[timr]

You're talking about reduction in statistical variance due to replication of measurement (and then averaging). I'm talking about what happens when they extrapolate from that value by a huge factor (which is what they've done, and the silly article does egregiously).

The paper isn't clear what they mean when they said "~25% within-sample coefficient of variation", so I can't directly address what you're asking, but it's tangential to the point I'm making. My naïve interpretation is that they did an ANOVA, and reported the within-group variance, or something similar.

All I'm saying in my footnote is that, whatever the final point estimate, scaling it by a factor of C will affect the variance of the final sample distribution by C^2. So for example, if you have an 8% variance on the measurement at ug/g, and you scale it by 1300 (for 1300g; what the interwebs tells me is the mass of a standard human brain), then you'd expect the variance of the scaled measurement to be 1300^2 * 8%.

That makes a ton of assumptions that probably don't hold in practice -- and I expect the real error to be larger -- but illustrates the point.

[myrmidon]

I think there is some kind of mixup, you can not scale up the variance percentages quadratically:

If you do a small-scale measurement, say you get result of 5g, with a standard deviation of 0.2g. That means the variance is 0.04 g^2.

If you then scale the setup up by 1000 (=> getting 5kg as expected value), then the variance scales to 1000^2 * 0.04 = 40000 g^2.

BUT the standard deviation is still 200g. The relative uncertainty is NOT increasing quadratically!

(another sanity check: if you change the units by a factor of 1000, your variance must not increase, relatively).

But maybe I misunderstood your point?

[timr]

> If you then scale the setup up by 1000 (=> getting 5kg as expected value), then the variance scales to 1000^2 * 0.04 = 40000 g^2.

They didn't "scale the setup". They made a small-scale measurement, then extrapolated from that result by many orders of magnitude. They didn't grind up whole brains and measure the plastic content.

Imagine the experiment as a draw from a normal distribution (the distribution is irrelevant; it's just easier to visualize). You then multiply that sample by 10,000. What is the variance of the resulting sample distribution?

[myrmidon]

> What is the variance of the resulting sample distribution?

Relatively? The same. Yes it scales quadratically, but that is just because variance has such a weird unit.

Just consider standard deviation (which has the same physical unit as what you are measuring, and can be substituted for variance conceptually): This increases linearly when you scale up the sample.

An example: Say you take 20 blood samples (5 ml), and find that they contain 4.5 ml water, with a standard deviation of 0.1 ml over your samples.

From that, your best guess for the whole human (5 liters, i.e. x1000) has to be 4.5 liters water, with standard deviation scaled up to 0.1 liters (or what would you argue for, and why?)