Monday, February 22, 2010

Hitting the right wall in Vancouver?

I'm sure that many of you, as I did, tuned in Sunday night to watch the Canada-USA hockey game at the 2010 Vancouver Olympics. On a weekend when our athletes were expected to take home between five and nine medals, but finished with only one, the loss suffered by our team seemed symptomatic of a dream slipping away. This wasn't supposed to happen. With improving medal counts since 1980, and over $100 million spent on the best training money could buy, this was supposed to be our year. It was to be as much of a coming out party as the Beijing games were to China in 2008. But instead of enjoying a victory celebration, we're in fourth place with a week still to go. Many reasons for the result to date have been suggested. Perhaps the extra pressure put on the athletes by the home crowd is causing an abnormally high choke rate? Perhaps our team is relying on too many veterans.

But when you really get down into the numbers, could a paradoxical conclusion spring up, indicating that Canada's athletes are performing like never before? Stephen J. Gould in his masterpiece "Full House" discusses how the death of .400 hitting in baseball actually indicates an increase in excellence in the sport. Thus, could the results to date indicate that for our athletes, it's the so-called "right wall" of human performance that is keeping us from owning the podium?

To give you some context by what I mean by a "right wall," I need to take you back to high school. We all remember the bell-curve, shown below, which is used to distribute grades from a test with frequency on the y-axis and score on the x-axis.

Scientists call this kind of curve a natural distribution, because it's the distribution of values that arises in many systems in nature. On the graph there are two "walls." A score of 0 is the left wall, since no value lower then 0 can be obtained on the test. Similarly, a score of 100 is the right wall, because, no matter how smart, no one can score higher. The reason that teachers curve grades is that for an ideal test with infinite test takers, the marks of a class will be distributed along this normal curve, as each individual within the class has slightly different abilities and study habits.

In this test, there are no zeros (which indicate the test is too hard) and no hundreds (which indicate the test is too easy) given out and the peak of the curve is well away from either wall. The reason for this is that all students have been taught in the subject matter, and there is a typical amount of material that is remembered with some remembering less and others remembering more.

But what happens in situations close to one of the walls? For instance, if we were to consider the windspeed in Toronto, you could say with confidence that the winds are often light, but rarely zero and rarely stormy. Since the typical, or most likely windspeed is small compared with the storms, the distribution becomes distorted by the presence of the left wall :

Note also in this illustration the presence of what is called, in statistics, the long tail, trailing off to the right, which shows the likelihood of extreme, but infrequent events, such as hurricanes. As the wind speed gets greater, the frequency of that event becomes less and less. For instance, we might have a fairly windy day each week, a few bad storms a year, some remarkable storms every few years, and a devastating event (such as hurricane hazel in 1954) once in a lifetime.

Is there a right wall when it comes to wind? It turns out that there is a limit to how much energy the sun can pump into the winds, and thus there is a maximum wind speed:

This gets really interesting when we consider the problem of watching the winds for a specific period of time. The longer we watch the winds, (a day, a week, a year, a lifetime) the higher the record wind speed we will observe. In fact, over time, the record wind speed that we have observed up to that time will tend to approach or asymptote to the right wall which indicates the maximum value (shown in red):

What makes this concept applicable to athletics is that there is a maximum limit, or "right wall" to human performance as well. No one will run the 100m in a second, or jump over a bar 100 meters high, at least not without significant surgery or bioengineering. But progressively, athletes (who are the extreme values for our human distribution) will approach the limits of their sports. As they do, successive records will become closer and closer to the last record, and the competitive edge of each elite athlete over their peers gets smaller. As mentioned above, Stephen J. Gould invokes this principle in his book "Full House" to describe how the death of .400 hitting in baseball paradoxically implies that today's batters are the best that have ever played the game precisely because the possible edge to be had over their rivals is so small. Thus over time, it becomes more difficult to have a dominating performance.

But how does an athlete approach the right wall? They get there in two ways. First, they are highly trained. Second, most athletes possess an innate skill in their event: they have faster reflexes that allow a faster start, have a body type well suited to the motion of a sport, and so on. While any athlete can improve with training, most of us will never beat Kristina Groves on ice, no matter how hard we work at it. In fact, many of you may remember that in 2008, as much attention was focused on Michael Phelps' physical proportions as on his training regimen. You can't teach that. Finally, as far as the winter Olympics are concerned, the level of sophistication of equipment can make a huge difference, pushing out the right wall (see the discussion of the implications of the introduction of the clap skate in long-track speed skating below).

The first of these factors, better training, can be influenced by national governments and Olympic committees. The second aspect, increased innate ability, can also be improved by making the sport more visible and therefore increasing the pool of people from which the athletes are drawn. This happens because in a bigger pool of people, like when you watch the wind for longer, the existence of extreme cases become more likely.

