Q: In what ways are the US News & World Report rankings for colleges flawed?

About a quarter of the U.S. News formula is an opinion poll of university administrators (presidents, provosts and deans) and high school college counselors about their views on the reputations of the colleges.

One criticism would be, does this really speak highly for the validity of the results, when 23% of the result comes from administrators at competing universities and high school employees? Does a 45-year-old guidance counselor at Evanston high school or a 60-year-old dean at the University of Chicago really have any idea whether you'll get a better undergraduate education at Stanford, Harvard, Penn or Yale if you go there in 2011?

And, of course, can a national university really have a single, unitary reputation score? Surely the kind of student who would thrive at Caltech (the #1 school in the country a decade ago, despite offering no BA degree) is not the same as the student who would thrive studying medieval literature at Yale.

But #2, like all components of the U.S. News formula, there is no margin of error on the results of the opinion poll! The rankings are calculated as if every input -- the competitor and high-school employee view of a school's "reputation," its graduation rate, the average class size -- were absolutely certain. That is not so. 

In addition to statistical error, there's also a substantial systematic error in some of the parameters -- e.g. the "average class size" has a lot of slop in what you count as a class (just lectures? lectures and discussion sections? lectures, discussion sessions, and tutorials?). So does the graduation rate, etc. These figures should have error bars on them too.

I have discussed this briefly with Bob Morse, the guy at U.S. News who calculates the rankings, but he wasn't receptive to the idea that they should put appropriate error bars on all the inputs and propagate the uncertainty to the outputs, marking statistical ties as appropriate. (I suspect these statistical ties might cross substantial swaths of the final rankings, which may partly explain why U.S. News wouldn't be excited to try to sell magazines with that technique -- who wants to announce a nine-way tie for 1st place?) His position was that they assume the data coming from the schools is right, and they don't waste time worrying about what the rankings would be if the supplied figures weren't right.

Q: Could a flatlander utilize flatland technology to perceive something in the third dimension?

There are some big differences between living in a plane, versus living in Euclidean 3D and merely being able to perceive only a plane slice. Here's one: the wave equation in three dimensions is "nondispersive," meaning all frequencies travel at the same speed and the region of influence of an impulse is just an expanding circle. So if you say something 100 meters away, or transmit a signal from your radio, I will hear a delayed, quieter version of what you said.

In two dimensions (or any even number of dimensions), the wave equation is dispersive! You do not just hear a delayed, quieter version of stimuli at a distance; the sound itself is actually changed by traveling through the medium. The region of influence isn't a circle; it's a filled-in circle. Imagine throwing a rock into a pond -- the resulting ripples aren't just at the outside of the circle. The rock continues influencing the interior of the circle even after the first news of its plop has passed by. Similarly, the sound of thunder, constrained to a near-2D slice of the earth's atmosphere, is not just a bang when you hear it from far away. It gets transformed into a rolling sound because the different frequencies travel at different speeds.

So a flatlander who understood partial differential equations (this may be a small number of them...) could distinguish between these two possibilities (2d world, versus 2d slice of 3d world) by observing the behavior of waves as they propagate.

(NB: The proof that the wave equation is dispersive in even-dimensional Euclidean spaces and nondispersive in odd-dimensional spaces is really frickin' hard! I have seen it in a monograph and did not understand it at all.)