Baseball has always been a sport with a rich history of using statistics to understand and analyze who are the best players and teams. For a long time, the popular baseball stats were fairly rudimentary. After the Society for American Baseball Research (SABR) was established in 1971, the development of more sophisticated stats became known as “sabermetrics.”

One of the early analytics formulas was Bill James’ Pythagorean expectation, which was an attempt to predict the number of games a team should win given the number of runs scored and runs allowed. His original formula for a team’s winning percentage was

where RS and RA stand for runs scored and allowed, respectively. The reason James called it Pythagorean expectation was because the denominator was reminiscent of the Pythagorean formula relating the lengths of the three sides of a right triangle, a² + b² = c².

It was quickly realized that the exponent “2” in the formula was not optimal, and that Pythagorean expectation for winning percentage could be generalized to

The exponent  α =1.83 works better when applied to many decades of data. Even better, for a given year, the exponent can be adjusted depending on the overall run environment, which means the average number of runs scored per game across all of major league baseball. But for our purposes here, we will use α = 1.83 .

In practice, a team may win more or less than their Pythagorean expectation in any given year, but when averaged across all teams, the Pythagorean expectation is very accurate. It has been shown that the Pythagorean expectation for one year is a better predictor of the following year’s results than simply wins. For example, if a team has fewer wins in a season than their Pythagorean expectation, it means they were “unlucky” and will have a tendency to perform better in the next season.

You can apply calculus to the Pythagorean expectation formula to determine how many more runs a typical team would have to score in a season to produce one more win:

where RSavg is the average runs scored per game. For example, when a team scores 4.5 runs per game (about the average in the last few years), the Pythagorean expectation says that it would take approximately 10 additional runs in a season to create one more victory, which is the source of the popular sabermetrics yardstick that “10 runs equals 1 win.” While it may seem that 10 additional runs should produce more than just 1 win, remember that in some games the team is already ahead, and in others the team is more than 1 run behind, and so adding a run in those games does not affect the number of wins. When those 10 runs are randomly spread across the whole season, the net effect is to add just 1 win on average.

Finally, another sabermetric concept is wins above replacement (WAR), sometimes also called Wins Above Replacement Player (WARP). WAR for a given player is an estimate of how many extra wins that player was directly responsible for beyond what is expected for a so-called replacement player, which is a minimally qualified player that is just below major league level. A replacement-level player can also be thought of as the player who is brought up from the minors when a major league player is injured.

WAR is calculated by first determining how many runs a player is directly responsible for, and then applying the 10 runs = 1 win relationship to calculate the WAR for that player. The runs attributable to a player are not just what they score or drive in, which can depend on factors other than the skill of the player, but rather how many runs they would contribute to a typical team when applying their batting stats to generic baseball situations. There are also methods for calculating the runs added or subtracted for base-running, fielding, and pitching. As a rule of thumb, an average everyday player in major league baseball has a WAR of 2.0 for a full season, an all-star 5.0, and a league MVP around 10. Roughly speaking, a team of replacement players is expected to win just 50 games in a 162-game season, so the combined WAR for a team with an 81-81 record should be about 31.

The basic currency for Pythagorean expectation and WAR is runs. This means that these analyses are determined basically by a single parameter, and two teams that score the same average number of runs should split their games against each other evenly in the long run. But is that necessarily the case? I will investigate this question in Part 2.