Tuesday, March 28, 2006

 

Statistics and Stereotypes

As a social scientist, I’ve taught classes that involved data. Not math, per se – I never really taught them how to manipulate the data in any meaningful way – but just references to percentages, proportions, etc. (Yes, I’ve also taught composition. I’ve had a weird freakin’ career.)

Anyway, if I can ever solve this one (or steal a solution from my faithful readers, hint, hint...), I might actually get through to a cluster of students I’ve never really reached. Oh, faithful blogsophere, I ask your assistance...

In the extended triple overtime after the 2000 election, a student asked me why the red/blue map was so red, if the election was so close. I considered that a great question, and responded that one reason was that Democrats are likelier to live in cities, so a lot of votes are crammed into a small space. She asked why Democrats are likelier to live in cities. I responded that one reason was that party affiliation often correlates with race, and so areas with high concentrations of African-Americans could be expected to vote Democratic, and that usually means cities. The exchange after that:

Student: So you’re saying black people are Democrats?

DD: No, but the Democrats usually win 80 to 90 percent of the black vote. So if a city is mostly black, you could be pretty confident that it would vote Democratic.

Student: That’s not true. I think it’s up to the individual.

Huh?

I’ve had variations on this conversation many times, and that student response always brings me up short.

Is there a reliable way to prevent this apparent confusion of description and prescription? Is it that they don’t understand what percentages mean? I’m perplexed. I think the student is trying to be morally virtuous and reject stereotyping, which prevents her from actually trying to understand what I’m saying. Once the student decides that you’re evil, of course, no amount of explanation is going to work. Is there a way to describe demographics without triggering a defensive anti-stereotyping response in students?

Comments:
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My experience of this is that students "don't believe" in statistical probablity. Like it's a matter of faith, rather than, um, just-plain-the-way-things-are.

I usually talk about correlation versus causation. But maybe this just confuses things further.
 
So I'm new at blogger and somehow deleted my comment! ARGGGHHH. As maggiemay says, most students don't believe in probability. BUt that is the underlying explaination. If only 20 out of 100 african americans identify with the republicans (for whaterver reason... I'd ask the studen to ignore why that is the case for a minute), then its reasonable to believe that a city like Detroit which has say 800,000 african americans and 200,000 non african americans will see at least 64% of the votes go towards democrats.

In my last comment I also pointed out that we don't weight votes by how big a yard is, we weight them by person. So, if on average those who vote blue (minorities in many cases) have smaller yards concentrated in cities, then a small proportion of a map will be blue.
 
You might find this site helpful:
http://www.geog.ucsb.edu/~sara/html/mapping/election/map.html
 
I'd say something like, assuming each person makes their own decision as to how to vote, if 8 out of 10 of them decide the same way, then you have the results shown.

The problem is one of understanding probability. I also have trouble communicating logical possibility vs. probability.
 
there are quite a few aspects to this, but here's one. There's a big difference between a sample and an individual perspective and there are times when we each take both. There's a famous social psych paper (I can't remember by whom now) that begins with the example of a student complaining to a professor about the unfairness of an exam. From the professor's point of view, the exam was fair (because many other students did well); from the student's perspective it's not (s/he did well on all the other exams taken that semester). These are inherently different perspectives, and it's difficult to figure out a priori which is right.

If you take the perspective of an individual voter, the role of party identification may not loom that large. That might even be right. For example, even in your 80/20 case, how confident would you be to identify *who* would vote against the tide? If you could bet your retirement on the vote of a single voter with those odds (say you get 9 times whatever you've saved up if you're right), would you?

Getting people to take the sample/statistical perspective is a really important cognitive shift and one that doesn't come naturally.

And that's not always wrong. Right now I'm listening to an audiobook about the fall of Long Term Capital Management and the hubris of these economists is just profoundly stunnind got me.
 
Try telling your students that it is much like the actuarial data that insurance companies use. An actuary can't say whether a given female-Hispanic-nonsmoker-under-the-age-of-forty will die next week, but he can suggest how likely it is. Given a large enough population, the actuary can tell the insurance company how many funerals to expect each month.

And, of course, those predictions will change with time, if enough individuals make individual decisions that change the math. That's what keeps actuaries employed. (It used to be that most Southern blacks voted Republican, after all.)
 
Dicty -- I love the insurance example! That might actually work, since traditional-age students are completely obsessed with the cost of car insurance. Thanks!
 
Dicty, I'm going to steal that example, too. I face this problem with my 101 students, and even some of my graduate students have problems making the leap to statistical modeling.
 
The maps from coyote librarian look like good aids for visual learners, at least regarding the disproportionality in the land area. Another example that students may enjoy is casino gambling. In each game the probability that the house will win is greater than 50% (as far as I know), but individuals win often enough for casinos to attract a crowd. Each individual can hope to win some money, but the probabilities ensure that the casinos will make money, since enough people will lose in the long run.
 
