A lot of my research has been driven by the fact that the mean of the distribution and the variance of the distribution usually have different impacts on welfare; see this and that, for example.
Imagine that your income next year is going to be a random number of dollars in the $30,000 to $120,000 interval, a scenario that isn’t completely crazy if you are, say, a farmer in a middle-income country.
Assuming your welfare increases with income, an increase in the mean of your random income is a good thing. In other words, holding the [$30,000, $120,000] interval constant, you would prefer your expected income to be equal $90,000 rather than $60,000.
Alternatively, whether you prefer a more variable income or not depends on your risk preferences. So holding constant both the [$30,000, $120,000] interval as well as your expected income of $90,000, an increase in the variance of your income might affect you differently than it would someone else. If your income is normally distributed with a standard deviation of $5,000, then in about 19 out of 20 cases, your income will lie in the [$80,000, $100,000] interval. If your income is normally distributed with a standard deviation of $10,000, then in about 19 out of 20 cases, your income will lie in the [$70,000, $110,000] interval. So which do you prefer: A $5,000 standard deviation, or a $10,000 standard deviation?
There is no right answer, as what’s right for you depends on your risk preferences. A risk-neutral individual will be indifferent between the two. A risk-averse individual will prefer a $5,000 standard deviation to the $10,000 deviation (and, to the extent that it is feasible, they would prefer a $0 standard deviation, which is why crop insurance is hugely popular among farmers!) A risk-loving individual will prefer the $10,000 standard deviation.
Sounds sensible, doesn’t it? Except it often happens that people have a hard time thinking through the difference between the mean and variance of a distribution. This is especially true for college students who, by virtue of being more educated than the vast majority of the population, really should know better:
Imagine two course sections with the following grading schemes:
Section A: 4 assignments worth 5% each for a total of 20%; 35% midterm; 45% final.
Section B: 4 assignments worth 10% each for a total of 40%; 60% final.
In my experience, students often reason: “Section B places more weight on assignments. I can work with my friends and use other resources, and get good marks on the assignments. Therefore it’ll be easier to get a good grade in Section B.”
But professors know students work together on assignments, and it’s almost impossible to tell who has done the work and who has just copied it. They put weight on the assignments, so that students have an incentive to complete them. But profs don’t want the assignments to have much influence on the students’ final grade.
The best way to diminish the effective influence of assignments is to give everyone more or less the same grade – say 80 percent. That way assignments just scale up the class average, and students’ relative position in the class is primarily determined by their score in the examinations.
That is from a longer post by Frances Woolley over at the Worthwhile Canadian Initiative blog, about how students often completely miss the obvious fact that what matters is not so much their absolute score, but where they rank relative to the mean, or as I once heard someone say in college: “It doesn’t matter that you be good at this, all that matters is that you be better than most others at it.”
I remember all of the students in a course I was teaching four or five years ago failing to answer one of the questions on a midterm. That question, I think was worth 5 points. When I told the students that I had given everyone five additional, “bonus” points because that question had proved too difficult, a number of students thought that this was really, really great news.
But those students should really have known better, however, since I had already told them that I would curve the grades and the mean would be a B+. Those 5 free points on that midterm merely gave them the illusion that they were doing better, since it only moved the distribution a bit to the right, without changing the variance or the rank-ordering of students.
* Except for a Poisson distribution, that is.