# Some Obesity Math

When I was in graduate school, we had an old book laying around titled something like How to Lie with Statistics.  I don't remember ever actually reading the book but the title says it all: one can tell two entirely different stories depending on how one chooses to report the numbers.

I fear that much of the rhetoric surrounding obesity has fallen into this trap.  According to CDC data, the average weight of men aged 40-49 has increased roughly 15 lbs in the past 10 to 15 years (compare data in this publication to this one showing the average weight going from about 187 lbs in 1988-1994 to about 202 lbs in 2003-2006; I should also note that more recent data shows these weight gains leveling off).

Fifteen pounds doesn't seem like a huge number to me (I've personally lost and gained much more than this amount in my adult life).  So, how is it that this small to medium sized increase in average weight gets translated into a message that there is an epidemic?  Part of the answer is that when scientists translate averages into prevalence rates, the numbers look a lot scarier.

Stick with me while I illustrate with an example.

Let’s take a hypothetical population of men whose average weight is 180 lbs.  Suppose, men’s weights vary in the population according to a normal distribution with a standard deviation of 30 lbs.  This means roughly 68% of the men will have weights between 150 lbs and 210 lbs.  Suppose also, for convenience sake, that all the men are the same height: 5 ft 10 inches.

Obesity is defined as BMI greater than 30 (BMI is weight in kg divided by height in meters squared).  In our hypothetical example, where everyone is the same height, a man will be obese if he weighs more than 209 lbs.  Moreover, given our assumptions about the normal distribution, we can readily project that 16.6% of men in this population will be obese (1 minus the cdf of a normal distribution with mean 180 and standard deviation of 30 evaluated at the point 209 is 0.166).

What if all men gain a paltry 5 lbs?  The average weight goes from 180 up to 185 lbs.  Yet, (again given the assumption of the normal distribution), obesity prevalence will go up from 16.6% to 21.1%.

Thus, we have what most of us would consider a rather trivial gain in weight (an increase in 5 lbs or a 2.8% increase in weight); however, we have what appears to be a rather dramatic increase in obesity prevalence (prevalence goes up 4.48% or a 27% increase in prevalence of obesity!).

If we run through the same example again assuming men gain an average of 10 lbs, we can find that obesity prevalence goes up almost 58% even though weight only increased 5.5%!

Both statistics are "true" but they tell very different stories.