Thinking It Through: Even Simple Concepts Can Become Complex in Context
From Richard C. Larson
Think of math. In gearing up for standardized math tests, we want our students to know (or memorize) so many things: sine, cosine, tangent; solutions to polynomials; matrices; trigonometric identities; etc., etc. Cram it in; spew it out!

And what about true understanding of what they are doing? Take averages. Simple, right? If I have ten numbers, I add them together and divide by 10, and I have the average. “Turn the crank!” Memorization done. Well, how would a math student respond to Garrison Keeler’s weekly musing about fictional Lake Wobegon, in Minnesota: “Lake Wobegon, where all the women are strong, all the men good looking and all the children are above average.” How does our student respond? Well, at first glance, it is impossible for all the children to be above average. The teacher can demonstrate this by taking student volunteers to the front of the class, arranging them in order of height, and showing that the average is somewhere between the maximum and minimum heights. But wait! Suppose the children of Lake Wobegon really are smart, smarter then most of the students in Minnesota! Then it is possible for all the children of Lake Wobegon to be above average when the average is computed across all students throughout the state. So, statements about averages must be carefully crafted to know the reference population against which the average is taken. Do we teach this?

Consider a tasty treat: chocolate chip cookies. Suppose we have a most unusual cookie baker. He bakes two kinds of cookies: cookies with only one chocolate chip and others with nine! In any large batch of cookies, we can expect 50% of each type of cookie. To the baker and to the customer buying cookies, the average number of chips per cookie is five, up to ‘industry standards’! But suppose one day you wake up and find yourself a chocolate chip inside a chocolate chip cookie! You look around. Chances are that you find 8 “sibling chips” in your cookie. In fact, there is a 90% chance that you find yourself in a cookie with 9 rather than 1 chocolate chip. So, if you were to do this fantasy transformation of a random chip into a chocolate chip cookie every morning, 90% of the time you‘d end up with 8 sibling chips and 10% of the time you'd be “an only child”! Over time you would estimate that the average number of chocolate chips in a cookie is 9*(9/10) + 1*(1/10) = 82/10 = 8.2. You and the baker would have distinctly different averages! And both are correct!

Why do we care about chocolate chip cookies? Well, yes, they are delicious, especially those with 9 chips! But, mathematically, aren’t we doing the same probability experiment every time we go to a movie theater or to the mall, board an airplane, or commute to work? There is a sampling bias that finds us more often in busier-than-average situations. Many of us experience movie theaters as 70 to 80% filled – because we tend to go at popular times: Friday and Saturday nights. Managers of movie theaters bemoan the fact that upwards of 95% of seats offered for sale over the course of a week go unsold. Both averages are correct. Or consider two airplanes: one with 180 seats occupied and the other with 20 seats occupied. Average occupancy from the airline’s point of view is 100 paying passengers per flight. But you as a random passenger are 9 times more likely to be on the crowded airplane. Maybe everyone on the planes magically transforms themselves into chocolate chips!

We are speaking of averages, not a difficult concept. And we have only mentioned two of myriad complications with averages. And think of the additional complications with variances, modes and medians, not to mention entire distributions. How much of this do we teach? Or, maybe we need not concern ourselves with teaching each and every complication, but we should focus on preparing the student to adjust to these complications as she encounters them and then figures them out on her own. Isn’t that goal better than memorization and parroting back on a test?

If you found these "Flaws of Averages" interesting, then you might enjoy two popular BLOSSOMS video lessons on the topic, both done collaboratively by Dr. Rhonda Jordan and Dr. Daniel Livengood: Flaws of Averages and Averages: Still Flawed.

Richard Larson is the Mitsui Professor of Engineering Systems at MIT. He is also the Director of MIT LINC and the Principal Investigator of MIT BLOSSOMS.

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