At the start of the National Football League (NFL) 2014 season, I listened to several ESPN sports commentators talk about Peyton Manning, one of the greatest quarterbacks to ever play in the NFL. The NFL has a formula to rate how quarterbacks do in each game and in the season as a whole based on the percentage of passes completed, average yards gained per throw, percentage of passes that were touchdowns, and percentage of passes that were intercepted. ESPN has an even more complicated system that it calls the Total Quarterback Rating (Total QBR).

Even as great a quarterback as Peyton Manning has his ups and downs. Figure 1 shows Payton’s Total QBR for each of the 16 regular-season games in 2013. The week-to-week fluctuations are a reflection of how much randomness there is in athletic performances. Sometimes, a quarterback is sick or banged up. Sometimes, the play calling works, sometimes it doesn’t. Sometimes a pass is deflected, sometimes it isn’t. Sometimes the ball slips or is dropped. Sometimes, a fumbler’s team recovers the ball; sometimes, it doesn’t. Sometimes, the official throws a flag, sometimes he doesn’t. The commonplace observation, “On any given Sunday, any team can beat any other team,” reflects the reality that there is a lot of randomness in football games.

Even as great a quarterback as Peyton Manning has his ups and downs. Figure 1 shows Payton’s Total QBR for each of the 16 regular-season games in 2013. The week-to-week fluctuations are a reflection of how much randomness there is in athletic performances. Sometimes, a quarterback is sick or banged up. Sometimes, the play calling works, sometimes it doesn’t. Sometimes a pass is deflected, sometimes it isn’t. Sometimes the ball slips or is dropped. Sometimes, a fumbler’s team recovers the ball; sometimes, it doesn’t. Sometimes, the official throws a flag, sometimes he doesn’t. The commonplace observation, “On any given Sunday, any team can beat any other team,” reflects the reality that there is a lot of randomness in football games.

It is not surprising that after Peyton’s worst week, there was nowhere to go but up and that after his best week there was nowhere to go but down. It is unusual that he swung from worst to best between games 11 and 12, but it sometimes does happen.

Manning had an incredible year in 2013, arguably one of the best years in his 15-year career (to that point). He threw 55 touchdown passes with only 10 interceptions. The next closest in touchdowns was Drew Brees with 39. Manning’s Total QBR was 82.9; the next closest ratings were Philip Rivers (71.1) and Drew Brees (70.5).

Looking forward to the 2014 season, the commentators talked about Manning’s age (38), his pass receivers (Demaryius Thomas, Wes Welker, Julius Thomas, and Emmanuel Sanders), the running backs, and the offensive line. They predicted he would throw 48 touchdown passes and 12 interceptions and, once again, be the top NFL quarterback by a wide margin. For those fans who play Fantasy Football, they predicted that Manning would have 368 fantasy points, well above the predictions for Aaron Rodgers (347 points) and Drew Brees (329).

They didn’t talk about luck—about how unusually good or bad performances typically involve some fortune or misfortune. No one said a word about how Manning might have had good more good luck than bad in 2013. The more extreme is the good luck, the less likely it is to be repeated. The more Manning benefited from good luck in 2013, the less likely he was to have so much good luck, or even more, in 2014. Since Manning, at age 37, had one of his best years ever in 2013, I reasoned that good luck must have had a lot to do with it. So, I posted a blog before the 2014 season started titled, “Peyton Manning is Likely to Regress to the Mean.” I ended the blog with this prediction:

Manning had an incredible year in 2013, arguably one of the best years in his 15-year career (to that point). He threw 55 touchdown passes with only 10 interceptions. The next closest in touchdowns was Drew Brees with 39. Manning’s Total QBR was 82.9; the next closest ratings were Philip Rivers (71.1) and Drew Brees (70.5).

Looking forward to the 2014 season, the commentators talked about Manning’s age (38), his pass receivers (Demaryius Thomas, Wes Welker, Julius Thomas, and Emmanuel Sanders), the running backs, and the offensive line. They predicted he would throw 48 touchdown passes and 12 interceptions and, once again, be the top NFL quarterback by a wide margin. For those fans who play Fantasy Football, they predicted that Manning would have 368 fantasy points, well above the predictions for Aaron Rodgers (347 points) and Drew Brees (329).

They didn’t talk about luck—about how unusually good or bad performances typically involve some fortune or misfortune. No one said a word about how Manning might have had good more good luck than bad in 2013. The more extreme is the good luck, the less likely it is to be repeated. The more Manning benefited from good luck in 2013, the less likely he was to have so much good luck, or even more, in 2014. Since Manning, at age 37, had one of his best years ever in 2013, I reasoned that good luck must have had a lot to do with it. So, I posted a blog before the 2014 season started titled, “Peyton Manning is Likely to Regress to the Mean.” I ended the blog with this prediction:

*Peyton Manning’s phenomenal 2013 season surely benefited from more good luck than bad. Defensive players slipping, offensive players not slipping. Defensive players making bad guesses, offensive players making good guesses. Fumbles lost and recovered. Passes caught and dropped. Holding penalties called and not called. The list is very long. Luck—good and bad—is why the best team doesn’t win every game, why player stats go up and down from one game to the next.*

Manning is a Hall-of-Fame quarterback, but 2013 was not a below-average season for him. Manning is surely not as good as he seemed last year, and almost certainly will not do as well this year. You can take that to the bank.

