Another Demonstration that Smaller is Better. And Worse Too.

Small countries, like small schools, and better and worse

No one can master the substance of every single public policy area, but it is possible to learn analytic rules that can be applied to almost any issue. I will be forever grateful to Wainer and Zwerling for teaching me one such rule, which jumped into my mind when I read Christopher Caldwell’s analysis of the economics of small countries.

Caldwell’s case for smaller countries having economic advantages rests on his noting — accurately — that small countries are well-represented at the top of the list of per capita GDP. He cites Qatar, Norway, Brunei, Singapore and Luxemborg as examples of top-ranking countries which have small populations.

Enter Wainer and Zwerling, who documented how many educational reformers came to believe that small schools are best because they often appeared at the top of rankings (e.g., on reading scores). But what these reformers forgot to do was look at the bottom of the same lists, where small schools were also over-represented. The reason is not substantive but statistical: Small samples are more prone to extreme scores.

Caldwell triggered my “Wainer and Zwerling reflex”, which led me to investigate further. Guess what? Many of the countries at the bottom of the list of GDP per capita also have small populations: Central African Republic, Burundi, Liberia and Togo for example.

I don’t intend this observation to invalidate Caldwell’s whole article, which has some good points to make. Consider it more an endorsement of an analytic tactic for critical reading: When someone tells you that the small are clustered at the top of some ranking list, check out the bottom of the list before being convinced that something substantive is at work.

Author: Keith Humphreys

Keith Humphreys is the Esther Ting Memorial Professor of Psychiatry at Stanford University and an Honorary Professor of Psychiatry at Kings College London. His research, teaching and writing have focused on addictive disorders, self-help organizations (e.g., breast cancer support groups, Alcoholics Anonymous), evaluation research methods, and public policy related to health care, mental illness, veterans, drugs, crime and correctional systems. Professor Humphreys' over 300 scholarly articles, monographs and books have been cited over thirteen thousand times by scientific colleagues. He is a regular contributor to Washington Post and has also written for the New York Times, Wall Street Journal, Washington Monthly, San Francisco Chronicle, The Guardian (UK), The Telegraph (UK), Times Higher Education (UK), Crossbow (UK) and other media outlets.

16 thoughts on “Another Demonstration that Smaller is Better. And Worse Too.”

  1. So what is going on here could well be the Law of Large Numbers: measures that use larger quantities are more likely to be closer to the mean; large deviations from the mean are more common when the sample size is smaller.

  2. Arguably, there could be more here than just statistics and large numbers: small countries could be more easily subjected to the extreme influence of a single person or small group of people, leading in to a higher likelihood of unconventional policy decisions, for good or for ill (take Singapore, or Venezuela).

    I don’t think this idea quite works (it’s not like small countries necessarily have small parliaments, or small cabinets), but I think it’s at leat provocative.

  3. Not unrelatedly:
    where Gladwell (not Caldwell!) talks about men and black people being more variable than women and white people, hence overrepresented at top and bottom. (He wrote this before Larry Summers got in such trouble for saying similar things.)

    1. This is also one of the arguments in David Epstein’s ‘The Sports Gene’ (just finished reading, an excellent book). I remain unconvinced – it’s true that more diversity is found in Africa than outside, but (just giving my impressions, this is far from being an area of expertise):
      1) A lot of that is about really small populations in Africa that are not really what anyone talking conventionally means by ‘black.’ San and Hadza and Pygmies and such, groups that today give N in the few thousands but which contain lots of deep genetic variation. And they are outgroups to most African people as well.
      2) My impression was this more-diversity-in Africa notion was established for neutral variation, e.g. where changing a base pair doesn’t change the coded amino acid or protein. I don’t think it’s known to be the case for variations that actually ramify phenotypically. One reason such cases are harder to study is that selection typically filters “bad” variations out, reducing diversity.

      1. I should probably be asking my implied questions at gene expression, but here goes.

        My impression of the more-diversity-in Africa notion was that the part to focus on is the less-diversity-outside of Africa: founder effects from populations outside of Africa being descendants of migrants from just a few parts of the African populations over the last couple of 1000 centuries, with insufficient time having passed for non-harmful mutation to have restored the variation.* I am fairly confident that this is true for pre-Colombian populations in the Americas (but I’ve never heard anything about Australian aboriginal genetic diversity ???).

        *Obviously the discoveries about the homo sapiens genome parts of the Neanderthal and Denisovan genomes indicate other sources of genetic diversity in non-African populations, but currently these contributions appear to be small.

        1. Obviously the discoveries about the homo sapiens genome‘s containing parts of the Neanderthal and Denisovan genomes

  4. I don’t think the sample size hypothesis applies here (or at least not solely). Smaller schools performing both better and worse than bigger schools is a rather straightforward consequence of the Central Limit Theorem [1].

    But that’s not the case with the economies of small countries. Small countries have genuinely differently structured economies compared to medium and big countries, neither of which can be adequately described or approximated as a sum of independent random variables, so you can’t just throw the CLT at it and expect the math to work out. The CLT may still contribute to the overall effect, and you may still get a result where small countries are more likely to perform both better and worse, but the simple probabilistic explanation is insufficient.

    [1] Strictly speaking, not the classical CLT, where the random variables need to be independent and identically distributed, but one of the more modern variants with weaker conditions.

    1. If you think about randomness in distribution of valuable resources, a physically small country is less likely to contain a jackpot underground.

      1. A jackpot underground is not necessarily going to lead to a rich country. It depends a lot on who your neighbors and friends and oligarchs are.

        But the other thing you have to remember here is that you’re not just doing sampling (biased or otherwise), you’re doing sampling without replacement. My hypothesis for small schools (not so much for small countries) is that the administrations of such schools are (in one way or another) selecting the enrollment they want according to some criterion: if it’s academic excellence, you get one result; if it’s willingness to send kids to a lousy school you get another. But as the school gets bigger, administrations effectively have to relax their selection standards, simply because there aren’t that many outliers in the available population.

      2. That’s why I said it could still contribute to the overall effect, even if it’s not solely responsible. But the CLT, on which the hypothesis is based, requires a sum of independent random variables. Not a random variable mixed in with, say, the results of a difficult to analyze stochastic process describing components with unknown interdependencies and feedback loops.

        1. I would be hard pressed to rate how much variation comes from true randomness vs, say, bad or good choices.

          Another factor that may be effectively random – a small country may be devastated by random a natural disaster in a way that would not happen a larger country.

          1. MobiusKlein: I would be hard pressed to rate how much variation comes from true randomness vs, say, bad or good choices.

            You don’t need “true randomness”. You don’t really get that outside of quantum mechanics. What you need is something that’s adequately modeled by a random variable.

            MobiusKlein: Another factor that may be effectively random – a small country may be devastated by random a natural disaster in a way that would not happen a larger country.

            Doesn’t matter. What you need for the CLT isn’t just “some randomness”. You need randomness that has a specific structure, otherwise you can’t use the theorem. You may still arrive at the same result in a different fashion, of course, but you’d have to explain how that would work.

  5. For some reason this reminds me of the stories you used to see about how much worse your return on investment would have been if you had been out of the stock market on its ten best days. Funny, but I don’t ever recall seeing a story on how much better you would do if you had been out of the market on its ten worst days.

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