College kids are better than monkeys

A misleading headline from MSN: College Kids and Monkeys About Equal on Math.

This is in reference to the work of Elizabeth Brannon, who actually claims that certain monkeys have an intuitive "number sense" which is as good as humans; see for example this article. Despite what those of us who teach college students may occasionally think, they are better at mathematics than monkeys.

Tall people are smarter?

Men are more intelligent than women because they're taller, say psychologists Satoshi Kanazawa and Diane J. Reyniers.

I don't feel like picking this one apart. Have fun.

Schwitzgebel and Cushman's "moral sense test"

Eric Schwitzgebel (a philosopher at U.C. Riverside) and Fiery Cushman (a psychologist at Harvard) have designed a "Moral Sense Test" that asks respondents for their takes on various moral dilemmas. They're looking to compare the responses of philosophers and non-philosophers, so they've asked me to post a link to their test from this blog. They say that people who have taken other versions of this test have found it interesting to ponder the moral dilemmas they ask about. The test should take about 15-20 minutes and can be found here.

The test says "Please do not discuss the questions with anybody else, or consult any texts or outside material, while you are taking the test." I suspect that some of my readers will want to comment on the test, so if you intend to take the test, please don't read the comments to this post until after you've done so.

Some quick statistics on the calibration quiz

On Saturday I gave a quiz from Ian Ayres' book Super Crunchers which asked you to provide 90% confidence intervals for ten numerical questions with well-defined answers. Roughly speaking, you should select your answers so that you expect to get nine of the questions right and you believe you're equally likely to have gotten each of them wrong.

Nineteen people have taken the quiz.
Out of the 190 individual answers received, 97 were correct -- slightly over half. The distribution of scores on the quiz is as follows:

Score	2	3	4	5	6	7	8	9	10
Number of people	1	4	3	4	2	3	1	0	1

In short, the respondents as a group confirm Ayres' claim that "almost everyone who answers these questions has the opposite problem of overconfidence -- they can't help themselves from reporting ranges that are too small." Ayres cites a book by J. Edward Russo and Paul J. H. Schoemaker, Decision Traps: Ten Barriers to Brilliant Decision-Making and How to Overcome Them, which I haven't read; supposedly "most" people get between three and six questions right. I'm actually soewhat surprised that you as a group don't seem all that different from the general population.

I have some other comments -- which questions seem particularly difficult or easy, what we might say about confidence intervals other than 90 percent -- but I'm hoping more people might answer, so I'll wait for that. (Although if the remaining answers are suspiciously better-calibrated that the answers so far, that might turn out to be not such a good idea.)

A prediction-making quiz

I just read Ian Ayres' book Super Crunchers, which talks about how the large amounts of data that are now routinely collected enable better predictions than before. Sort of like Freakonomics but a bit more statistical. (Although all the math is hidden -- but I knew that going in.)

Now, there was a recent article The End of Theory which predicts that we don't need theories, we can just mine our data for correlations; I don't believe this. And Ayres talks about how some predictive models need human input -- for example, a model for predicting how Supreme Court justices will vote needs people to read previous input on the cases in order to decide whether the ruling being appealed was liberal or conservative, and also to determine what the major issues involved in the case are. But he ponts out that people are bad at predicting things because we are overconfident about our predictions.

This piqued my curiosity. Here's a quiz; I want to see how good you are at calibrating your own predictions. (This is taken from Ayres' book, p. 113.) For each of the following ten questions, give a range that you are 90 percent confident contains the correct answer. Ayres' test implicitly uses English units, but if you want to use metric (which I suspect a lot of you are more comfortable in) that's fine; I'll convert.

So, for example, if one of the questions were "What is the population of Philadelphia?", and you gave the numbers "1.2 million, 1.6 million", that would indicate that you believe with probability 90 percent that the population of Philadelphia is in that interval. (The 2006 Census estimate for this, by the way, is 1,448,394.)

Your goal is to get exactly nine of these right. Yes, I know that sounds weird! But the point is that if you get all ten right, you're proabably underestimating your own abilities to predict things. If you get eight or less, you're probably overestimating them.

Send your answers to me at izzycat AT gmail DOT com; don't leave them in comments.

Here are the questions:
1. How old was Martin Luther King, Jr. at death?
2. What is the length of the Nile River?
3. How many countries belong to OPEC?
4. How many books are there in the Old Testament?
5. What is the diameter of the moon?
6. What is the weight of an empty Boeing 747-400?
7. In what year was Mozart born?
8. What is the gestation period of an Asian elephant?
9. What is the air distance from London to Tokyo?
10. What is the depth of the deepest known point in the ocean?