Unfortunately, this visibility is limited by the size of the country. Thus, with equal training and visibility you would expect all countries to perform roughly in proportion to the size of their populations. Instead what we find is that some countries perform out of proportion. If we look just at the countries that won medals in 2006, they have a combined population of 2.42 billion (1.12 billion neglecting china). Thus, neglecting China, Germany (7.2% pop, 11.5% medals) and Canada (3.0% pop 9.5% medals) over performed significantly, while the USA (26.8% pop, 9.9% medals) and Russia (12.5% pop, 8.7% medals) under performed significantly. Part of the reason for this imbalance is that in a country like the USA, a large portion of the population is found in warmer areas, so the effective size of the pool is not limited by overall population, but instead by population located in cold areas. Similarly, in poorer countries, the pool is also reduced by a lack of well-funded sporting programs (particularly for the Winter Olympics due to a higher dependence on expensive equipment and tracks).

Other small countries succeed by choosing to specialize. The Netherlands is a good example where speed skating is king. The dutch have now won over 100 medals and have an advantage relative to their population as a result (1.4% pop, 3.6% medals).

But what about Canada? Well, an important observation (made by Brian Williams of CTV) is that of the 24 medals won by Canada in Turin, 18 were won by women. Reduce their contribution to that of the men (6 medals each) and we get down to something like the population proportion with 4.7% of the medals. Furthermore, history has shown that, for social reasons, the approach to the right wall for men occurs before the approach for women (see Gould's "Full House" for the example of the marathon). However, in Canada, women in amateur sport are trained as well as the men. Could they therefore have had an advantage over other countries?

To examine this factor, let's take a look at long-track speed skating since it is one of the few events which has a long history at the Olympics and whose form has remained more or less consistent with time (as opposed to alpine events which are staged on a different course each Olympics). Fully half of Canada's medals in 2006 came from this classification (long and short track). Thus it makes some sense to look at the progression of world records in a longer and a shorter length for both genders. Here each is shown relative to the current world record:

The curve shows several relative plateaus (which indicate approaching the right wall) followed by jumps. Many of these correspond to changes in technology which have extended human capability. For instance, the 1997 introduction of the clap skate, which resulted in a 5% improvement in times for both men and women. However, while women were making larger strides until the later 1970s, since then their progress has been similar to the progress made by the men. Thus we can only conclude that, at least in speed skating, immaturity of the sport cannot be to blame.

But looking at the progression of world records does not tell the whole story, as it increases the influence of outliers. One sparkling performance that breaks a world record will bias that metric for years to come. This is the error made by Paul Kedrosky ( who argues that the lack of a plateau in the men's 100m dash world record progression indicates the presence of doping or some other chemical edge that moves the right wall further out. Thus it makes more sense to look at the performance of larger samples over time to detect the trends within the elite level of a sport.

A good choice here is the single-distance world championships which were run between 1996 and 2008. For these, I have tabulated the 1000m and 5000m (since the men do not run the 3000m here) and averaged the top ten finishers:

Again the Women and Men hang together, with the introduction of the clap skate being the largest change. The bouncing around of the recent value is an altitude effect which can make up to 2 seconds of difference. It is possible in this figure to appreciate better the plateauing effect hidden within the world record progression. Thus, it seems that the sport is fairly mature.

However, this last measure also makes one important omission, that of the variance between the top ten. If the variance is small, then anyone on any given day can win. If the variance is large, then there is a clear advantage to peaking at the event. Those who do can win many more medals then chance would predict.

What we find here is that for Women, particularly in the longer distances, prior to 2005, is that there was a greater difference between the top ten then for the men. This could have contributed to the ability of a particular Woman to dominate the event in previous Olympics in a way that is more difficult to do today. Surprisingly, this effect is not seen to the same extent in the shorter distances.

Adding to this, is the availability of medals in speed skating, as opposed to other sports. A dominating moguls skier takes home but one medal, a dominating hockey player could generate none, if not surrounded by others of their caliber. But for a dominating speed skater, 5 or more is a definite possibility.

Combine these factors together along with three others. First, the small-number statistics of events in which you get but one chance to perform. Second, the third place cut-off of medaling which hides the true distribution (and can seriously affect the medal count of a country with abundant top 5 finishes but few top 3). Thirdly, the vast increase in the number of medals awarded since 1980, when Canada's total began to increase.

Altogether, this suggests that the 2006 medal count may have been a bit high, while the 2010 medal count a bit low. But both represent a solid performance and a continuing improvement on the average, perhaps even an over-achievement for such a small country. This priciple is further reinforced by below-expected achievement during the first week and the above-expected achievement of the last few days. On the whole, it seems that our athletes are up there at the right wall of human performance, and the world has now, largely, joined them. This is something to be proud of, and to look forward to in future competitions.

*Update: Debra Black of the Toronto Star makes an excellent point that, unlike for men who dominate professional sport, the olympics is the biggest athletic stage for canadian women, and thus may attract more top female athletes.

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