**BMI (Body Mass Index)**

In its 5 January 2013 issue the *Economist* published this letter from
me:

*
SIR - The body-mass index that you
(and the National Health Service) count on to assess obesity
is a bizarre measure. We live in a three-dimensional world,
yet the BMI is defined as weight divided by height squared.
It was invented in the 1840s, before calculators,
when a formula had to be very simple to be usable.
As a consequence of this ill-founded definition,
millions of short people think they are thinner than they are,
and millions of tall people think they are fatter.
Nick Trefethen
Professor of numerical analysis
University of Oxford
*

*
*

The appearance of this admittedly strongly
worded letter has led to communications from
many people around the world, and I would like to
explain a formula that I think would be worth considering
as an alternative.

Current formula: BMI = weight(kg)/height(m)^2 = 703*weight(lb)/height(in)^2.

The oddity is the appearance of that exponent 2, though our world is three-dimensional. You might think that the exponent should simply be 3, but that doesn't match the data at all. It has been known for a long time that people don't scale in a perfectly linear fashion as they grow. I propose that a better approximation to the actual sizes and shapes of healthy bodies might be given by an exponent of 2.5. So here is the formula I think is worth considering as an alternative to the standard BMI:

New formula: BMI = 1.3*weight(kg)/height(m)^2.5 = 5734*weight(lb)/height(in)^2.5

The numbers 1.3 and 5734 are designed make the BMI reading unchanged for an adult of average height, which I take to be about 66.5 inches, i.e., 1.69 meters. (The square root of 1.69 is 1.3.) To find your "New BMI", try the New BMI Calculator written by Nick Hale.

Are these numbers "right"? No! -- for human beings are complicated, and any BMI formula will deliver just one number. No single number can be right, and indeed, the extreme reliance of today's medical and insurance establishments on a simple formula worries me a great deal. But perhaps this revised formula might reflect better than the standard one how the weights of healthy adults really depend on their heights.

Would it make a difference? Yes indeed. Roughly speaking, each person 6 feet tall would lose a point from their BMI reading, and each person 5 feet tall would gain a point. That's millions of people. If the new numbers gave a more accurate indication of actual health issues, this could be a significant change for the better.

What about the density of muscle vs. fat? We hear about this frequently
in discussions of BMI (including in the *Economist* article I was
responding to, which mentioned Olympic weight lifters),
but it's a smaller effect. Muscle is about 18% denser than
fat. This means that if you heroically exercised so much
that you converted 10% of your body volume from fat to muscle
(wow!), your BMI
reading would go up just 1.8%. That's much less than the corrections just
mentioned for short or tall people.

What about men vs. women? This is a fascinating question. On average women are about 8% shorter then men, and it follows that if we switched from current BMI to new BMI, a typical woman's reading would go up 2% and a typical man's reading would go down 2%. That may sound small, but it's around half a BMI point, so it would carry many people across the normal/overweight or overweight/obese boundaries (25 and 30, respectively). Sometimes it is said that women can get away with higher BMI readings than men before suffering the health effects of obesity. If true, is this partly an anomaly caused by the current definition of BMI? I don't know.

I must finish by emphasizing that I am an applied mathematician, not
a doctor or an epidemiologist.
The new formulas proposed
above are not based on epidemiological studies and they
may not be a improvement, for all kinds of reasons.
To get a sense of the complexity of the BMI issue, a good
place to start is the Wikipedia article on the subject, and for
a key epidemiological paper supporting the use of the
exponent 2.0, see S. B. Heymsfield et al., American Journal of
Clinical Nutrition, 2007. Obesity will be one of the biggest
health issues worldwide of the 21st century. If great reliance
is going to be placed on a single formula for assessing it, the
justification of that formula deserves careful discussion.

---

[Prof. Alain Goriely of Oxford has contributed the following fascinating remarks. It seems the inventor of the BMI formula himself mentioned the exponent 2.5!]

The BMI was first discussed by Quetelet, the first Belgian scientist (he published his research on the weight of men at different ages in 1832, 2 years after the creation of Belgium, so indeed the first one).

The BMI was known for many years as the Quetelet Index until it was renamed the BMI by the American Ancel Keys. But Quetelet himself was well aware of the [complexity of the] choice of scaling. Here is a quote from his book "A Treatise on Man and the Development of his Faculties", 1842:

*
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.
*

[In the next paragraph, however, Quetelet does indeed suggest the exponent 2 for fully grown adults.]