The average North Korean mathematician

This is a link post for https://www.telescopic-turnip.net/essays/the-average-north-korean-mathematician/

Here are the top-fifteen countries ranked by how well their teams do at the International Math Olympiads:

When I first saw this ranking, I was surprised to see that North Koreans have such an impressive track record, especially when you factor in their relatively small population. One possible interpretation is that East Asians are just particularly good at mathematics, just like in the stereotypes, even when they live in one of the world's worst dictatorships.

But I don't believe that. In fact, I believe North Koreans are, on average, particularly bad at math. More than 40% of the population is undernourished. Many of the students involved in the IMOs grew up in the 1990s, during the March of Suffering, when hundreds of thousands of North Koreans died of famine. That is not exactly the best context to learn mathematics, not to mention the direct effect of nutrients on the brain. There does not seem to be a lot of famous North Korean mathematicians either (there is actually a candidate from the North Korean IMO team who managed to escape during the 2016 Olympiads in Hong-Kong. He is now living in South Korea. I wish him to become a famous mathematician). Thus, realistically, if all 18 years-old from North Korea were to take a math test, they would probably score much worse than their South Korean neighbors. And yet, Best Korea reaches almost the same score with only half the source population. What is their secret?

This piece on the current state of mathematics in North Korea gives it away:

“The entire nation suffered greatly during and after the March of Suffering, when the economy collapsed. Yet, North Korea maintained its educational system, focusing on the gifted and special schools such as the First High Schools to preserve the next generation. The limited resources were concentrated towards gifted students. Students were tested and selected at the end of elementary school.” 

In that second interpretation, the primary concern of the North Korean government is to produce a few very brilliant students every year, who will bring back medals from the Olympiads and make the country look good. The rest of the population's skills at mathematics are less of a concern.

When we receive new information, we update our beliefs to keep them compatible with the new observations, doing an informal version of Bayesian updating. Before learning about the North Korean IMO team, my prior beliefs were something like most of the country is starving and their education is mostly propaganda, there is no way they can be good at math. After seeing the IMO results, I had to update. In the first interpretation, we update the mean – the average math skill is higher than I previously thought. In the second interpretation, we leave the mean untouched, but we make the upper tail of the distribution heavier. Most North Koreans are not particularly good at math, but a few of them are heavily nurtured for the sole purpose of winning medals at the IMO. As we will see later in this article, this problem has some pretty important consequences for how we understand society, and those who ignore it might take pretty bad policy decisions.

But first, let's break it apart and see how it really works. There will be a few formulas, but nothing that can hurt you, I promise. Consider a probability distribution where the outcome $x$ happens with probability $p\left(x\right)$. For any integer $n$, the formula below gives what we call the $n$th moment of a distribution, centered on $\mu$.

To put it simply, moments describe how things are distributed around a center. For example, if a planet is rotating around its center of mass, you can use moments to describe how its mass is distributed around it. But here I will only talk about their use in statistics, where each moment encodes one particular characteristic of a probability distribution. Let's sketch some plots to see what it is all about.

First moment: replace n with 1 and μ with 0 in the previous formula. We get

which is – suprise – the definition of the mean. Changing the first moment just shifts the distribution towards higher or lower values, while keeping the same shape.

Second moment: for n = 2, we get

If we set μ to be (arbitrarily, for simplicity) equal to the mean, we obtain the definition of the variance! The second moment around the mean describes how values are spread away from the average, while the mean remains constant.

Third moment (n = 3): the third moment essentially describes how skewed (asymmetric) the distribution is.

Fourth moment (n = 4): this describes how leptokurtic or platykurtic your distribution is, that is, how extreme the extreme values are.

You could go on to higher n, each time bringing in more detail about what the distribution really looks like, until you end up with a perfect description of the distribution. By only mentioning the first few moments, you can describe a population with only a few numbers (rather than infinite), but it only gives a simplified version of the true distribution, as on the left graph below:

Say you want to describe the height of humans. As everybody knows, height follows a normal distribution, so you could just give the mean and standard deviation of human height, and get a fairly accurate description of the distribution. But there is always a wise-ass in the back of the room to point out that the normal distribution is defined over $R$, so for a large enough population, some humans will have a negative height. The problem here is that we only gave information about the first two moments and neglected all the higher ones. As it turns out, humans are only viable within a certain range of height, below or above which people don't survive. This erodes the tails of the distribution, effectively making it more platykurtic (If I can get even one reader to use the word platykurtic in real life, I'll consider this article a success).

Let's come back to the remarkable scores of North Koreans at the Math Olympiads. What these scores teach us is not that North Korean high-schoolers are really good at math, but that many of the high-schoolers who are really good at math are North Koreans. On the distribution plots, it would translate to something like this:

With North Koreans in purple and another country that does worse in the IMOs (say, France), in black. So you are looking at the tails and try to infer something about the rest of the distribution. Recall the plots above. Which one could it be?

