Math Mutation 58: On Average, Things Are Average You may have heard of the "Sports Illustrated Cover Jinx", the legend that athetes who are featured on the cover of Sports Illustrated tend to suffer a decline in their careers afterwards. Naturally, we all like to think that divine powers are punishing these superstars for their excessive egos. After all, if the universe was fair, it would be *you* on that cover instead, right? But if you look at the careers of the cover atheletes, there really may seem to be some kind of jinx effect: many athletes really do seem to achieve less after their cover appearances. So what's going on here? Actually, the explanation lies in simple mathematics. The phenomenon influencing this situation is known as "regression towards the mean". This is a fancy way of stating that if you take any measurable quantity that has a sizable chance component, and you look at it again after you get a really high value, the next value you see will probably be lower. There's no magic here: *all* values have a higher chance of being close to average, assuming typical random distrubutions, so any given value you take will be most likely to be close to average. So if you get a really high value, the next one will almost certainly be lower-- not because of any type of curse, but simply because on average, the value will be close to average. So what does this mean in terms of athletic performance? Well, I don't want to diminish the drive and effort of individual athletes, and it's certainly the case that superstars are usually very good at their games, with or without a Sports Illustrated cover. But I think we can all agree that there are numerous chance factors in any sports competition: the weather, the winds, the particular unpredictable strategies of the opposing team, the random fumbles that may or may not occur at opportune moments, the distracting yells from fans, or the intriguing mathematical podcasts from the previous day that are still distracting the athletes' thoughts. Together, these add up to a noticeable chance component in any athletic competition, and can add up to a sizable chance factor over a number of games. So, if an athlete has an exceptional season that lands him on the cover of Sports Illustrated, that means that these random components of his performance all happened to line up to create a very high positive "plus factor" adding to the pure-talent part of his performance. Due to regression towards the mean, this plus factor is very unlikely to reach the same high positive level the following season: chances are that it will be close to average, as is the probablility in any given season. Thus, while the athlete does not get any less talented, the random factors are probably not going to add up in his favor during the season after his Sports Illustrated cover. And it will look like the athlete is jinxed-- but actually, all that has happened is that an exceptional streak of luck has not repeated itself. Regression towards the mean can pop up in many areas of your daily life, if you look for it. You probably won't have two car accidents for two days in a row-- while you might credit this fact to consciously improving your driving after the first accident, it may just be that on average days, your bad driving isn't quite bad enough to cause an accident. Following an exceptional score on a test in school, you may be disappointed to find that your following tests are not so hot; maybe your studying slacked off, or maybe you just got lucky that one time. If you feel especially snotty one day and take a homeopathic remedy, you'll probably feel better the next day-- not because homeopathic remedies do anything, but because you're likely to be close to your average health on any given day, unless truly suffering from a serious illness. Whenever you try to analyze some type of cause-effect relationship in your daily life, you should try to think about whether regression towards the mean might be the best explanation. And this has been your Math Mutation for today. References:
  • Regression Towards the Mean at Wikipedia