Tyranny of the Center

Tyranny of the Center: a favorite phrase of mine that I keep threatening to write about. Here is a first and brief stab, inspired by my having recently used it in a comment on ThinkThankThunk.

In elementary statistics, you learn about measures of center, especially mean, median, and mode. These are important values; they stand in for the whole set of data and make it easier to deal with, especially when we make comparisons. Are we heavier now than we were 30 years ago? You bet: the average (i.e., mean) weight has gone up. Would you rather live in Shady Glen than Vulture Gulch? Sure, but the median home price is a lot higher.

We often forget, however, that the mean or median, although useful in many ways, does not necessarily reflect individual cases. You could very well find a cheap home in Shady Glen or a skinny person in 2010. Nevertheless, it is true that on average we’re fatter now—so when we picture the situation, we tend to think that everyone is fatter.

One of my goals is to immunize my students against this tendency to assume that all the individuals in a data set are just like some center value; I think it is a good habit of mind to try to look at the whole distribution whenever possible. Let’s look at a couple situations so you can see why I care so much.

Income by Sex

Suppose we sample 1000 people from the 2000 U S Census, and ask whether the men earned more than the women. That’s the stereotype, right? And, on the average, it’s correct. Here is a table:

Personal income by sex. Men have more.
Personal Income by Sex, sample of 1000, 2000 US Census. All figures in this post were done using Fathom.

The top number in each cell is the mean, the bottom is the median. It’s clear that men earn more. There are no fancy counterintuitive shenanigans going on—no Simpson’s paradox, no Monty Hall game—so what am I complaining about?

Here is my beef: a single summary number, an aggregate statistic such as the mean, always eliminates information. That’s its nature, the whole point. We make the mean so that we don’t have to talk about each individual in the data set. But if we use only the mean, and forget to look at the distribution, we might miss something important. Here are two more representations of the same data, a histogram and a box plot:

Income by sex histogram
Income by Sex, a histogram. Same data. The vertical scale is density; don't worry about it.
Income by sex box plot
Same data, this time in a box plot.

The men are still earning more than the women on average, but the distribution gives a richer picture. In particular, there are a lot of men who are not earning very much. Not as many as there are women, to be sure, but seeing the distribution squashes the misapprehension you may have that all men earn more than all women.

Rather than elaborating on this point, I’ll just let it stew for now. Okay, one elaboration: I think that learning to look at the whole distibution is important for being a good citizen in a land where we’re bombarded with stats, many of which are generated by people and institutions with agendas. One of the ways you can be more forceful about your point when it’s a little sketchy is to choose some aggregate statistic—often but not always a measure of center—that makes your point better than the data as a whole do.

[A small note: the astute reader may rightly point out that this is why we teach about center and spread, not just center. Shouldn’t we report interquartile range (IQR) or standard deviation? To that I say two—no, three!—things: (1) You’re right; but (2) a table with the IQR is still pretty opaque, whereas a picture tells the story, and since we have the data and a computer to plot it, why not show it all? and (3) this distribution is so skewed that a measure like standard deviation can be misleading. And a fanatical devotion to the Pope. Onward!]

Assessment (soap box alert)

We are, as a society, trained to reduce complexity, ideally to a single number. That has survival value—you don’t care what kind of saber-toothed tiger it is, you just gotta run—but today many things are just more complex. Nowhere is this more pernicious than in school. We add up the scores on tests to get a single score. We average the scores on tests and quizzes, and weight them to get a grade. Then we average the grades over different courses to get a GPA. We let the kid with the highest GPA be Valedictorian.

All along the way, at every step, we compare.

  • Everybody knows that the valedictorian is the “best student.”
  • If you get an A in English and a B in Math, you’re good at English and not so good in Math.
  • If you get an 85 on the test and Aloysius got an 81, you did better and know more.

Every time you wind up on the short end of a comparison, you feel worse or more alienated. That’s bad enough, but the kicker is that it’s a lie. Humans are multidimensional, and it’s not fair to collapse them to a number. It’s like giving cuisines a rating; it doesn’t make sense. Which is better, Italian or Chinese? It depends. What do you feel like eating?

I’m not advocating a touchy-feely everyone-has-something-to-offer philosophy (though everyone does); there is such a thing as a right answer, and some work is better than others, just as there is good Italian food and crummy Italian food. And I know that sometimes we just can’t look at the details; we want to make a decision quickly. But hey, we’re talking about kids here, and learning, the future of our country, and our profession! That’s why I’m looking at standards-based grading for this year: I want something that shows me the complexity of a kid’s understanding rather than reducing it to some average.

Put another way, it’s seductively easy to think of a kid as “a B student.” Easy but deeply wrong. We may have to bend over and give a kid a B because we have to assign a letter at the end of a semester, but I want to find a way to help keep myself from using the letter in my everyday opinion of these kids, and something that focuses on many dimensions of understanding may be just the ticket.

Author: Tim Erickson

Math-science ed freelancer and sometime math and science teacher. Currently working on various projects.

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