Also:
1. feel free to forward this quiz to other people. (I encourage it, although there's a non-negligible chance I might regret this if I get too many answers. I'll survive.)
2. if you have stories about how you made your guess, send them to me; I may use them in a future post.
I'm not going to post the answers; none of them are hard to find. Once answers stop coming in I'll make a post about how good you are at making these predictions.

Number sense?

Numbers Guy, from the March 3, 2008 issue of The New Yorker, on human beings' intuitive "number sense".

Interestingly, we're quicker at comparing numbers that are further apart, which seems to imply some intuitive sense of "number line" -- and our number line perhaps has some sort of strange metric (not the usual one) in which the distance between 2 and 3 is longer than the distance between 7 and 8, since we're faster at saying 3 is larger than 2 than we are at saying 8 is larger than 7. (See, for example, yesterday's post on hyperbolic discounting which exploits a similar phenomenon with respect to timelines.)

And here's an interesting fact:

Because Chinese number words are so brief—they take less than a quarter of a second to say, on average, compared with a third of a second for English—the average Chinese speaker has a memory span of nine digits, versus seven digits for English speakers. (Speakers of the marvellously efficient Cantonese dialect, common in Hong Kong, can juggle ten digits in active memory.)

I wonder if this sort of thing is true in general -- does this correlation extend beyond just a pair of languages? Wikipedia's article on the "seven plus or minus two" phenomenon backs this up -- the actual limit is something like two seconds of speech -- although unfortunately there's no citation to follow.

(Via Seed's Daily Zeitgeist.)

Hyperbolic discounting

Greg at The Everything Seminar posts about "hyperbolic discounting". Roughly speaking, theoretically one should value a payoff of an amount P of money at time t in the future with the same value as Pe^-rt today, where r is the inflation rate. But people actually seem to value P at that time in the future like they value P/(1+ct) today for some constant c. People probably have a decent sense of what the inflation rate is; thus Pe^-rt and P/(1+ct) should agree locally, so I'd guess c = r. (Set the two expressions equal to each other, so e^rt = 1 + ct, and take the first two terms of the Taylor series.)

Greg writes:

I think something like a logarithmic measure on actual time might give the hyperbolic discounting model.

That's true. Let's say we live at time 0; the correct (exponential discounting) value of a payoff of 1 at time t is e^-rt. The value of a payoff of 1 at time T under hyperbolic discounting is 1/(1+rT). Setting these equal, we get

.
Solving for each variable in terms of the other,

So roughly speaking, from looking at the first equation, the discounting that people actually use instinctively is obtained by taking the logarithm of the time T they're discounting over (up to some scaling, which really just sets the units of time), and then applying the correct (exponential) model. This reminds me of a logarithmic timeline, but in reverse. People see the period from, say, 16 to 32 years ago as being as long as the period from 32 to 64 years ago. This is also why I don't believe in a technological singularity even though I'd like to; the arguments often seem to be based on "look! lots has changed in the past hundred years, more than changed in the hundred years before that!" but our memories of "change" are somewhat selective.

Monkeys can do math. Sort of.

Monkeys and college students equal at mental math? (Reuters)

Monkeys and college students were asked to do addition problems mentally, by being shown two sets of dots and then being asked to pick out a third set of dots that had the same number of dots as the first two sets combined. Apparently (according to the Reuters article) the college students were told not to count or verbalize as they did the math; this doesn't seem to make a huge difference, though, since average about one second so it really wouldn't have been practical to do so. This seems like an unfair handicap; we're so used to doing math verbally that doing it any other way is difficult. I also wonder if calculator use among humans has contributed to this. What would have happened if the same study were done fifty years ago?

The headline is a bit inaccurate, though; the monkeys responded equally quickly to the problems, but they were less accurate (94% accuracy for humans, 76% for monkeys).

All snarking aside, though, it's interesting that there are parts of mathematics that appear to not depend on linguistic abilities. And it seems a bit surprising that monkeys would be nearly as good at these tasks as college students, because the college students have had quite a bit more mathematical experience! But the task in question had to be done quickly enough that it really couldn't be verbalized, and most of the students' mathematical experience has been mediated through language.

The actual article is available online: Jessica F. Cantlon*, Elizabeth M. Brannon, "Basic Math in Monkeys and College Students", PLoS Biology 5(12): e328. (It's nice to be able to actually read the article that the journalists are hastily generalizing about! Often a subscription is required to do so, which is incredibly frustrating.)

