The above chart shows the line of best fit predicting cranial capacity from head circumference in 121 skulls. The authors of the paper created the following formula to predict cranial capacity from head circumference:
Expected cranial capacity = 5.43 (head circumference) – 1346
However this formula predicts negative cranial capacity for the smallest circumferences in the scatter plot. Why? Because to make the relationship linear, the authors excluded the 11 smallest crania.
Although this paper is a major contribution to the field, the authors apparently lacked the skills to calculate non-linear relationships so Pumpkin Person decided to do it for them, creating a revised formula (using all 121 skulls):
cranial capacity = 0.0080(head circumference)^2 – 1.9(head circumference) + 158
So while the authors’ formula predicted a 200 mm head circumference would have a -260 cc capacity (physically impossible) my formula predicts 98 cc capacity (consistent with the line of best fit) because my formula can adapt to the curving shape of the relationship.
It’s important to note that the formulas above are both based on skulls so when applying them to living heads an adjustment needs to be made. For example, when J.P. Rushton estimated the cranial capacity of living army staff using Lee & Pearson’s regression equations using head length, head breadth and head height, he deducted 11 mm for fat and skin around the skull for each measurement since he was applying a formula derived from skulls on living heads.
However Lee & Pearson noted that there was no obvious way to adjust for fat and skin around the skull when using circumference to predict capacity.
However it occurred to Pumpkin Person that if head length and head width are deducted 11 mm, and these are more or less equivalent to head diameter, then perhaps one should convert circumference into diameter, subtract 11 mm, and then convert back to circumference before applying circumference formulas.