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Deceptive Per Captia Rankings for Brain Cancer

We're working on a project to produce a report on the ratings that students give to their professors at the end of the semester. There is a big concern by the math people on the committee that we will give some professors an unfair bad (or good) rap because of the variability in these rankings. They don't want to report the ratings as a mean (average). Instead, they want to plot an uncertainty range. 

I was reading a book this week (How Not to Be Wrong by Jordan Ellenberg) that provided a great example of the risk of ranking things when there is uncertainty. It can lead to erroneous conclusions. Here is a summary of his argument that appeared in an NPR interview. Perhaps this sort of example will be helpful for the committee to share when teaching the general faculty about the new instrument.

If you take a rare disease like brain cancer and you look at its rate of incidents in different states, there are very big differences. And so you might say, "Well, I should go where this form of cancer is the rarest. Clearly something's going on in that state that is preventative against that disease." But when you look at the numbers, they're rather strange because at the very top of the list you see South Dakota with an extremely elevated rate of brain cancer, but if you look at the bottom, you see North Dakota with almost none. So that's very strange because South Dakota and North Dakota are not actually all that different.

But when you look at those numbers a little more closely, what you notice is that the states at the top of the list [South Dakota, Nebraska, Alaska, Delaware, Maine] and the states at the bottom of the list [Wyoming, Vermont, North Dakota and Hawaii, and the District of Columbia] have something in common, which is that they are very small. ... So basically hardly anybody lives in those states; that's what they have in common. And a sort of fundamental principle is that when you compute rates, the smaller the state, or ... the smaller the sample size, the more variation is going to be created just by random chance.

This seems analogous to the problem of small class sizes for ratings which cause us to draw a longer uncertainty bar on the report. One disgruntled student in a small class can cause a disproportionate movement on the class average. 

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