Oliver spends a good bit of time debunking the Body Mass Index (BMI) as a measure of overweightness/obesity or health in general. And rightly so; BMI is flawed in a variety of ways. But I don’t think this is one of them:
Today, the same BMI formula is applied to men and women, despite the fact that, because men are taller on average, the formula automatically gives them a lower BMI. (BMI is a ratio of weight to height-squared, so the taller one is, the lower one’s relative BMI will be.) [p. 94-95]It would make sense to have a different BMI formulas for men and women simply because they have different physiologies (for instance, women just naturally carry more body fat). But it’s untrue that BMI is lower for taller people, so long as their bodies are proportionate to their height. In fact, BMI will be higher for such people. This is because weight increases with the cube of the proportional increase in height.
Suppose person A has an “ideal” BMI. For simplicity, measure him in three dimensions: height (h), width from side-to-side (a), and width from front-to-back (b). Person A’s total volume is hab.1 Treat weight as a scalar function of volume; that is, weight = d*volume.2 So A’s weight is w = dhab. His BMI is weight over height-squared; thus:
BMI = dhab/(h^2) = dab/hNow, suppose person B is taller than A, but with the exact same proportions. Given the same proportions, this person should be considered equally healthy and therefore “ideal” as well. If person B’s height is k times B’s height, then B’s side-to-side width must be k times A’s side-to-side width, and B’s front-to-back width must be k times A’s front-to-back width. So B’s total volume is (kh)(ka)(kb) = (k^3)(hab), and B’s weight is (k^3)(dhab). And thus:
BMI = (k^3)(dhab)/(k^2)(h^2) = kdab/hNote that person B’s BMI is k times larger than person A’s, despite A and B both having exactly the same bodily proportions. This is certainly a problem for BMI, since it gives two people with identical proportions different results; but the error is not in the direction indicated by Oliver.
1. Yes, this is basically treating the person like a rectangular-based column, but nothing important hinges on that. If you like, imagine taking the person and breaking him into thousands of little Lego-like pieces and doing the same calculation on each one.
2. Weight = mass x force due to gravity, and mass = volume x density, and so weight = volume x density x force due to gravity. Let the scalar d = density x force due to gravity.