Introducing the Expected Win Differential

From athletes to coaches to fans, people in the sports world all have their own theories on what a win-loss record says about a team. Some think it’s that simple, like one of the all-time great NFL head coaches in Bill Parcells saying, “you are what your record says you are.” Others think there is much more to that. Those dissenting opinions often come from the mathematics fanatics, like baseball sabermatricians and stat-heavy football publications. Of course, you can count TABMathletics in with those who believe that a team’s win-loss record doesn’t fully represent its quality of play. Heck, this website wouldn’t exist if we thought the win-loss record said it all.

Over the years, many of the number-crunchers found ways to show how a record can be misleading. This movement started with some simple theories, including those made by another famous Bill. Shortly after leaving the United States Army in the early 1970’s, Bill James began writing baseball articles that focused more on statistical analysis and less on the game’s narrative. It started a revolution that took nearly four decades to gain steam in the Mainstream Sports Media. While many publications are presently giving analysis that goes beyond what James offered to the public, readers who love fundamental sports analytics must respect the Bill James classics. The one that best applies to football is the Pythagorean Win Differential (PWD). While first applied to baseball analysis, this stat was effectively adjusted to use points to assess a football team’s quality of play.

The concept behind this in baseball is quite simple: a team’s record does not always reflect the team’s ability to score runs and limit runs scored. Statistical theory says that run differential is actually a better indicator of quality than the win-loss record. Simply, with more runs scored/allowed tallied than wins/losses, it is more likely that the team “converges” to its mean production (or quality of play) when looking at the run differential. James applied this theory to the Pythagorean Theorem in an attempt to analyze run differential. This created an “expected” record than can be compared to the true record. One can compare the two records to create the Pythagorean Win Differential.

When James created the Pythagorean expectation formula, he used 1.83 as the exponent instead of the 2 used for the true mathematical formula. When using this same mentality for points scored and allowed in football, publications like Pro Football Reference and Football Outsiders switched the exponent to 2.37 to adjust for the mean points scored per team per season. However, these publications haven’t adjusted the exponent to reflect the changes in mean scoring in the NFL in recent years. The TABRankings thus used 2.4 as a more accurate indicator of quality. After these adjustments, there seems to be a strong indicator of which teams overachieve or underachieve relative to their record. Applying this to the 2012 season, the chart below shows the Pythagorean Win Differential for each team.

Table 1: Pythagorean Win Differential, 2012 season

PythW: Pythagorean Wins, or (Points Scored ^ 2.4) / ((Points Scored ^ 2.4) + (Points Allowed ^ 2.4))

Given the numbers and the year-to-year correlation, it looks like there’s some merit to thinking Pythaogrean wins are more telling of a team’s quality than true wins. Let’s compare. The correlation between each team’s 2011 true wins and 2012 true wins was 0.2753, which shows a weak positive relationship. Meanwhile, the correlation between each team’s 2011 Pythagorean wins and 2012 true wins was 0.3138, which shows a slightly stronger positive relationship.

These results suggest that the Pythagorean adjustment could be effective, but it clearly isn’t enough. That’s where other stats that highly impact win-loss record may come into play. Take turnover margin for instance. Teams with the turnover margin advantage finished with a 162-42 regular season record and 8-0 postseason record in 2012 (per Cold, Hard Football Facts). With an approximately 80 percent correlation to victory, this stat definitely plays a big role in the outcomes of games. However, turnovers only account for a generally low percentage of play. For example, NFL teams threw an interception on only 2.63 percent of pass attempts in 2012. Therefore, the turnover can be classified as a play that may overstate the overall quality of a team’s play.

To matters even more interesting, the turnovers don’t tend to correlate well from one year to the next. The mean correlation between takeaways and giveaways from one year to the next was 0.1825 during the 1998-2003 season (per Jim Armstrong). In theory, turnovers only give us a one-year snapshot that overvalues or undervalues a team’s win-loss record.

Here’s why we developed the Turnover Win Impact Regression (TWIr). This stat is designed to estimate the record-based regression the average team will face, given its turnover margin from any given season. Of course, TWIr must must work under the same unit as PWD (re: points). Four points per turnover is the most commonly accepted equivalency (per Hidden Game of Football), so that conversion will be used for this study. That conversion will be combined with the regression of Armstrong’s mean correlation coefficient (re: 0.8175) and a 38 points-per game-differential equivalency (per 2012-scoring-based PWD results) to create the TWIr formula.

Check out the results, as the TWIr formula for the 2012 season is applied to each team.

Table 2: Turnover Win Impact Regression

TWIr: 0.8175 * (“Points” / 38)

Since this the first year, we will have to revisit the regression coefficient in the near future. Perhaps a few years will indicate if we’ve made the nice moves in developing this formula. For now though, we will use PWD – TWIr to create the Expected Win Differential (EWD) for each team in the upcoming 2013 season. In theory, this could lead to a few different scenarios. Some teams may have a bad PWD that is explained a bad turnover margin, which would lead to a neutral EWD. A similar result will occur from a good PWD combining with a good turnover margin. However, in other cases, either PWD or TWIr will single-handedly lead to a notably good or bad EWD. We will make those notably good or bad results determined by the threshold of regression of ±1.454 wins. This is determined by finding 80 percent of one unit’s worth of (PWD + TWIr). Teams above the threshold are denoted in blue, and they will be likely to positively regress in win-loss record in 2013. Teams below the threshold are denoted in red, and they will be likely to negatively regress in 2013.

Check out the table below to see where your team stands! In total, 11 teams are outside the threshold of regression.

Table 3: Expected Win Differential

EWD: PWD – TWIr; Note: The threshold for regression is ±1.454 wins.

For the sake of this study, we will only the monitor how the teams in blue and red fare. We also will monitor how the formula does as well, because there could be some necessary tweaks in the future. (For example, teams tend to regress less on offense than defense. This could mean we will need to weigh regression more in favor of takeaways.) If this formula does well, we will use EWD for each NFL Preview. If not, we will make some adjustments to figure out the best way to predict general scoring-based regression.

A team’s win-loss record normally changes every year. The 2013 season will test to see if EWD shows how those changes will occur.