Actually, I don't think anyone has quite hit the nail on the head. I too encounter this sort of thing all the time. This type of student has a deep committment to individualism (as do most), but also is half-hearted/conflicted about the usual liberal platitudes. So, you combine Bush v. Gore + black people + people expressing strong preferences (voting) in large numbers and the result is a wacky mix: "Wait, wait, what am I supposed to say? Bush...um..ahh...black people...um..ahh...people in cities...um..ahh...people voting in large numbers against...um..ahh...Oh, I got it, it is really up to the individual...you know, like, it is a personal issue! So, support for dominant US creed? Check. Don't say anything bad about black people? Check. Don't reveal that I am from boonies and all my family voted for Bush? Check. Yea, THATS the ticket. I see it ALL the time!
 
You could suggest they look at some parts of Freakonomics (Levitt,Dubner). One might not like (or agree with) some of their analyses, but they do quite a decent job of explaining how black or white (or other characteristics) can simply be instrumental variable for the real reason of how people vote/raise their kids/study in school, etc.

I do find it is written in layman's terms, the causality/correlation is also explained to some degree. Maybe they enjoy it.
 
To get back to the initial problem, that is, why the 2000 red/blue map looked so red when actually Gore got half a million more votes than Bush: Land doesn't vote. People vote.

A student can see this by looking at a "population cartogram," a map with the states redrawn so that a state's size is proportional to its population. Such a map can be found on http://www-personal.umich.edu/~mejn/election/, the third map down.

In our family we have an instinctive understanding of math and it annoyed us that people were fooled by the 2000 election red/blue map into thinking that Bush won a large majority of votes, when actually he lost the popular vote. We referred to the red/blue map as the Cow Vote Map, because the states are scaled more closely to their cattle population than their voter population.

-- Cardinal Fang
 
Bush's approval rating among African-Americans is 2. Two. That is to say, two percent of black Americans think Bush is doing a good job.

So I think it's safe to say that yes, black people are Democrats.

-- Cardinal Fang
 
There are some other points to consider here.

First, my guess is that the student is bothered by the apparent causal relationship in your explanation. I know I'm simplifying this, but they might think you're saying that someone will vote Democratic because they're African American.

They might be looking at your explaination in the same way that you would look at statement like this:

Everyone dies after eating peanuts.

While, technically that statement is true, (eventually, we all do, right?), it just doesn't sit well because the two actions, eating peanuts and dying aren't necessarily related (at least not for everyone).

So, the point that I'm coming to here is that there are probably multiple characteristics of cities that correlate with voting patterns and it's hard to know which characteristics caused the observation.

For example, people in cities tend, on the whole, to be more highly educated than people in rural areas. And, there is a strong correlation between education and voting patterns. Needless to say, the more education you have, the more likely you are to vote Democratic.
 
I have two examples I use for this -- the first is the insurance example, and the second (if I have 10 minutes) is to talk about the expected value of the roll of a six-sided die. Since that expected value is 3.5, but it is, of course, impossible to roll that, it seems to bring home the difference between statistical expectation and the results of a single event. The two together seem to be sufficient to plant the basic seeds of how probability works, which is a good thing.
 
Hmmm...

It seems we keep forgetting a couple key points.

1. This has nothing to do with sampling. The election is a measure of the total population (the only population that matters is the population of votes.) There is no sampling error introduced in elections, no deviation frome the mean--none of that. Remember, statistics is designed to help us infer something about the population when we cannot actually measure the whole population.

2. The typical maps were so largely red because a) in 2000 they didn't do shading, thus a 50%+1 vote margin for one candidate would receive that candidate's color--period. b) Many of the maps were developed that way based on the electoral college vote-again, not a shaded measure, but an all one color measure.

3. Sampling only comes into play when we seek to understand the correlation between numbers of a population that have a given race, and their political affiliation. This is usually done by asking a series of questions, in a poll, and then looking at the paired data "From Venus, Green Party" for instance. Yes, the data then reaches some "conclusions" and must be shared with some understanding of the sampling error, and sampling methodology, and the like--but these sorts of things are designed to identify trends, not individual behavior.

4. No one likes to have discussions that seem to imply prejudging of individuals, and saying that someone is "____" (race) and thus highly likely to be "_____"(party) leads one to assume that when engaging that person. And while we all want to think we are free of prejudging, we all do it. The blog "Targuman: Inside Higher Ed :: The Real Bias in the Classroom" references an article where students pre-judge professors based on what they THINK a professor's views are, rather than judging each individual.


For instance, based on this discussion, I of course would continue to say that not all academics are liberals, but those that commented here generally seem to be (and perhaps even are somewhat prejudging of people who come from rural areas (refered to in a pejorative as "boonies" by the way...) and thus assuming they voted for Bush.

Bottom Line: Tell your students to please treat each person you meet as an individual, and make few assumptions about them before getting to know them. But tell them to understand that in the aggregate certain views are "dominant."
 
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