The data is about the NFL standings of National Football Conference (NFC) in 2017 – 2018. There are 16 teams in the NFC including Minnesota Vikings, LA Rams, Dallas Cowboys, Green Bay Packers and the defending champions, Philadelphia Eagles. The data set was obtained from http://www.espn.com/nfl/standings/_/season/2017/group/conference. I did not include data for the American Football Conference (AFC) because the Cleveland Browns lost all their games during that season. (But they’ve won one game this season so lets see how the season goes.)

For each team, the number of games won (W), the number of games lost (L), the total number of points scored during the season (PF) and  the total number of points allowed during the season (PA) were collected. The data is shown below;

The Pythagorean formula (described first by Bill James in the context of baseball) relates the number of points scored by a team and the points allowed to the winning percentage. Thus W/L  = (P/PA)^k where k is a constant that is dependent on the particular sport. Taking log, we can express the formula as log(W/L) = k log (P/PA). I created a scatterplot of log(W/L) against log(P/PA) and overlayed the best fitting line to estimate the value of k. I also constructed a plot of the residuals against log(P/PA).

The top panel is the graph of log(W/L) against log(P/PA). The line is the least-squares fit with slope of 3.005. Though the points deviate from the best fitted line, most of the points lie in
a narrow band around the line.

The bottom panel is the graph of residuals. Most of the residuals lie between -0.4 to 0.4. This means, the percent deviations of the actual ratio of wins to losses (or winning percentage) from
the ideal ones range between -40% and 40%. The largest residual is about 0.6068 means the ratio of wins to losses (or winning percentage) for the team, Arizona Cardinals is about 61% larger than the ideal ratio of wins to losses (or winning percentage).  I don’t think there were any teams that were significantly unusual but the Giants scored less touch downs and allowed quite a number of touchdowns against their team.