Answer: just by looking at the extreme values, you cannot possibly tell, because any of these plots would potentially match. In Bayesian terms, each moment of the distribution has its own prior, and when you encounter new information, you could in principle update any of them to match the new data. So how can we make sure we are not updating the wrong moment? When you have a large representative sample that reflects the entire distribution, this is easy. When you only have information about the top 10 extreme values, it is impossible. This is unfortunate because the extreme values are precisely what gets all our attention – most of what we see in the media is about the most talented athletes, the most dishonest politicians, the craziest people, the most violent criminals, and so forth. Thus, when we hear new information about extreme cases, it's important to be careful about which moment to update.

This problem also occurs in reverse – in the same way looking at the tails doesn't tell you anything about the average, looking at the average doesn't tell you anything about the tails. An example: on a typical year, more Americans die from falling than from viral infections. So one could argue that we should dedicate more resources to prevent falls than viral infections. Except the number of deaths from falls is fairly stable (you will never have a pandemic of people starting to slip in their bathtubs 100 times more than usual). On the other hand, virus transmission is a multiplicative process, so most outbreaks will be mostly harmless (remember how SARS-cov-1 killed less than 1000 people, those were the days) but a few of them will be really bad. In other words, yearly deaths from falls have a higher mean than deaths from viruses, but since the latter are highly skewed and leptokurtic, they might deserve more attention. (For a detailed analysis of this, just ask Nassim Taleb.)

There are a lot of other interesting things to say about the moments of a probability distribution, like the deep connection between them and the partition function in statistical thermodynamics, or the fact that in my drawings the purple line always crosses the black like exactly $n$ times. But these are for nerds, and it's time to move on to the secret topic of this article. Let's talk about SEX AND VIOLENCE.

This will not come as a surprise: most criminals are men. In the USA, men represent 93% of the prison population. Of course, discrimination in the justice system explains some part of the gap, but I doubt it accounts for the whole 9-fold difference. Accordingly, it is a solid cultural stereotypes that men use violence and women use communication. Everybody knows that. Nevertheless, having just read the previous paragraphs, you wonder: “are we really updating the right moment?”

A recent meta-analysis by Thöni et al. sheds some light on the question. Published in the journal Pyschological Science, it synthesizes 23 studies (with >8000 participants), about gender differences in cooperation. In such studies, participants play cooperation games against each other. These games are essentially a multiplayer, continuous version of the Prisoner's Dilemma – players can choose to be more or less cooperative, with possible strategies ranging from total selfishness to total selflessness.

So, in cooperation games, we expect women to cooperate more often than men, right? After all, women are socialized to be caring, supportive and empathetic, while men are taught to be selfish and dominant, aren't they? To find out, Thöni et al aligned all of these studies on a single cooperativeness scale, and compared the scores of men and women. Here are the averages, for three different game variants:

This is strange. On average, men and women are just equally cooperative. If society really allows men to behave selfishly, it should be visible somewhere in all these studies. I mean, where are all the criminals/rapists/politicians? It's undeniable that most of them are men, right?

The problem with the graph above is that it only shows averages, so it misses the most important information – that men's level of cooperation is much more variable than women's. So if you zoom on the people who were either very selfish or very cooperative, you find a wild majority of men. If you zoom on people who kind-of cooperated but were also kind-of selfish, you find predominantly women.

As I'm sure you've noticed, the title of the Thöni et al paper says “evolutionary perspective”. As far as I'm concerned, I'm fairly skeptical about evolutionary psychology, since it is one of the fields with the worst track record of reproducibility ever. To be fair, a good part of evpsych is just regular psychology where the researchers added a little bit of speculative evolutionary varnish to make it look more exciting. This aside, real evpsych is apparently not so bad. But that's not the important part of the paper – what matters is that there is increasingly strong evidence that men are indeed more variable than women in behaviors like cooperation. Whether it is due to hormones, culture, discrimination or cultural evolution is up to debate and I don't think the current data is remotely sufficient to answer this question.

(Side note: if you must read one paper on the topic, I recommend this German study where they measure the testosterone level of fans of a football team, then have them play Prisoner's Dilemma against fans of a rival team. I wouldn't draw any strong conclusion from this just yet, but it's a fun read.)

The thing is, men are not only found to be more variable in cooperation, but in tons of other things. These include aggression, exam grades, PISA scores, all kinds of cognitive tests, personality, creativity, vocational interests and even some neuroanatomical features. In the last few years, support for the greater male variability hypothesis has accumulated, so much that it is no longer possible to claim to understand gender or masculinity without taking it into account.

Alas, that's not how stereotyping works. Instead, we see news report showing all these male criminals, and assume that our society turns men into violent and selfish creatures and call them toxic [Here is Dworkin: “Men are distinguished from women by their commitment to do violence rather than to be victimized by it. Men are rewarded for learning the practice of violence in virtually any sphere of activity by money, admiration, recognition, respect, and the genuflection of others honoring their sacred and proven masculinity.” Remember – in the above study, the majority of unconditional cooperators were men]. Internet people make up a hashtag to ridicule those who complain about the generalization. We see all these male IMO medalists, and – depending on your favorite political tradition – either assume that men have an unfair advantage in maths, or that they are inherently better at it. The former worldview serves as a basis for public policy. The question of which moment to update rarely even comes up.

