What does this paper mean by mean square error?

Question

I'm reading a paper about estimating fetal weight by ultrasound and other techniques. In Table 2 they give descriptive statistics, to wit: of all the examiners, the one with the highest mean square error (E3, an untrained ultrasound resident) has 21.64. But with N = 204 samples, that would allow for an error of at most $$ \sqrt{204\cdot21.64\,\text{g}^2} \approx 66.44\,\text{g} $$ if all the error were concentrated in a single measurement. Indeed, from the maximum and minimum alone it seems like the mean square error for E3 should be at least $$ \frac{300^2+542^2}{204} > 1881 $$ so I must be missing something.

Answer 1

As far as I can tell, you are correct. If the fetus that E3 estimated to be lightest ($\hat{y}=2,500$) was the actual lightest one ($y=2,200$), and the same for the heaviest one ($\hat{y}=5,592$ - I love the four significant figures - and $y=5,050$), and E3 got everything else completely correct, then the mean squared error would indeed be

$$ \frac{300^2+542^2}{204}\approx 1,881, $$

and any other assumption can only increase the MSE.

Note that this also applies to the other examiners, e.g., for E1, we should get an MSE of at least

$$ \frac{150^2+160^2}{204}\approx 236. $$

I don't see anything you could be missing, and I would assume an error in calculations. I'd recommend you contact the authors.