Gah! This is why understanding math is important!

[barnabas_truman]

I just realized that "Body Mass Index" is defined as mass divided by height squared;
BMI = m / h^2.
This leads me to wonder if Adolphe Quetelet was an idiot (hint: certainly not, but I can't think of a better explanation), and, more to the point, why anybody ever thought his formula was a good idea.

Allow me to explain. This is going to involve a bit of math and physics, so buckle down and be prepared to look things up or ask questions as needed.

Suppose, for the sake of the argument, that every body has the same proportions and the same density.

Then the mass of a body is its volume times density:
m = d * V
where d is density and V is volume.
But volume is proportional to height cubed:
V = k * h^3
where k is some proportionality constant.
We're assuming for the moment that both d and k are the same for everyone.

If we substitute these into Quetelet's formula, we find that
BMI = d * k * h^3 / h^2
Two of the h's cancel out, leaving
BMI = d * k * h
In other words, BMI is equal to a constant times one's height!

Body Mass Index is a measure of height, not of obesity! What were they thinking??


Of course, the assumptions that d (density) and k (proportions) are the same for everyone isn't true at all in reality. So BMI is really measuring the product of density, height, and a constant involving body proportions. But why allow height to be a factor? The end result of this is that, all other things being equal, tall people have higher BMI. Isn't the whole point of dividing by height to cancel out the influence of height entirely?

On the other hand, if BMI is defined as mass divided by height CUBED, the extra h-factor also cancels, and BMI becomes the product of d and k, which actually means something. Why isn't it defined this way to begin with?

Alternatively, mass divided by (height squared times shoulder width) might work even better--that at least attempts to scale it individually for different body structures, something that BMI ignores entirely.


P.S. It should also be noted that Quetelet's goal was to apply rigorous methods from statistics to the study of populations, and that he never intended BMI to be used to diagnose individuals. However, I still can't imagine why he wouldn't at least take the square-cube law into account!

Comments

[mutive.livejournal.com]

Well, for better or worse, unless you use correlations, insurance is close to meaningless. *shrugs* I'm still pissed off that as a woman I have to pay significantly more than a man does because I *might* become pregnant. But, eh, there it is.