# Does my BMI look big in this?

At some point – whether it's at the doctors, at the gym, or online – all of us have probably encountered the Body Mass Index.

Body Mass Index (BMI) is derived from a simple mathematical formula, devised by Belgian scientist Adolphe Quetelet in the 1830s, that divides a person's weight in kilograms by their height in metres squared to arrive at an estimate of an individual's body fat.

It's supposed to provide an approximate measure to help judge if someone has a healthy weight – and indicate, for instance, if they are obese. But as 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^{3} would be appropriate. They don't! However, weight/height^{2} 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^{2.5}. Certainly if you plot typical weights of people against their heights, the result comes out closer to height^{2.5} than height^{2}.'

Sticking with the current formula, he says, leads to confusion and misinformation: 'Because of that height^{2} 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.'

Quetelet's formula was invented at time when there were no calculators or computers so it's perhaps little wonder he opted for something so simple. What's stranger, perhaps, is why institutions such as the NHS, the Department of Health, and the National Obesity Observatory continue to use the same flawed formula today.

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!'

So what's the alternative and what difference would changing the formula make to the medical measure of BMI?

Nick proposes a new formula [more detail here] where BMI = 1.3*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' would gain a point in their readings, and millions of people of height around 6' 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.'

Alain comments: 'So according to Quetelet the scaling is 3 for babies (babies are spheres), 2 for kids (kids grow more like celery sticks, as we know), then 5/2=2.5 for grownups (beefing up so to speak). It seems Quetelet never cared about obesity (not a big issue in the 1840's).'

*Nick Trefethen is Professor of Numerical Analysis at the University of Oxford.*

*Alain Goriely is Professor of Mathematical Modelling at the University of Oxford.*