Commenter pumpkinhead doubts Oprah’s head is as big as I say it it, writing:

 Given this information of 1358 cc cranial capacity(for white females mind you) and 91 SD, Oprah would be 7.45 SD above average making her 1 in the tens of trillions.

I don’t want to argue about Oprah’s head size here since we’re already having that argument in the previous thread, but rather I want to make a more general point.

The problem with pumpkinhead’s argument is it assumes cranial capacity fits the Gaussian curve which is a reasonable assumption for non-pathological heads, but there’s just one problem with it.  If head circumference is normally distributed, and head circumference has a cubic relationship with cranial capacity, then extreme cranial capacity can not be normally distributed.

For example, the late J.P. Rushton suggested that at least in very young children, cranial capacity ( cm3 ) = circumfence3/118.4.  This is the formula for a volume of a hemisphere, which as Rushton acknowledged, is an oversimplification, because the cranium is not a perfect hemisphere and there is massive individual variation in head shape.

But let’s imagine a world where all crania were perfect hemispheres.  I randomly generated 60 crania and these had a mean of 54.7 cm (SD 1.28 cm which because of my small sample size, is slightly smaller than the actual U.S. female SD).  In this sample, the one in several billion level (+6.3 SD) would be reached at 62.76 cm.

Now what happens when all the circumferences are converted into cranial capacity using the hemisphere formula.  The mean becomes 1387 cm3  and the SD becomes 97.  What this means is that a head that is +6.3 SD in circumference (62.38 cm), becomes +7.2 SD in cranial capacity (2088 cm3 ).  So the cubic effect of going from circumference to volume make extreme humans seem superhuman.

Of course our crania are not perfect hemispheres, but the cubic effect still applies.