# Winning Percentage Estimators

**Winning Percentage Estimators** are methods which estimate a team's winning percentage based on the number of runs they score and allow. Such methods are sometimes used as a predictor of future team winning percentage, as well as to convert individual player's run contributions into wins. W% estimators usually are based on either run differential or run ratio, although hybrid methods and those based on distributions of runs scored and allowed also exist.

## Differential Estimators[edit]

Differential methods of estimating W% use a team's run differential, runs scored minus runs allowed. They generally take the form:

W% = s*(R - RA)/G + .500 = (R - RA)/G/RPW + .500

Where s is the slope, as the form is a linear equation of the type y = mx + b. The y-intercept is .500, since the expectation for a team that allows as many runs as it scores is .500.

The slope s is the number of marginal wins per marginal run. Its reciprocal is the number of marginal runs per marginal win (Runs Per Win). The slope can be held constant, as in the common rule of thumb that RPW = 10, but more accurate results can be obtained by allowing the slope to change based on the total number of runs scored and allowed per game for the team (RPG).

Examples of differential estimators with changing slopes include:

- The very simple formula W% = (R - RA)/(R + RA) + .5 sets RPW = RPG
- The formula W% = .91*(R - RA)/(R + RA) + .5, developed by Ben V-L is equivalent to setting RPW = RPG/.91
- Pete Palmer sets RPW = 10*sqrt(runs per inning) ~= 10*sqrt(RPG/9)

## Ratio Estimators[edit]

Ratio-based methods of estimating W% are fueled by the team's run ratio, runs scored/runs allowed. The most famous of this class is Bill James' Pythagorean Theorem, which raises run ratio to a constant exponent (often set at 2 although James found optimal accuracy at 1.83) to approximate win/loss ratio. However, there are other examples of ratio estimators:

- Percentage Baseball
- Bill Kross estimated W% as R/(2*RA) for teams with R>RA and 1 - RA/(2*R) for teams with RA<R. These formulas can be derived from the Pythagorean Theorem by finding the tangent line of the win/loss ratio function at the point R = RA, and are also equivalent to James' idea of "doubling the edge", i.e. that a team that scores 10% more runs than its opponents should win 20% more games.
- Clay Davenport of
*Baseball Prospectus*developed a version of the Pythagorean Theorem that replaced the fixed exponent with a floating exponent based on RPG. This variation is known as Pythagenport. - David Smyth also developed a floating exponent formula that performed better for very low scoring environments. This variation is best known as PythagenPat.