean formula to determine how many runs a player needs to create (or prevent) for that team to win one more game than it would otherwise.

If a team scores 810 runs and allows 810 runs, we can predict a winning percentage of .500. That is, the team should go 81-81 over a 162-game season.

For a team that scores 820 runs but still allows 810, the Pythagorean formula predicts a winning percentage of .506. This corresponds to a record of 82-80. In other words, 10 additional runs are needed to turn one loss into a win.

This is the math that underpins modern player valuation techniques such as Wins Above Replacement (WAR). By analyzing thousands of games over the last 50+ years, we calculate how many runs each play creates. For example, a double creates about 0.85 runs. A home run creates about 1.40 runs.

Then, if the above team adds a player in the off-season who contributes 21 more doubles and 30 more home runs than the player he replaces, that will create an additional 59.85 runs (21 * 0.85 + 30 * 1.40). Since every 10 runs creates an additional win, this player will add about 6 wins to the team's projected won-loss record.

The same math is used for pitching and defense. If a center fielder with great range converts 20 doubles into outs over the course of a season, he has prevented the opposition from scoring about 17 runs -- the equivalent of 1.7 wins.