[dododo]

if it is a power law in the sense that a few companies return a lot, whilst most return a little, how does that make the mean irrelevant?

the mean of a power law of this form is more skewed by the few companies returning a lot than it is by the majority. assuming that more than 50% of the companies return a little, the median would be irrelevant, but the mean is very sensitive to outliers.

[tedunangst]

Consider four companies' returns: [1, 1, 1, 100]. The median, 1, is a reasonable predictor for the next company's return. The mean, 25, doesn't seem like a likely outcome.

[dododo]

there is not a normative way to discriminate between those two predictions.

it comes down to what kind of loss you incur for making an incorrect prediction. for mean, it's a squared loss (which penalizes you for being wrong in the tails more heavily) but for a median it's the absolute loss.

the argument being made was that fb is an outlier and (as I read it) the mean didn't reflect it, so the maths was "lazy" as it didn't take that into account. actually the mean does take these outliers into account (since it's corresponding loss function penalizes you harshly for being wrong in the tails) and so is probably not too unreasonable at predicting on outlier companies (certainly better than the median).