Manning is a Hall-of-Fame quarterback, but 2013 was not a below-average season for him. Manning is surely not as good as he seemed last year, and almost certainly will not do as well this year. You can take that to the bank.

I was right. Not because I know a lot about football, but because I know something about luck and regression to the mean.

Manning regressed to the mean in 2014. Instead of the predicted 48 touchdowns with only 12 interceptions, he had 39 touchdowns and 15 interceptions. Instead of leading the league with 368 fantasy points, Peyton finished fourth with 307 points, behind Aaron Rodgers (342), Andrew Luck (336), and Russell Wilson (312). Peyton’s Total QBR was 77.4 and he finished third, well behind Tony Romo’s 82.7 and Aaron Rodgers’ 82.6.

Peyton didn’t have a bad year. He was still one of the top quarterbacks in the league. But he didn’t have as much good luck as the year before and he didn’t finish in first place.

It’s not just Manning who regresses to the mean. Figure 2 is a scatterplot for the 2013 and 2014 seasons for those players who played both seasons. The average Total QBR increased from 58.6 to 63.5 between 2013 to 2014, so Z-scores are used in Figure 2 to make the data comparable. If there was no regression to the mean, players who did very well one season would be equally likely to do better or worse the next season. the QBRs would be evenly scattered about the 45-degree line drawn through the origin with a slope of one.

Manning regressed to the mean in 2014. Instead of the predicted 48 touchdowns with only 12 interceptions, he had 39 touchdowns and 15 interceptions. Instead of leading the league with 368 fantasy points, Peyton finished fourth with 307 points, behind Aaron Rodgers (342), Andrew Luck (336), and Russell Wilson (312). Peyton’s Total QBR was 77.4 and he finished third, well behind Tony Romo’s 82.7 and Aaron Rodgers’ 82.6.

Peyton didn’t have a bad year. He was still one of the top quarterbacks in the league. But he didn’t have as much good luck as the year before and he didn’t finish in first place.

It’s not just Manning who regresses to the mean. Figure 2 is a scatterplot for the 2013 and 2014 seasons for those players who played both seasons. The average Total QBR increased from 58.6 to 63.5 between 2013 to 2014, so Z-scores are used in Figure 2 to make the data comparable. If there was no regression to the mean, players who did very well one season would be equally likely to do better or worse the next season. the QBRs would be evenly scattered about the 45-degree line drawn through the origin with a slope of one.

There is regression to the mean in that players who did exceptionally well in 2013 tended to be below the line in 2014, while the reverse was true of those who did poorly in 2013. Specifically, all four of the top quarterbacks in 2013 were closer to the mean in 2014, as were three of the four lowest rated quarterbacks.

It is not that the best performing players had their skills deteriorate in 2014 while the worst performing quarterbacks had their skills improve. It is that the best performing players had good luck in 2013 while the worst performing quarterbacks had bad luck.

The role of luck is confirmed by the fact that there is a similar regression going the other direction, from 2014 to 2013, since regression to the mean has nothing to do with players’ abilities evolving over time. It has everything to do with performances fluctuating about abilities because of good and bad luck. Three of the top for quarterbacks in 2014 were closer to the mean in 2013, as were all four of the lowest rated quarterbacks.

The correlation between 2013 and 2014 QBRs is 0.46, which means that a quarterback who is one standard deviation from the mean in either season is predicted to be 0.46 standard deviations from the mean in the other season.

Peyton Manning is human and, like other humans, is susceptible to regression to the mean.

It is not that the best performing players had their skills deteriorate in 2014 while the worst performing quarterbacks had their skills improve. It is that the best performing players had good luck in 2013 while the worst performing quarterbacks had bad luck.

The role of luck is confirmed by the fact that there is a similar regression going the other direction, from 2014 to 2013, since regression to the mean has nothing to do with players’ abilities evolving over time. It has everything to do with performances fluctuating about abilities because of good and bad luck. Three of the top for quarterbacks in 2014 were closer to the mean in 2013, as were all four of the lowest rated quarterbacks.

The correlation between 2013 and 2014 QBRs is 0.46, which means that a quarterback who is one standard deviation from the mean in either season is predicted to be 0.46 standard deviations from the mean in the other season.

Peyton Manning is human and, like other humans, is susceptible to regression to the mean.