bmi

BMI - Body Mass Index.  Calculated by taking weight in kg and dividing by height in meters squared.

I've seen lots of criticisms of BMI (most of them justified) but I've never seen the most obvious one. It is not dimensionless.

Being dimensionless means, duh, not having dimensions.  When we talk about quantities they either have dimensions (or units) or they don't.  Science likes dimensionless quantities because they have no dimensions.  The quantities aren't dependent on a particular geometry or environment.  There are lots of them.

BMI is not dimensionless.  It has units of kg per meter squared. A better calculation for BMI, let's call it BMIa, is weight divided by height cubed.

Now you'll say well that isn't dimensionless either (kg per meter cubed).  True.  But we can make it dimensionless by dividing by a density.  A good one would be to use a constant density of an "average" person.  The problem with the BMI dimensions is you can't get rid of them.  There's no kg per meter squared quantity in the universe.  There's no weight per area that applies.

The reason that BMIa can become dimensionless is that it makes sense physically.  Weight is a function of density multiplied by volume.  One characteristics length scale for humans is height.  So we use height cubed to get a volume.  It's self consistent.

It's still not perfect.  The perfect way to do it would be to divide weight by our body volume.  Or we could approximate it by taking height times width times depth.  This would just be our density.  Which is ultimately what determines if we are fat or not.  Dividing by an average human density would then give us a nice index.  If you were average density then you'd be BMI = 1.  Denser would give a BMI > 1.

The effect of the incorrect BMI calculation is that tall people suffer.  Take me for example.  I'm 173lbs and 6'2".  My BMI is 22.3.  That's a normal to high BMI but I'm very thin boned and have almost no fat on me so why would I be so close to being "overweight" (BMI > 25).  Well, because I'm tall.

So this has the effect of distorting the research when looking across countries when average country heights are radically different.