The Olympic poset

During the Olympics (yeah, I know, you've all forgotten about the Olympics), there was much argument about whether the right thing to do, if we want to determine which country "won the Olympics", is to count the country that got the most gold medals (China), the most medals (the US), or something else -- say a points system that allocates three points for a gold, two for a silver, and one for a bronze.

Simon Tatham has prepared a Hasse diagram of the medal table. The main idea is:

So we want to say that one country has done strictly better than another if the medal score of the latter can be transformed into the former by a sequence of medal additions and medal upgrades.

This gives a partial order on the countries.

Alternatively, we could say country A does strictly better than country B if and only if A gets more points than B under all weighting schemes in which we assign x points for a gold, y points for a silver, and z for a bronze, with x ≥ y ≥ z ≥ 0. This seems like it's equivalent to Tatham's order, but I haven't thought that hard about it.

One could extend this to include the population of a country; the natural order there would be that A did strictly better than B if the medal score of B can be transformed into that of A by a sequence of medal additions, medal upgrades, and population reductions. The idea here, of course, is that if two countries win the same assortment of medals, the one with lower population did better. But going there is dangerous; do we then take into account GDP? Popularity of sports in general in the country? The fact that the particular set of sports in the Olympics is more popular in some countries than others?

Trying to explain the Olympic gymnastics tiebreaker

Nastia Liukin of the USA wins silver on the uneven bars; He Kexin of China wins gold. This is news because the two of them had the same score. I've seen a lot of bad explanations of how the tiebreaker works, and implications that it involves some Big Scary Mathematics.

The way gymnastics scoring currently works is that each contestant receives a score for the difficulty of their routine (I think this is open-ended), called the "A score", which is essentially the sum of the difficulties of the various things they attempted to do. Then six judges give a score out of 10, in multiples of 0.1, for how well they did it; the lowest and highest scores are thrown out and the other four are averaged, and this is the "B score". The two scores are added to give the score for that routine.

Both Liukin and He received 16.725 points -- so they're tied, right? Wrong. The first tiebreaker, in this case, is that the contestant who had the higher A score wins -- which rewards the contestant that attempts a more difficult routine. But both had A score 7.700, B score 9.025.

The impression I got (watching NBC's broadcast last night) is that if there's still a tie, then the B scores given by the four middle judges are looked at individually. In this case, for He the six judges gave 9.3, 9.1, 9.1, 9.0, 8.9, 8.9; for Liukin they were 9.3, 9.1, 9.0, 9.0, 9.0, 8.8. In both cases the middle four scores add up to 36.1. The lowest of these scores (so the second-lowest of the original scores) is thrown out. This leaves 27.2 for He, 27.1 for Liukin, so He wins. See the tiebreaker page at the official Beijing Olympics site; there's no explanation here, but he various numbers shown there seem to bear it out. Note that instead of reporting a score of x, they sometimes use 10-x, which is the number of points deducted from the highest possible B score, which is 10.0. This explains the phrasing in some sources that refers to an "average of deductions".

I'm not sure what the logic behind this is. At first I thought that it rewarded inconsistency -- the competitor who has their scores more tightly clustered will probably have a higher second-lowest score. But this isn't the right interpretation, because the scores weren't received on different routines, but on different people's measurements of the same routine -- so does the tiebreaker reward having a routine which is hard to score? Also, it was stated many times that there are no ties in the current scoring system, but what would have happened had He and Liukin received identical scores from each judge?

The math here isn't that hard; I think the big flaw was that nobody seemed to know what the rules were.

Variance in Olympic events

It's often claimed that the reason that there are many more men than women in certain academic disciplines (mathematics is one, but that's not the point of this post) is not that men and women have different mean abilities, but rather that the standard deviation of male ability is larger than the standard deviation of female ability. (Of course, it is unwise to espouse these views publicly, for political reasons; that's what got Larry Summers in a lot of trouble.)

It occurs to me, having watched lots of the Olympics in the last few days, that something similar might be true in athletic events. I'm not claiming that men and women are physically identical (I'm not blind), or that their average performance in physical feats is the same. But it may be the case that the difference between the very best men and the very best women in physical feats (say, times in some sort of race, because these are the most easily quantified) is larger than the difference between the average man and the average woman, because there could be more variance among men than women.

Is there any evidence for this? I'm obviously not a student of this sort of thing (in fact, I don't even know what "this sort of thing" is called, although it's clearly some subfield of biology or medicine).

Oh, and Jordan Ellenberg wrote an explanation of why the new gymnastics scoring system is good. I'm glad he did, because I'd had a feeling it was better than the old system but was having trouble articulating why.

Lucky babies redux

Two babies born at 8:08 am on 8/8/08, weighing eight pounds, eight ounces, both in the United States.

How many would you expect?

The 2007 crude birth rate for the US is 14.2 per 1000, per year; the estimated US population is 304,843,316. The product of these is 4,328,775 births per year, or 8.25 births per minute.

From here I can find the distribution of birth weights (in Norway, 1992-1998 -- better figures would be appreciated). About six percent of babies weigh between 3850 and 3950 grams, which is a 3.5-ounce-wide interval; thus about 6%/3.5 = 1.7% of babies weigh 8 pounds, 8 ounces (to the nearest ounce) at birth.

So the expected number of babies born in the US at that particular minute, at that weight, is about 1.7% of 8.25, or 0.14.

There were two. The probability of this happening, assuming births are a Poisson process, is about one in 112. I wouldn't trust this number too much, because birth weights are supposedly growing with time and the Norwegian distribution is probably different from the US distribution.

So if I had to guess, people at the hospitals are fudging the numbers; if we were being totally honest, those babies might turn out to have been born at 8:09 am and weighed eight pounds, seven ounces, or something like that. Not that there's anything wrong with that.

(This post borrows a lot from a post I just remembered I made, lucky babies, about babies born on July 7, 2007 at 7:07 and weighing seven pounds, seven ounces -- but those were fictional babies.)

Big numbers confuse the New York Times

From Chinese basketball builds towards podium (August 9):

"With 300,000 million people playing basketball across the country — roughly the same number as the population of the United States..."

Really? I mean, it's almost a cliche by this point that There Are Lots Of People In China -- but I didn't know there were quite that many.

Things the Olympics broadcast won't tell you

Things we're not hearing about during coverage of Olympic swimming: apparently the construction of the "Water Cube" (the Beijing National Aquatics Center) is based on the Weaire-Phelan sturucture. More specifically, the Weaire-Phelan structure is apparently the best known solution to the problem of partitioning space into cells of equal volume with minimal suface area. The edges of the cells in this structure make up the steel frame of the building; in order to make a more "organic"-looking pattern the pattern was sliced at an oblique angle.

Somehow I missed this in the New York Times on Tuesday. See also the Guardian from 2004 and Science News a few weeks ago.