This makes me wonder whether this process of looking at the extremes then updating our beliefs about the mean is just the normal way we learn. If that is the case, how many other things are we missing?

19 comments

Comments sorted by top scores.

p_max_bivariate_montecarlo <- function(n,r,iters=60000) {
    library(MASS) # for 'mvrnorm'
    mean(replicate(iters, {
                sample <- mvrnorm(n, mu=c(0,0), Sigma=matrix(c(1, r, r, 1), nrow=2))
                which.max(sample[,1])==which.max(sample[,2])
                })) }

find_r_for_topp <- function(n, topp_target=0.5) {
  r_solver <- function(r) {
     topp <- p_max_bivariate_montecarlo(n, r)
     return(abs(topp_target - topp))
   }
  optim(0.925, r_solver, method="Brent", lower=0, upper=1)$par
  }
library(parallel); library(plyr) # parallelism
rs <- ldply(mclapply(2:193, find_r_for_topp))$V1
# c(0.0204794413, 0.4175067131, 0.5690806174, 0.6098019663, 0.6994770020, 0.7302042200, 0.7517989571, 0.7652371794, 0.7824824776, 0.7928299227, 0.7911903664, 0.8068905240, 0.8177673342, 0.8260679686, 0.8301939461, 0.8258472869, 0.8314810573, 0.8457114147, 0.8477265340, 0.8599239760, 0.8541010795, 0.8539345369, 0.8578597015, 0.8581440013, 0.8584451493, 0.8612079626, 0.8640382310, 0.8693895810, 0.8681881832, 0.8540880634, 0.8688769562, 0.8734774025, 0.8762371597, 0.8737293740, 0.8791205385, 0.8798232797, 0.8780001174, 0.8813006544, 0.8795942424, 0.8809224752, 0.8789012597, 0.8826072026, 0.8820106235, 0.8833360963, 0.8880434608, 0.8865534542, 0.8860206658, 0.8909726985, 0.8918581133, 0.8896068426, 0.8931125753, 0.8915826504, 0.8881211032, 0.8860882133, 0.8857084275, 0.8962766690, 0.8921903730, 0.8942188090, 0.8969666799, 0.8926138586, 0.8971690171, 0.8946804108, 0.8973194094, 0.8942509678, 0.8999695035, 0.8965944860, 0.8961380935, 0.8940129777, 0.9032449177, 0.9008863181, 0.9032217868, 0.9005629127, 0.9020274591, 0.8959058553, 0.9021526115, 0.9039115040, 0.9011588080, 0.9035249155, 0.9017018519, 0.9055311291, 0.9050712304, 0.9090986369, 0.9102189075, 0.9058648333, 0.9062347968, 0.9036232208, 0.9098300563, 0.9104166481, 0.9082378601, 0.9097509415, 0.9072401723, 0.9110669707, 0.9097015650, 0.9095392911, 0.9104547321, 0.9109965730, 0.9105344751, 0.9113974777, 0.9098016391, 0.9108745395, 0.9096074058, 0.9101558716, 0.9114150600, 0.9098197705, 0.9140866653, 0.9110598057, 0.9098305291, 0.9126945140, 0.9116794250, 0.9098304525, 0.9162597410, 0.9138880049, 0.9166744242, 0.9115174937, 0.9098300563, 0.9137427958, 0.9154025570, 0.9098300563, 0.9153743094, 0.9121638454, 0.9098300563, 0.9124202538, 0.9150891460, 0.9155692284, 0.9154097048, 0.9148239514, 0.9135391377, 0.9134265701, 0.9184868581, 0.9155030511, 0.9160840080, 0.9156142020, 0.9180363741, 0.9133847724, 0.9178412895, 0.9164848154, 0.9185051043, 0.9186443572, 0.9163631983, 0.9067252079, 0.9171935358, 0.9068669658, 0.9172083988, 0.9216221015, 0.9173032657, 0.9161656322, 0.9193769687, 0.9196134184, 0.9189703040, 0.9168335043, 0.9208238293, 0.9176496818, 0.9177692888, 0.9193447026, 0.9083813817, 0.9171593478, 0.9207227165, 0.9215861226, 0.9094130225, 0.9197835707, 0.9175185705, 0.9207226893, 0.9213173454, 0.9211625233, 0.9187349438, 0.9094856342, 0.9218536229, 0.9213765908, 0.9216097564, 0.9215764567, 0.9098389885, 0.9098265564, 0.9217230988, 0.9219802481, 0.9226050491, 0.9174997507, 0.9098423672, 0.9208316851, 0.9219666398, 0.9213117029, 0.9227359249, 0.9107645063, 0.9217438628, 0.9225905693, 0.9220370631, 0.9259721234, 0.9225535447, 0.9249239773, 0.9256348191, 0.9232228035, 0.9101015711, 0.9253350470)
library(ggplot2)
qplot(1:192, rs) + coord_cartesian(ylim=c(0,1)) + ylab("Necessary bivariate correlation (Pearson's r)") + xlab("Population size") + ggtitle("Necessary correlation strength for ~50% chance of double-max on 2 correlated variables (Monte Carlo)") + theme_linedraw(base_size = 24) + geom_point(size=4)