## Saturday, December 5, 2015

### A Fine Line

“There is a fine line between numerator and denominator,” says Math Addict’s post on Facebook and my new sweatshirt from SunFrog.  “Only a fraction of you will understand this.”  An in-joke for math nerds who know that the numerator on top is divided by the denominator on the bottom to define a fraction.
Not only is there a fine line dividing numerator from denominator, there is a line dividing numerator thinking from denominator thinking.  Numerator thinking oversimplifies because it leaves out proportionality and probability that denominator thinking accounts for. In other words, sometimes you can’t just add, subtract, or multiply the given numbers to figure out what’s going on, you have to divide.
People who say, “I don’t want to travel abroad, what with all those terrorists,” lack denominator thinking with its understanding of probability.  For instance, your chance of being killed by a gun in our own country is more than a thousand times greater than being killed by an overseas terrorist.

From CNN:  http://www.cnn.com/2015/10/02/us/oregon-shooting-terrorism-gun-violence/
Using numbers from the Centers for Disease Control and Prevention, we found that from 2001 to 2013, 406,496 people died by firearms on U.S. soil. (2013 is the most recent year CDC data for deaths by firearms is available.) This data covered all manners of death, including homicide, accident and suicide.
According to the U.S. State Department, the number of U.S. citizens killed overseas as a result of incidents of terrorism from 2001 to 2013 was 350.
In addition, we compiled all terrorism incidents inside the U.S. and found that between 2001 and 2013, there were 3,030 people killed in domestic acts of terrorism. This brings the total to 3,380.

Another example of numerator thinking is the warning that chocolate is toxic to dogs!  Yes, but how toxic depends on how much the dog eats, how strong is the chocolate is, and how big the dog is.   http://www.petmd.com/dog/chocolate-toxicity   As has been known for 500 years, the dose makes the poison.  Ingesting minuscule amounts of a substance is unlikely to be harmful.
Proponents of a cause can use oversimplified numerator thinking to persuade you of claims that are plausible but not true.  For instance here’s how to “prove” just as many vaccinated people get sick as unvaccinated people do. Let’s say 900 people are vaccinated for a disease and 100 are not vaccinated.  The disease affects 90% of those exposed so 90 unvaccinated people get sick.  If the vaccine is 90% effective, 90 vaccinated people still get sick. Therefore, since both 90 vaccinated and 90 unvaccinated people get sick, vaccination doesn’t work! This anti-vaccination argument conveniently leaves out the 810 people out of the 900 vaccinated who stayed well.
My favorite example of bogus numerator thinking is the food miles fallacy.  Fifty miles is less than 1600 miles so it should save fuel to eat food harvested within a 50 mile radius than from 1600 miles away, says the numerator thinker.  However, a Prius that gets 50 miles to the gallon, drives a 50 mile round trip to a local farm, and carries 50 pounds of food uses three times as much gasoline per pound of food transported (the critical statistic) as a semi that gets four miles to the gallon, drives 1600 miles, but carries 60,000 pounds of food. You have to use denominator thinking and divide by the weight of food carried, in order not to be fooled by this claim that’s plausible but not true.