# BMI Magic

The body mass index (BMI) is a heuristic proxy for estimating human body fat based on an individual's weight and height. BMI does not actually measure the percentage of body fat. It was devised between 1830 and 1850 by the Belgian polymath Adolphe Quetelet during the course of developing social physics. It has been for awhile as an indicator of "how fat am I". What if it is wrong? BMI divides a person's weight in kilograms by their height in meters squared to arrive at an estimate of an individual's body fat.The problem lies with the height/weight ratio.

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Nick Trefethen of Oxford University's Mathematical Institute pointed out in a recent letter to The Economist the basic formula BMI relies on is flawed:

"If all three dimensions of a human being scaled equally as they grew, then a formula of the form weight/height cubed would be appropriate. They don't! However, weight/height squared is not realistic either," Nick tells me.

"A better approximation to a complex reality, which is the reform I wish could be adopted, would be weight/height

. Certainly if you plot typical weights of people against their heights, the result comes out closer to height ^{2.5}
than height squared."

Sticking with the current formula, he says, leads to confusion and misinformation: "Because of that height square term, the BMI divides the weight by too large a number for short people and too small a number for tall people. So short people are misled into thinking they are thinner than they are, and tall people are misled into thinking they are fatter than they are."

The reason for its survival may be that all the various agencies have agreed on it and, Nick says, "nobody wants to rock the boat."

It highlights, perhaps, how uncritical many of us are of the mathematics behind widely-used measures. There are probably many more flawed formulas out there but as Nick comments "it would be hard to compete with this one in impact in a world approaching a billion obese people!"

Nick proposes a new formula [more detail here] where BMI = 1.3 x weight(kg)/height(m)^{2.5}= 5734*weight(lb)/height(in)^{2.5}

"Suppose we changed that exponent from 2.0 to 2.5 and adjusted the constant so that an average-height person did not change in BMI. Suddenly millions of people of height around 5 foot would gain a point in their readings, and millions of people of height around 6 foot would lose a point," Nick explains.

"In our overweight world, such changes would distress some short people and please some tall people, but the number they'd be using would be closer to the truth and good information must surely be good for health in the long run."

Intriguingly, it's likely that Quetelet would have approved of using the 2.5 exponent. Alain Goriely, also of Oxford University's Mathematical Institute, says that Quetelet himself was well aware of the wrong choice of scaling.

In 1842 Quetelet wrote in A Treatise on Man and the Development of his Faculties:

"If man increased equally in all 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 first year after birth; then the proportion we have just pointed out is pretty regularly observed.

"But after this period, and until near the age of puberty, weight increases nearly as the square of the height. The development of weight again becomes very rapid at puberty, and almost stops after 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 supporting the specific gravity constant, that the transverse growth of man is less than the vertical."

BMI is used now differently for children. It is calculated the same way as for adults, but then compared to typical values for other children of the same age. Instead of set thresholds for underweight and overweight, then, the BMI percentile allows comparison with children of the same sex and age. A BMI that is less than the 5th percentile is considered underweight and above the 95th percentile is considered obese for people 20 and under. People under 20 with a BMI between the 85th and 95th percentile are considered to be overweight.

For further information see BMI.

Body Fat image via Wikipedia.