Are we 2-D? BMI!

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.

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9 Responses to “Are we 2-D? BMI!”

  1. BMI and IPA « Coby Lubliner’s Blog Says:

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  2. Karen Says:

    Although I do agree BMI is flawed because it takes no account of gender, age, muscle mass or frame size, I do not agree that BMI should be cubed. Human beings do not necessarily grow equally in width and depth when they grow in height. You only have to compare photographs of very tall people with that of very short people, to see that on average shorter people look bigger in relation to their height than taller people and therefore do not typically scale up proportionately with height.

    If you are comparing a short and tall person with similar skeletal proportions in relation to their height then in this case the weight would probably be cubed, howeve you would probably be comparing a small framed short person with a large framed tall person (frame size being in relation to height). This would not be what is average, it would be comparing the extremes.

    Professional basketball players are heavier because they are men and because they are muscular. Most men have higher BMI’s than women and most muscular men have higher BMI’s than non muscular men. Although rare, if a 6ft7 woman had a BMI of 26 she would probably be overweight unless she was muscular.

    At the end of the day BMI squared is far more comparable to body fat percentage than BMI cubed, the exact expondent may be somewhere in between but probably closer to BMI cubed, i.e. 2.2.

  3. bmi-formula Says:

    Thanks for posting this article. I’ve been looking for the point when they changed the BMI standards, now I’ll be better able to explain the insanity of it all, to people.

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  6. Electric Switch Says:

    we have different office chairs in the office and i love to use those office chairs with clothe cover :,`

  7. Brian Says:

    Haha, yes I agree that we are indeed 3-D and not 2-D life forms (although the BMI calculation as it is used today suggests that we are 2-D). As to whether or not our weight is proportional to the square or the cube of our height, well, I would guess that it is in fact somewhere in between. I do agree, the BMI calculated for tall people always seems to be unreasonably high. But using the cube instead of the square seems to turn it around backwards too much. Might I propose we use a factor of 2.5 when comparing peoples’ weights and heights? For those of you who don’t know what we are talking about here… go back to school.

  8. Dom Thorpe Says:

    I’ve recently blogged about the BMI and how it doesn’t seem to work and when I thought about it it seemed obvious that the equation should involve a cube function rather than a square.

    Tried explaining it to my girlfriend unsuccessfully however after doing the math and thinking that I may have made a new discovery I thought I should look online to see if anyone else has written about the same thing. Not surprising that I wasn’t the first to think of it.

    It seems ludicrous that anyone ever considered that a square function should be used rather than a cube function. The question is “How do we change the present thinking so it is in line with our method of thinking?”

    Let’s make it our mission to eliminate the system we use now and replace it with the 3D version!

  9. by clicking here Says:

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