Why is BMI related to the square of height?

Question

I'm aware that the U.S. CDC defines BMI (Body Mass Index) as mass (in kg) divided by height squared (in meters squared.) The CDC then defines what ranges are considered underweight, healthy, or overweight broadly based on BMI (with different ranges by gender.)

My question is: Why the square?

As any mechanical or civil engineer (or just anyone who is vaguely familiar with those topics) will know, if you scale something up proportionally, mass will scale with the cube of the increase in a particular dimension, not the square. Because, for a given material, mass is proportional to volume.

Furthermore, since material strength usually scales with the cross-sectional area (not volume,) which is proportional only to the square of a particular dimension, making something larger usually requires scaling up supporting structures more than proportionally in order to maintain the same safety margins. (See: Square-Cube Law)

So, why are these principals not applied when dealing with human anatomy? Is there some reason why it isn't believed that humans should scale proportionally (or even super-proportionally in order to maintain similar load-bearing characteristics?)

Answer 1

It turns out the answer to this is fairly simple, and but not terribly easy to find.

In the 1800s a Belgian named Adolphe Quetelet (1796–1874) performed some statistics on the weights and heights of people in Europe and Britain (chiefly France and Britain) and published his seminal work, Sur l'homme et le développement de ses facultés, ou Essai de physique sociale (Treatise on Man and the Development of his Faculties, or Essays on Social Physics) in 1835. The relevant section for this can be found as a PDF for free (I believe) at the Wiley Publishing journal Obesity Research

In this work, he examined the growth rates and dimensions of people throughout their lifetime, coming to the conclusion that after 30 people reached their maximum height and and after 40 for men and 50 for women, they reached their maximum weights.

He has this to say about the developmental relationship between weight and height. Emphasis mine in all cases:

If man increased equally in all his dimensions, his weight at different ages would be as the cube of his height. Now, this is not what we really observe. The increase of weight is slower, except during the fist year after birth; then the proportion which we have just pointed out is pretty regularly observed. But after this period, and until near the age of puberty, the weight increases nearly as the square of the height. The development of the weight again becomes very rapid at the time of puberty, and almost stops at the twenty-fifth year. In general, we do not err much when we assume that, during development, the squares of the weight at different ages are as the fifth powers of the height; which naturally leads to this conclusion, in supposing the specific gravity constant, that the transverse growth of man is less than the vertical.

He then goes on to state:

However, if we compare two individuals who are fully developed and well-formed with each other, to ascertain the relations existing between the weight and stature, we shall find that the weight of developed per- sons, of different heights, is nearly as the square of the stature. Whence it naturally follows, that a transverse section, giving both the breadth and thickness, is just proportioned to the height of the individual.

and continues after a table of statures and weights:

Thus, the stature of men and women, fully developed and well-formed, varied in the proportion of five to six nearly: it is almost the same with the ratios of the weight to the stature of the two sexes: whence it naturally follows, as we have already said above, that the weight is in proportion to the square of the stature.

and finally reaches a list of conclusions, with the 7th being:

7. After the full development of individuals of both sexes, the weight is almost as the square of the stature. From the two preceding relations, we infer, that increase in height is greater than the transverse increase, including breadth and thickness.

This work was then converted into the BMI measurement in 1974 by Ancel Keys in the Journal of Chronic Diseases. In this article they state:

In spite of the fact that it is easy to show that the body form does not remain constant with increasing length, the ponderal index, or the similar Rohrer index,W/H³, has been rather widely used....

...In the present paper it will be shown, in confirmation of some recent conclusions of others, that in this respect the ratio W/H² is clearly better than the ponderal index. It is proposed that this ratio, W/H², be termed the body mass index.

Answer 2

BMI isn't a "real" thing, it's an arbitrary measure meant to capture some aspect of "overweightness". The exponent has traditionally been chosen as "2" because that roughly fit with data observed. In the Keys et al 1972 paper that established BMI, they did try other exponents 1 and 3, but found 2 to be the best correlate of body fat.

Cubed relationships between length and weight are only an approximation, and species vary quite a bit on how close they are to the "cube rule". For a real-world application, I've come across this in the context of sport fishing, where it is easy to measure a fish's length but sometimes more difficult (you need a scale; the fish needs to hold still) and possibly harmful to the fish to weigh them; you can estimate an approximate weight from a species-specific formula, however. Wikipedia has a page on this with some examples for different species.

For humans, we simply don't tend to follow a cubed relationship. A XX% change in height for humans is not associated with an equivalent XX% change in width or "depth", it's associated with something a bit less.

There have been some suggestions to use a different exponent than 2, though, because with the current formula, BMI tends to not track well with adiposity or health outcomes for the tallest or shortest individuals. A barrier to making this change is agreeing on which one to use and the inertia of a publication record on the old measure. Here are a few examples, though, where people have investigated whether a different exponent for height and/or weight would better index a healthy/unhealthy body composition (I'm sure there are many more; not all ultimately recommend against the current scaling):

Foster, D., Karloff, H., & Shirley, K. E. (2016). How well does the standard body mass index or variations with a different exponent predict human lifespan?. Obesity, 24(2), 469-475.

Garn, S. M., Leonard, W. R., & Hawthorne, V. M. (1986). Three limitations of the body mass index. The American journal of clinical nutrition, 44(6), 996-997.

Tjeertes, E., Hoeks, S., van Vugt, J. L. A., Stolker, R. J., & Hoofwijk, A. (2017). The new body mass index formula; not validated as a predictor of outcome in a large cohort study of patients undergoing general surgery. Clinical nutrition ESPEN, 22, 24-27.

Xu, Y., Yan, W., & Cheung, Y. B. (2015). Body shape indices and cardiometabolic risk in adolescents. Annals of Human Biology, 42(1), 70-75.

Answer 3

While the answers of Bob and Bryan are perfectly valid, let me attempt to provide some intuition from a physics/statistics standpoint.

Generally, when a quantity is proportional to some other quantity, raised to the n-th power, this n power is related to the effective number of degrees of freedom of the independent variable.

For example, in the inverse square law, the number 2 arises from the fact that energy disperses in all space uniformly, and there is no degeneracy between any dimensions. The result is that consecutive wavefronts form spherical shells, and the surface area (of the sphere) is proportional to r^2.

Conversely, humans can't expand in all 3 dimensions equally. The effective number of degrees of freedom for a human is about 2. One comes from the height, which is almost free to change. The other one comes from the combination of 'width' and 'depth'. These two are strongly degenerate - the technical term is that they co-vary almost completely, thus lowering the effective number of degrees of freedom.

Note, the fact that width and depth are almost degenerate means that there is a linear relation between them. So that third dimension doesn't really play a role, because the human body is not free to grow in all space.

How do we know that? That's empirical - it's just that the effective number of degrees of freedom for a human is closer to 2 than it is to 3. 3 is simply the upper bound when the object grows uniformly in all directions.