links for 5 August

The Fermi Paradox is back, via Slashdot. The Fermi paradox, for those of you who don't know, is basically the following question: if there are so many examples of extraterrestrial intelligence, as a lot of people believe, how come none of them have contacted us yet? This ties in to one of my favorite overmathematizations, the Drake equation, which computes the number of extraterrestrial civilizations in our galaxy by multiplying seven factors, most of which we have no good idea of. The result is a number with a ridiculously huge margin of error; depending on who you ask, the number of extraterrestrial civilizations that we might be able to here could be anywhere between zero and a million or so. Good expositions of the Drake equation usually point out that we have no way of predicting, for example, the average lifetime of a civilization. One particularly interesting resolution I've seen of the Fermi paradox is that other civilizations decide that they just don't care about talking to other species and spend all their time looking at the local equivalents of Internet pornography and reality television. I'm not saying I believe this, just that I've heard it. A bit more plausible, I think, is the idea that civilizations evolve so quickly that a civilization that was where we'll be in the year 3000 (if we don't kill ourselves first) wouldn't be interested in talking to us. (If you think 3000 is too soon, substitute some year further in the future.) I think it would be interesting to talk to a civilization that was where we were a thousand years ago, but a lot of people believe that the evolution of civilization is accelerating; Ray Kurzweil is probably the best-known exponent of this idea, called the Singularity. I'm a bit suspicious of it because a lot of the arguments seem to rely on the fact that we remember what happened in the recent past much better than what happened in the distant past.

What autistic girls are made of, by Emily Bazelon in today's New York Times. Disorders on the autistic spectrum are usually thought of has being uniquely the province of boys, but they happen to girls too. There are researchers who think of autism as being an "extreme male brain", and if that's true it kind of makes sense that it would be more common among males than females. Also, apparently it's harder to be an autistic woman than an autistic man because women are expected to understand social networks better than men; I'm kind of curious if this has always been true or if it's a historical accident. Vaguely relatedly, Who's a Nerd, Anyway? by Benjamin Nugent from last Sunday's NYT; people who are considered nerds are "hyperwhite", according to the linguist Mary Bucholtz. (This is "white" in a cultural sense, as in the way white Americans tend to act; I don't think the author intends to say that there's anything genetic about being a nerd.) What I find interesting is that this same tendency towards oversystemization can be called either hyperwhite or hypermale, despite the fact that we usually think of sex and race as being orthogonal to each other. Finally, Mark Liberman comments at Language Log on reactions to Nugent's article, and how in general non-scientific bloggers blogging about science, and non-scientific journalists writing newspaper articles about science, make fools of themselves.

The Probabilistic Method by Noah Snyder at Secret Blogging Seminar. I love when people find out that the probabilistic method exists. For those of you who aren't familiar with it, the probabilistic method is a method used to prove that a collection of objects contains some object with a certain property not by actually finding the thing but by just proving that if you pick an object from the collection, it has probability greater than zero of being the thing you're looking for. It's kind of a mindfuck, because many of its applications are in combinatorics and people expect there to be explicit constructions of things in combinatorics. It's possible to have a group of forty-two people such that there's no five of them who all know each other and no five who don't know each other. But I can't explicitly tell you which people in that group know each other and which don't. (This is an example of a Ramsey number.)

what does "losing 35 IQ points" mean?

The Gregarious Brain, in today's New York Times magazine. The article is about people who suffer from Williams syndrome, and focuses on the fact that people with Williams have trouble with abstract thought and have "exuberant gregariousness and near-normal language skills". These individuals don't have the best social skills, though, which makes one feel sorry for them -- they want to connect, but they can't.

This comes about because the part of their brains which deals with abstract thought (the dorsal part) are underdeveloped, but the parts dealing with language (the ventral part) are normally developed, because certain genes don't act during the formation of the brain.

What grabbed me, mathematically, was this:

These deficits generally erase about 35 points from whatever I.Q. the person would have inherited without the deletion. Since the average I.Q. is 100, this leaves most people with Williams with I.Q.’s in the 60s. Though some can hold simple jobs, they require assistance managing their lives.

Does this mean anything? Obviously "I. Q. points" are not something which just sits there in our brain. If my IQ is, say, 130, there aren't 130 little blobs sitting there in my brain which do my thinking for me -- or even 1.3 times as many such little blobs as the average person. (I know, you might be thinking that neurons are such "little blobs", but brain size isn't correlated very well with intelligence.) Furthermore, IQ scores are set up to have a Gaussian distribution, which I suspect is not the right thing to do. Perhaps the intelligence of individuals whose brains have developed "normally" are normally distributed, but the fact that there are a large number of disorders which "take away" IQ points makes me think there'd be a bump around, say, 60 or 70 IQ points -- a couple standard deviations below the median.

And that's only if intelligence is normally distributed to begin with. You'd expect that if intelligence were the sum of a bunch of independent effects, but I suspect there's some sort of synergy going on where "the whole is greater than the sum of its parts" -- a moderately above-average ability in, say, spatial reasoning and computational ability might make a better mathematician than someone who's really good at one of those and only average at the other. In general there are lots of complex skills which are made up of simpler skills in this way.

I suspect the central part of the distribution is approximately normal, though; the weirdness probably goes on with the very smart or the very stupid.