Statistical analysis in sports is getting more and more powerful all the time. Sports handicappers who are aware of the stats that are available and how they can be helpful in the handicapping process have a clear and obvious edge over others. Baseball stats guru Bill James is one of the biggest driving forces in the development of more potent statistical analysis. One of his concepts that has taken hold in baseball and is being used increasingly in basketball as well is Pythagorean expectation. It can be a powerful tool for handicappers.

In baseball terms, Pythagorean expectation is the number of games a team should have won – their win percentage – based on the number of runs that they have scored and allowed. The base formula that James developed is:

**Win percentage = (runs scored x runs scored) / ((runs scored x runs scored) + (runs allowed x runs allowed))**

or, in simpler terms:

**Win percentage = runs scored ^{2} / runs scored^{2} + runs allowed^{2}**

Incidentally, the name Pythagorean expectation sounds fancy and intimidating, but it apparently comes from the formula looking a bit like the Pythagorean formula we all learned in school.

So, why is this valuable for sports handicappers? By comparing the expected number of wins from the formula to the number of wins a teams actual wins we can potentially learn about the team. If the teams Pythagorean expectation is significantly lower than the number of wins they have then the team has enjoyed a fair bit of luck in the season. Luck doesn’t always hold up, so you could expect the team to lose games at a higher rate going forward. On the other hand, if the team’s Pythagorean expectation is much higher than their actual win percentage then they aren’t getting the breaks that they deserve base on their production. If they maintain their production levels then you could reasonably expect their win percentage to improve. In short, Pythagorean expectation is a quick way to assess if a team’s record is an accurate assessment of their play. The formula isn’t a crystal ball, so relying on it too heavily is a bad idea. It’s doesn’t provide clear answers, but it is a great, quick way for sports handicappers to spot major variations between what is happening and what should be happening. Once you spot those variations you can try to explain them and then see if that explanation leads to a betting opportunity.

Baseball statisticians have refined the formula in an attempt to make it more valuable and accurate. Much of the focus of that work has been to try to determine what the best exponent to use in the Pythagorean formula would be. You can check that out if you are interested, but it isn’t really necessary if you are looking to use Pythagorean expectation as a crude handicapping tool.

Pythagorean expectation is making its way into basketball betting as well. The first to apply to use the formula for basketball was Dean Oliver, a statistician who has turned his passion into front office jobs withe the Supersonics and Nuggets. Incidentally, Oliver’s book Basketball on Paper is a must read book for serious handicappers.

Pythagorean expectation in basketball works much like it does in baseball – points for and points against instead of runs scored and runs allowed. Because there are so many more points scored in basketball, though, the challenge with the the formula is determining what the exponents should be. When Oliver originally used the formula he used 14 as his exponent. John Hollinger, another famous basketball stats freak, uses 16.5. Both of those formulas refer to the NBA. The difference between 14 and 16.5 as exponents is significant, but because of the nature of the formula the difference between the outcomes is reasonably small. Tweaking the exponents is done just to try to refine the win percentage that is produced – to make it more accurate and therefore useful to the sports handicapper.

Pythagorean expectation is most commonly used from a handicapping perspective in basketball through the work of Ken Pomeroy in college basketball. Pomeroy’s ratings and statistics are heavily used by college basketball handicappers and oddsmakers. When Pomeroy applies the standard formula to college basketball he uses an exponent between 8 and 9.

Pomeroy doesn’t use the standard formula exactly, though. Instead, he uses adjusted offensive and defensive efficiencies in the place of points scored and points allowed. Offensive efficiency is the points a team scored per 100 possessions, and the defensive efficiency is the points they allow per 100 possessions. Adjusted offensive efficiency is when that efficiency has been changed so that it reflects how the team would perform against an average opponent on a neutral court. To further complicate things in an effort to produce better results, Pomeroy also weights his efficiencies so that more recent games are given more weight than those at the beginning of the season. When he uses adjusted efficiencies he uses 11.5 as the exponent.

Because of the refinements that Pomeroy has made to the Pythagorean expectation he is able to use the concept to produce the likelihood of each team winning when two teams play and the predicted final score. His approach is a long way from what James originally started with, but it is a powerful example of why statistics, when properly applied, can be so useful for sports handicappers.