Primer – What is a Pythagorean expectation?

Pythagorean expectation is the idea that you can calculate a team’s winning percentage based solely on its for and against. It originated with baseball nerds but, according to its Wikipedia article, has been adapted for other sports. It is also where the name of this site came from. Pythago is basically what Pythagoras would have been if he had been Australian.

To rip straight from Wikipedia –

“The basic formula is:

pythagorean expectation formula

where Win Ratio is the winning ratio generated by the formula. The expected number of wins would be the expected winning ratio multiplied by the number of games played.”

The principle is the same for the NRL but instead of squaring the numbers, we use an exponent of 1.9 because it’s more accurate than using 2.

Let’s take an example. By the end of the 1999 season, Western Suburbs Magpies had scored 285 points and conceded a massive 944. On that basis, the expected win ratio is calculated as follows:

WR = PF^1.9 / (PF^1.9 + PA^1.9)

WR = 285^1.9 / (285^1.9 + 944^1.9)

WR = 9.3%

Over a 24 game season, this equates to a Pythagorean expectation of 2.2 wins. The Magpies actually won 3 games that year, so Pythagoras either slightly underestimated them or they outperformed – and I use the term in the loosest possible sense.

Now the error is important because it tells us how accurate this method of forecasting is. From the sample of 300 for-and-against-s since 1998, Pythagoras had 140 expectations that were within one game of the team’s actual number of wins. Only a handful of expectations (13 in fact) were out by more than three games from the actual number of wins.

Overall, the Pythagorean expectation is reasonably accurate. The mean absolute error at the end of the season is 1.3 games or around 5%. This means that Pythagoras is, on average, off by about 1.3 games for each team. Some of that is due to the fact that the formula generates part-wins (decimals) and a team can only win in whole numbers, so there will always be a bit of discrepancy. This error will vary a bit from season to season. For example, the error comparing Pythagoras to the mean absolute error was 1.55 or about 6%, so a little higher than we would normally get, but still fairly accurate.

The main takeaway is that you can estimate a team’s number of wins based on their for and against with a reasonable degree of accuracy.