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.

It's That Time of Year Again

It's that time of year again.  The holidays are here, and you can't flip through the channels without eventually hitting a Christmas special.  We watched Miracle on 34th Street yesterday, and I was struck by the scene at the end where all the mail addressed to Santa gets shipped to the courtroom where Kris Kringle is on trial. One of the clerks in the mail room says "there must be about 50,000" letters to Santa Claus that they have to get rid of.  How many letters get delivered to the court room?

In the movie, 21 duffel bags full of letters are brought in and dumped on the judge's desk.  Judging by how the men carry the bags, they certainly weigh between 2 and 200 lbs, so we can reasonable assume there are about 20 lbs of letters to Santa in each bag.  According to the United States Postal website, a standard letter weighs less than 3.5 ounces.  Assuming a 1 ounce letter, we can easily compute the number of letters delivered,

# of letter = (21 bags) × (20 lbs/bag) × (16 ounces/lb) / (1 ounce/letter)

= 7000 letters.

At about 7000 letters, the mail clerk was correct to within about an order of magnitude.  Not too shabby.

A Couple of Updates...

Fermi Lives at Harwoon Union

It's been a busy semester.  A couple of months ago, I received a very nice email from Lisa Therrien and several other teachers at Harwood Union Middle School in Duxbury, VT.  Apparently Fermi fever is catchy, because Ms. Therrien is teaching her students how to make order of magnitude estimates.  I got invited to listen to several of her students describe their estimations over Skype.  It was a very enjoyable experience, and I can truly say that Harwood Union Middle School has some very creative young mathematical minds.  I only wish more teachers incorporated this style of thinking into their math and science classes.


Estimation as a Skeptic's Tool

In addition to Harwood Union School, I also got invited to speak for Ann Arbor Science & Skeptics in October.  It was a great group, and we discussed how estimation can be used as a skeptic's tool. Here's a few of the cool estimates that were suggested by the group:

• How much hair throughout the entire world is grown in one day? (Enough to cover about 10 square miles.)
• What was Forrest Gump’s average speed as he ran across the country? (About 5 cm/s.) 
• If the Sun disappeared, how quickly would the temperature drop on Earth? (About 0.1°C per century if you’re talking about the average temperature of the whole Earth, or about 20°C per day if you’re talking about just the surface temperature.)

The only downer of the trip for me was that I had to admit my secret shame.  I, physicist Aaron Santos, have never actually read a Carl Sagan book.  It's a problem that I will hopefully be remedied in the near future, but for now, feel free to throw your rotten tomatoes.

Shark Blood

"Nom nom nom nom nom!"

I've been seeing this commercial a lot lately, so I got curious.  How sensitive would a shark's nose have to be in order to detect a drop of blood from one mile away?

Let's say you took a drop of blood and dumped it in the ocean.  Because of diffusion, that drop would eventually spread out in all directions, much like a drop of red food coloring in a glass eventually spreads out and makes the entire glass red.  If the blood spreads out equally in all directions, then it will eventually cover a hemisphere 1 mile in radius.  This means the blood will be distributed over a volume of

V = 2 π R³

=2 · (3.14159) · (1 mile)³

= 2.6×10¹⁰ m³.

A drop of blood is about 1 mL.  This means the density of molecules in a 1 mile radius drops by a factor of about

(1 mL) / (2.6×10¹⁰ m³) = 3.8×10^-17.

That's very tiny.  I'm not exactly sure what part of the blood the shark is supposed to smell, since there are many components that make up blood.  Let's assume he's smelling the red blood cells.   A healthy red blood cell count is in the range 4.6-6.1 million per μL = 5×10^-9 per mL.  If the blood spreads out uniformly over this distance, there will be

(3.8×10^-17) · (5×10^-9 per mL) = 1 red blood cell every 5 m³.

That's a very tiny amount.  I highly doubt a shark, or any other creature, could detect something with this small a concentration.  Besides, the MythBusters already busted this myth during shark week. 


