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Comment by leto_ii | Hacker News Reader

I'm not sure this analysis is as surprising or meaningful as it might seem at first sight.

In the end I think what this tells us is that in the general population there are many distinct subpopulations with different distributions of running speed. I would expect that if you start conditioning on things (e.g. gender, age, level of fitness) things start looking a lot more normal, rather than log-normal.

You might not be able to beat the Freeze simply because you're not a young fit male who trains to run fast. Really nothing surprising here.

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I think the point is that all these traits are multiplicative, not summative. Your maximum running speed at the age of 40 is maybe 0.8 times what it was when you were 20, not your running speed at the age of 20 minus some. And when you have several multiplicative traits like age, gender, training history, all different aspects of genetics, etc., you get a lognormal distribution instead of a normal distribution, and with it a much larger variance between individuals.

If I'm not wrong you can't obtain a lognormal as a mixture of Gaussian distributions. The processes that generate these distributions are completely different. Gaussian distribution arise when the factors are additive, and lognormal when factors are multiplicative, so these are two completely different distributions. The same way I don't think you can get a normal distribution just by conditioning on an exponential distribution.

Say you get a lognormal by combining five gaussian-distributed factors (as in the past). Then you condition on four of those factors. The remaining factor, and hence your overall result, will be gaussian.

Now, maybe in real life there are far too many factors and we're only holding a few constant so the math doesn't apply, but I do think the math is valid.

Good point, I was wrong. However, it can also happen that the factors are not gaussian, and still the multiplication would follow a lognormal distribution by the CLT. So saying that "if you condition on enough factors the distribution will be normal" requires the assumption that the factors are normally distributed.

A gaussian mixture is a universal density approximator, so you can definitely approximate a lognormal with such a model.

I was thinking about this and how you might get a lognormal from normals. Peer comments I think have good suggestions.

The thing is a while back I took a look at the times achieved by competitive athletes at the 100m dash (from worldathletics.org), and they looked normally distributed. Intuitively that made sense to me - many other human attributes are normally distributed, so why not this?

Just a guess: all of those competitive athletes are in the far, far upper-right quadrant of the performance graph, where the curve flattens out. Their distribution will approximate to a normal curve, whereas zooming out to include everyone reveals a gaussian distribution.

Allan writes about stats in every day life. He's not trying to show something surprising?

He's explaining why something is, using stats.

You're probably overthinking it.

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