Statistics, politics, and how we understand the world, or, Are men taller than women?

by Alan Cohen

Ron Paul finally admitted he could not win the Republican presidential nomination in the US.  During this year’s Republican primary, we’ve watched a particular sort of political drama that has become typical. No, I’m not referring to the competition to be the most extreme candidate; I’m referring to the belief that somehow, against all odds, ones preferred candidate will come from behind, defy conventional wisdom, and win.

Rick Santorum, Newt Gingrich, and Ron Paul stayed in the race long after they had lost all real hope of winning. Many of their supporters encouraged this, and refused to concede defeat. This attitude is due, in my opinion, to a pervasive type of ignorance: statistical ignorance. This ignorance causes us to misjudge political races, and has lots of other bad societal consequences as well. Ask yourself the following question: if I flip a coin 100 times, how many heads will I observe?

Now ask it again, but confine yourself to the following four responses:

a) 50

b) 50, as long as the coin is fairly weighted

c) we can’t say for sure

d) we can’t say for sure, but we can precisely identify the probabilities of different results if we know how the coin is weighted

When presented with these four choices, it is obvious that d) is the best answer. However, most of us, when asked the question without the multiple choice options, will tend to respond more along the lines of a), b) or c). I think our brains are built (i.e., have evolved) to dichotomize things when possible. People are good or bad, with us or against us. We don’t deal well with uncertainty, and we even want certainty about uncertainty: either we know or we don’t know. (We are also sensitive to cheating, thus the appeal of response b). But d) is actually quite easy to demonstrate:

(For the geekier among you, it’s easy to play with this yourself in the free software R. The command hist(rbinom(10,100,0.5)) generates a histogram of a random binomial distribution with 100 fair flips of a coin (probability =0.5) 10 times, as in the first panel.)

So what does coin flipping have to do with politics? Well, the point is that the distributions above depend on the properties of the coin: that it is fairly weighted. A coin is a simple system, with only two possible outcomes and very few things that affect these outcomes. And regardless of how we change the weighting of the coin, we can still assign a very specific probability distribution to the two outcomes, as long as we know the weighting.

Political systems are more complex, but the same essentially holds true. We can identify that a certain state – Oklahoma, for example – has a certain overall tendency to vote Republican, and that this tendency is affected to a certain extent by the national political climate, local economic conditions, and so forth. We can also use polling data to estimate the outcomes of elections. Polls can sometimes be VERY wrong – as they were recently in the provincial elections in Alberta. But we have been doing polls for long enough now that we can actually estimate quite well how much error we can reasonably expect from polls in different situations: primary versus general elections, elections for senate versus president, one month versus six months before the election, and so forth.

Taken together, this means that with the knowledge of the relevant factors, we can build a statistical model to predict the outcome of elections, and that often the outcome is essentially determined with 100% certainty well before the election occurs. But more importantly, we can also estimate the uncertainty in our statistical model. It’s like saying that we predict that out of 100 coin flips, at least 40 will come up heads, and that the chance of being wrong is 1.75%.

In many cases, this will mean that there is a lot of uncertainty in the outcome of elections. And complex statisical models can get things wrong. But, to take an obvious example, the election of Barack Obama was considered big news, and was not generally viewed as a foregone conclusion until it happened. It certainly was big news, and even if it was almost certain to happen it was worth celebrating the moment with big headlines representing the historic nature of the event, but from a statistical perspective it was essentially a foregone conclusion a month or more in advance, barring some major scandal or unforeseen event.

In this context, it was bizarre  (though inevitable) to see McCain claiming right up to the end that he was going to win, and to see the “surprised” reactions of the media on election night. Everyone knew Obama was the favorite, but no one seemed to appreciate how much of a favorite, except statisticians at blogs like 538. And perhaps this is appropriate – we wouldn’t want the news media to have called the race for Obama weeks early, something that could depress turnout and change the outcome. But it nonetheless reinforces in the public’s mind the idea that anything can happen, that maybe there will be a last minute upset, and that we can ignore some of the fundamental statistical principles that govern the universe.

As mentioned earlier, this principle of “precise uncertainty” or “measurable uncertainty” can be counterintuitive. Our brains aren’t built to deal with it, though they can be trained. Nonetheless, I think increasing appreciation for it can help us understand many aspects of our world much better.

Here’s another example: Are men taller than women? Most of us will answer yes, but obviously there are some women who are taller than some men. Here’s an approximation of what we might expect:

With a distribution such as this, we can calculate the percentage of the time that men are taller than women, and by how much. Height may not seem that important, but imagine that we extend this principle to personality traits. There are raging debates in society about whether men and women are biologically different, or if it’s all cultural. Are men from Mars and women from Venus? The answer is not yes-no; the answer is that different personality traits have different frequency distributions in men and women, and that there are simultaneously (a) substantial overlap and (b) substantial differences in the averages. (When we look across cultures to see how much this changes, and when we look across species to see what sorts of male-female differences appear determined by natural selection, we can also arrive at reasonable estimates for what differences are due to nature versus nurture. Hint: it’s both!)

No one is surprised that there is overlap in height or personality between men and women, nor that there are average differences. But because our brains are not trained to appreciate probability distributions, we try to simplify the reality in counterproductive ways, pretending either that there are no differences or that the differences are absolute. This is one of the reasons I believe that better statistical education in schools could teach tolerance, compromise, and reason in many of our fundamental societal debates.