Quantitatively Thinking

John Oliver said it best: April 15 combines Americans two most-hated things: taxes and math.  I’ve been thinking about the latter recently after hearing a fascinating talk last weekend about quantitative literacy.

QL is meant to describe our ability to think with, and about, numbers.  QL doesn’t include  high-level math skills, but usually is meant to describe  our ability to understand percentages and proportions and basic mathematical operations.This is a really important type of literacy, of course, but I fear that the QL movement could benefit from merging QL with SL–Statistical Literacy.

No surprise, that, coming from this blog.  But let me tell you why.  The speaker began by saying that many Americans can’t figure out, given the amount of gas in their tank, how many miles they have to drive before they run out of gas.

This dumbfounded me.  If it were literally true, you’d see stalled cars every few blocks in Los Angeles.  (Now we see them only every 3 or 4 miles.)  But I also thought, wait, do I know how far I can drive before I run out of gas?  My gas gauge says I have half a tank left, and I think (but am not certain) that my tank holds 16 gallons.  That means I probably have 8 gallons left.  I can see I’ve driven about 200 miles since I last filled up because I remembered to hit that little mileage reset button that keeps track of such things.  And so I’m averaging 25 mpg. But I’m also planning a trip to San Diego in the next couple of days, and then I’ll be driving on the highway, and so my mileage will improve.  And that 25 mpg is just an average, and averages have variability, but I don’t really have a sense of the variability of that mean.  And this problem requires that I know my mpg in the future, and, well, of all the things you can predict, the future is the hardest.  And so, I’m left to conclude that I don’t really know when my car will run out gas.

Now while I don’t know the exact number of miles I can drive, I can estimate the value.  With a little more data I can measure the uncertainty in this estimate, too, and use that to decide, when the tank gets low, if I should push my luck (or push my car).

And that example, I think, illustrates a problem with the QL movement.  The issue is not that Americans don’t know how to calculate how far they can drive before their car runs out of gas, but that they don’t know how to estimate how far they can drive. This is not just mincing words. The actual problem from which the initial startling claim was made was something like this: “Your car gets 25 mpg and you have 8 gallons left in your tank.  How far can you drive before you run out of gas?”  In real life, the answer is “It depends.”  This is a situation that every first-year stats student should recognize contains variability.   (For those of you whose car tries to tell you how many miles you have left in your tank, you’ve probably experienced that pleasing event when you begin your trip with, say, 87 miles left in your tank and end your trip 10 miles later with 88 miles left in your tank.  And so you know first hand the variability in this system.) The correct response to this question is to try to estimate the miles you can drive, and to recognize assumptions you must make to do this estimation.  Instead, we are meant to go into “math mode” and recognize this not as a life-skills problem but  a Dreaded Word Problem.  One sign that you are dealing with a DWP is that there are implicit assumptions that you’re just supposed to know, and you’re supposed to ignore your own experience and plow ahead so that you can get the “right” answer, as opposed to the true answer. (Which is: “it depends”).

A better problem would provide us with data.  Perhaps we would see the distances travelled on 8 gallons the last 10 trips.  Or perhaps on just 5 gallons and then would have to estimate how far we could go, on average, with 8 gallons.  And we should be asked to state our assumptions and to consider the consequences if those assumptions are wrong.  In short, we should be performing a modeling activity, and not a DWP.  Here’s an example:  On my last 5 trips, on 10 gallons of gas I drove 252, 184, 300, 355, 205 miles.  I have 10 gallons left, and I must drive 200 miles.  Do I need to fill up? Explain.**

The point is that one reason QL seems to be such a problem is not because we can’t think about numbers, but that the questions that have been used to conclude that we can’t think about numbers are not reflective of real-life problems.  Instead, these questions are reflective of the DWP culture.  I should emphasize that this is just one reason.  I’ve seen first hand that many students wrestle with proportions and basic number-sense.  This sort of question that comes up often in intro stats — “I am 5 inches taller than average.  One standard deviation is 3 inches.  How many standard deviations above average am I?”  –is a real stumper for many students, and this is sad because by the time they get to college this sort of thing should be answerable through habit, and not require thinking through for the very first time. (Interestingly, if you change the 5 to a 6 it becomes much easier for some, but not for all.)

And so, while trying to ponder the perplexities of finding your tax bracket, be consoled that a great number of others —who really knows how many others? — are feeling the same QL anxiety as you.  But for a good reason:  tax problems are perhaps the rare examples of  DWPs that actually matter.

**suggestions for improving this problem are welcome!