Bonus Shark Blood Question #1: Even if a shark could detect concentrations this small, how long would it take a red blood cell to travel 1 mile?

Bonus Shark Blood Question #2: Where does a shark's blood come from if blood comes from bone marrow and sharks are cartilaginous fish?  (I know...this is not a numbers question, but I've always been curious.)

Mission Improbable

I watched Mission: Impossible yesterday and was reminded of two things: (1) there was a time when people thought Tom Cruise wasn't crazy and (2) there was a time when people thought the internet was.  Long before anyone dreamed of Facebook, Twitter, or rage comics, the internet was a pretty novel concept, and many people, myself included, weren't ready to dive in head first.  I haven't thought about this for awhile, but I'm pretty sure Mission: Impossible was the reason I decided to get a computer.  After all, if super spies used the internet, then it was probably pretty darn cool.  Who knows?  If I hadn't seen it, maybe I wouldn't be blogging this now.¹  Still, there's one thing that bugged me in my second viewing.  In the iconic scene where Cruise is dangling from a rope while trying to steal sensitive government data, an alarm will go off and ruin the mission if he touches the floor, makes a sound, or raises the temperature of the room by 1° F.  There were a few close calls on these fronts, but I was particularly skeptical of the temperature restriction.  How long would it take your body heat to raise the temperature of the room by 1° F?

Human bodies release heat at a rate of about P = 100 W.²  This heat will spread throughout the room, which has a volume of roughly 15 ft by 15 ft by 30 ft = 200 m³.   With a density of 1 kg/m³, the total mass of air in the room will be about m = 200 kg.  The specific heat of water is about c = 1 J/g·K.  From this we can fin the time it will take to raise the room by 1° F (~0.6 K),  

 t = m c ΔT / P

= (200 kg) · (1 J/g K) · (1° F) / (100 W)

= 20 minutes.

Apparently, they would have time as long as they were fairly quick about it.³

[1] Admittedly, if I had never seen the movie, I couldn't be writing this post right now, but you get the idea.
[2] See the post titled "Igloos Rock!"
[3] I've neglected the fact that the room is temperature controlled.  As the temperature rises, the temperature control will turn on the AC to return it to normal.  This may have given them even more time.

And That's the Tooth

Yes, I watch absolutely terrible movies.

While watching Family Guy the other day, I saw this.  The Tooth Fairy has been played by everyone from Eddie Murphy to The Rock, but this one got me thinking: How many teeth does the Tooth Fairy collect each year?¹

Humans are born with 20 baby teeth.²  Since you live about 80 years, your rate of tooth loss is about

(20 teeth per person) / (80 years)
= 0.25 teeth / year · person.

Not all of the world believes in the Tooth Fairy, so it's a safe bet that she doesn't visit every newly gap-toothed child.  Assuming only 10% of the world (roughly 700 million people) have been visited by the Tooth Fairy, then the total number of teeth that have been collected this year would be

(10%) · (1 year) · (700 million people) · (0.25 teeth / year · person) = 1.8×10⁷

That's 18 million teeth, or roughly enough teeth to fill a refrigerator. Apparently, our cartoon Tooth Fairy is fairly accurate.


[1] In case you're wondering, most of my inspiration for estimations comes from watching cartoons.
[2] I'm not including wisdom teeth in this calculation since (a) the Tooth Fairy generally doesn't collect these, and (b) when removed, they're considered biological waste, so the dentist doesn't give them to you.  As if having surgery isn't bad enough, they don't even let you keep them a souvenir, which, incidentally, belonged to you in the first place.  I only discovered this after having shoulder surgery where the doctors removed some small bone fragments.  I was particularly upset that I couldn't keep the bone chunks, because I figured the Shoulder Fairy probably paid a lot more than the Tooth Fairy.  Alas, it wasn't to be.  My mom tried to convince me that it wouldn't have mattered anyway because there's no such thing as a Shoulder Fairy, but I think she was just trying to cheer me up.