Friday, June 8, 2012

Grading and variance

Term is wrapping up for my classes here, and I'm suddenly being barraged with a lot of strange comments about grading. Students are concerned that not turning in a homework may lower them as much as half a letter grade in a class (each homework is worth about 4% of the grade). There are concerns about how high the median is of the midterm versus the final. While the questions are a bit odd for the third term in a sequence that has always been graded on an A-/B+ centered curve, it has started me thinking about how we grade and how we explain grading on a curve to our students.

Most of the undergrad coursed I've taught and TA'ed and taken have been graded as follows: 30% homework, 30% midterm, 40% final. If there are regular labs involved, or more midterms, the makeup changes, but it is roughly each component (labs, homeworks, each exam) makes up 1/n of the points of the course. This, prima facia, may lull one into thinking that if I do really well on the homeworks, but not so hot on one of the midterms, I may still do okay in the class. But this is actually not at all the case when grading on a curve.

When grading on a curve, in addition to the weighting I give the parts of the course, they are also weighed by the variance of the students' grades. If the homeworks, midterm, and final are each worth a third of the final score it seems like they all matter equally. However, this is not true. In a program where students are encouraged to collaborate and learn from each other in their homeowrks, most students get almost all of the homework points (say, a mean of 90 and a standard deviation of 3) then someone who does 2 st. devs above average has a 96, or 6 points above the average score.

I aim to have my exams have an average around 60 with a standard deviation of around 15. A half standard deviation is still 7.5 points here, more than the 2 standard deviations on the homework.

So, in my previous example, if a student performs 2 standard deviations above mean on the homework (does really well), but is half a standard deviation low on the midterm (does not so hot), it is not that she is still okay in the class, she is now below average.

Furthermore, if I make a mistake and make an exam too easy, which tends to move the standard deviation down as well as moving the average up, I effectively cause the exam to play a smaller role in determining a student's ranking in the class. Similarly a hard high-variance exam has a larger role. In other words, when my TAs and students want a final to be easier than the midterm, they are essentially asking for it to count less. Given that a student who has done well on the midterm is unlikely to make requests on the difficulty of the final, I'm sure this is not what they actually mean.

I had not really appreciated this subtlety when I was an undergraduate, nor in previous classes I had taught. I think it is likely that most of my students don't realize it either. Our course syllabi describe that 30% of one's grade comes from cumulative homework and the midterm each, and 40% from the final. But giving only this information, and little else is misleading to the class. "Do my homeworks count for nothing?" Yes, unfortunately, they count for little. The fact that you do your homeworks is what matters. This grading system is set up for you to do the homeworks, and learn from them. But the grades are primarily based on how you do on the exams.

This leaves me with the question of how to deal with this difference between the perception and reality of the grading process. On one hand, I can make easier exams, but at my graduate school, where the mean was at a B/B-, a mean in the 80s means that the difference between an A and an A- is about who makes fewer stupid/careless mistakes. That somehow doesn't seem fair.

I could recalculate the means and standard deviations of all three componenents to some predetermined point, but that would also over emphasize stupid/careless mistakes that a student would otherwise shrug off as "Don't I feel dumb. Ehn, it's just a point."

In actuality, I don't have a problem with how this grading scheme allocates grades, just with the gap between the perception and the reality of how much different parts of the course effect one's grade. Perhaps I should spend time on my course webpages explaining my grading scheme in detail, specifically how grading on a curve works, and how important variance is in the entire process.

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