Wednesday, January 18, 2012

Ballparking: Practical Math for Impractical Sports Questions

I have a new book.  It comes out in a few months.  You should buy it.  It will make you smile.

Fuzzy Numbers

We're halfway through January, which means most people have probably given up on their New Years resolutions.  For those who are still committed to going to the gym every day and losing that last three pounds, I salute you, but let me give you one bit of warning: Don't trust the nutritional labels on backs of packages.  If you're counting calories, you may find yourself disappointed.  One of the most difficult things to impress on students is that real world numbers aren't exact.  There's always some degree of uncertainty.  When you look on the back of that energy bar and it says "200 calories", it's probably not exactly 200 calories.  Perhaps it's 201 calories.  Or 197 calories.  Or 234 calories.  Just how far off is it?  In a science lab, you would never report a real world number without providing some error range.1  It may seem like I'm being a pedantic stickler here, but real world numbers are incomplete without an error range.  Saying "This chocolate covered bacon cake is only 137 calories", is not as truthful as saying "This chocolate covered bacon cake is only 137 calories plus or minus 6,723 calories" when there's an enormous error range.  Just how far off are nutritional labels?

To investigate how far off the reported numbers on a nutritional label are, I first bought a package of Mini Clif Bars.2  Given the limited amount of equipment at my disposal, it would be difficult to measure the number of calories in each bar directly.  However, it seems reasonable that the number of calories in a bar would be proportional to its weight, so you can estimate the caloric fluctuations by measuring the fluctuations in the mass.  Each bar had a listed weight of 28 g and was said to contain 100 calories.  The package contained 18 bars with weights ranging from 28.2 g to 33.3 g.  The bars had a mean weight of 30.7 g and a standard deviation of 1.3 g.  That's a 4% fluctuation away from the mean, which itself was almost 10% larger than the listed value.  Scaling these fluctuations up to the listed value of 100 calories, we find that the actual number of calories for a Mini Clif Bar is likely closer to 109.5±4.5 calories.

 A histogram for the different Mini Clif Bar masses (in grams) shows significant fluctuations away from the mean.

Let me be clear.  I'm not insinuating that the good people at Clif Bar & Company have devised a dastardly plot to plump up casual exercise enthusiasts.  If anything, I'm kinda happy I got more food than I paid for.  My only complaint is that nutritional labels don't include a range detailing how imprecise or "fuzzy" these numbers are.  No matter how careful a manufacturer is, there are always the fluctuations that will creep in, and we would do well acknowledge this fact.

[1] The term "error range" is a bit misleading because the word "error" implies a making mistake. "Oops...I really didn't mean to drop that now bloody hatchet down your underwear.  Please forgive my error."  In reality, error ranges have nothing to do with making mistakes.  All real world measurements have some degree of uncertainty, so perhaps it's better to think of them as "uncertainty" ranges  rather than "error" ranges.

[2] Admittedly, a rigorous study should include more than one box and a variety of different food types, but doing one sample is enough to illustrate my point.  Plus, I like Mini Clif Bars.