Once again, a rash of media articles about obesity in the United States has broken out. And once again, the obesity statistics are defined in terms of BMI. Here is an example, from forbes.com:
To determine which cities were the most obese, we looked at 2006 data on body mass index, or BMI, collected by the Centers for Disease Control’s Behavioral Risk Factor Surveillance System, which conducts phone interviews with residents of metropolitan areas about health issues, including obesity, diabetes and exercise.
In this case, participants report their height and weight, which survey analysts use to calculate a BMI. Those with a BMI between 18.5 and 24.9 are considered at a healthy weight, those with a BMI between 25 and 29.9 are considered overweight, and those with a BMI of 30 or higher are considered obese. About 32% of the nation is obese, according to the Centers for Disease Control; Memphis ranked above the national average at 34%
Never mind that the city that ranked third in obesity, Nashville, turned up among the 25 “fittest” (as opposed to “fattest”) in a different survey, this one by Men’s Fitness (and, as far as I can tell, not based on BMI). I am not interested in the results, only in the use of BMI. And, what’s more, after entering “obesity BMI” in a Yahoo news search, not one of the first ten articles that I clicked on included an actual definition of BMI.
The BMI, or body-mass index, is defined very simply as a person’s weight (in kilograms) divided by height (in meters) squared. Thus, since I weigh 66 kg (145 lb.) and stand 1.71 m (about 5 ft 7½ in), my BMI is 66÷1.71² ≈ 22.5.
Now anyone with any familiarity with physical science will recognize a quantity defined as force (such as weight) divided by length squared (or area) as representing pressure or stress. For example, for people of different sizes but with similar body proportions, the area of any portion of their body surface – for example, the portion that is in contact with a chair on which they may be sitting – will be proportional to the square of the height. If the chair bears a person’s full weight, then the average pressure on the chair’s seat, equal to the weight divided by the contact area, will be proportional to that person’s BMI.
It is precisely for this purpose – the design of office chairs – that the quantity now known as BMI was invented by the nineteenth-century Belgian mathematician Adolphe Quetelet.
But human bodies are three-dimensional, not two-dimensional. For people of different stature but similar geometric proportions, the body volume is proportional to the cube, not the square, of the height. And if the proportions of the various constituents of body mass (bone, muscle, fat etc.) are similar, then the weight is proportional to the volume, and consequently to the cube of the height. Consequently, what people who are geometrically and physiologically similar have in common is the weight divided by the height cubed, not squared.
What this means is that people with the same build will have a higher BMI if they are taller and a lower BMI if they are shorter. It has already been noted that very tall people who are quite fit — for example, professional basketball players — have BMI values that would rank them as overweight. Thus, an NBA guard who is two meters (about 6 ft 7 in) tall and who has the same build as I do would weigh 66×(2.0÷1.71)3 ≈ 106 kg (232 lbs) and his BMI would be 26.4, in the “overweight” range.
It has also been remarked that in populations that, on the average, are significantly shorter than European (or European-descended) ones, a lower overweight threshold is necessary. For Southeast Asians, for example, it’s 23 (as in this document from Singapore). Were the body types the same, this would be consistent with average height being about 8% less. In fact, the average adult height in China, for example, is 6–7% less than the average of white Americans. But the body types are in fact different (for example, the waist-hip ratio of Chinese men is 0.87 while that of white Americans is 0.98, as given here).
I have no doubt that if an index were defined on the basis of weight divided by height cubed, the discrepancies would become negligible.