## Wednesday, May 7, 2008

### Wednesday Math, Vol. 22: Will this be graded on the curve?

Teachers get asked that question all the time. Will this be graded on the curve? Most students who ask have no idea what it means, and no idea that teachers who "grade on the curve" don't always use the same method. Lemme 'splain as much as I can, and also 'splain what students usually mean by the question.

The "curve" is supposed to be the normal curve, also known as the bell-shaped curve. The idea of the bell-shaped curve is that lots of sets of data are "normally distributed", meaning that we can predict about how much of the data will be around the average, how much will be considerably higher than average and how much lower than average. On the chart above, average is the Greek letter mu and the standard deviation is the Greek letter sigma. Without going into gory detail, standard deviation tells us how spread out the data is. The old fashioned curve graders would give any student whose grade was within one standard deviation of the average, either high or low, a C, which is the blue region of the chart above. B meant between 1 and 2 standard deviations above the average and A meant 2 standard deviations above average or higher. Likewise, a D was the negative version of a B and an F was the negative version of an A. If the data was normally distributed, or close to it, this would mean about 2% of students would get an A, 14% would get Bs, 68% would get Cs, 14% would get Ds and 2% would get Fs.

Here's the thing. Grades on tests should NOT be normally distributed if a teacher is any good at teaching and gives anything like a fair test. There will be people who struggle with the material, maybe in math more than anywhere, but there can easily be a lot more people above average than below average, with the stragglers bringing the average down. For example, the average grade on a 100 point test can easily be around 75 or so. This means at best, a student can only be 25 points above average, but a student could be 75 points below average, though that is exceedingly rare. Students getting scores below 50 on such a test not only bring the average down, they make the scores more spread out and that raises the standard deviation. A 15 point standard deviation on a 100 point test is pretty common, and it could even be higher. If the average is 75 and the standard deviation 15, it would take 75+15 = 90 points just to get a B on the strict curve method, and A's would be impossible if the two standard deviation rule is followed, since 75+(2 x 15) = 75+30 = 105 points.

Instead of actually computing the average and standard deviation, both of which are very easy on a spreadsheet, some teacher pick percentages beforehand of who will get every type of grade and call that "grading on the curve". Under this system, the teacher might say "10% will get As, 20% will get Bs, 40% will get Cs, 20% will get Ds and 10% will fail." I would never do this, and if any teacher is reading this who has a system like this, I would politely tell them this is junk. This kind of strict predetermined grading system might make a huge distinction between people whose performances are actually very close to one another.

Another way of doing "the curve" is to decide that the average score is the split between C+ and B-. In other words, even if you are a tiny bit above average, you are a B student, and at average or below is a C student. This is sometimes called the grade inflation curve. I don't use the curve myself, but I dislike this method nearly as much as I dislike the "only x% of my students get As, no matter what" method.

When a student asks me, "Do you grade on the curve?", it means one of two things. The most common meaning is "I suck at math, but I think there are some people in class who suck worse than I do, so I think I will pull off a D, while the bottom 10% will get Fs." The second most common meaning is "I'm pretty good at math, but I might not be in the top 10%, so if you grade on the curve, I'm pretty much resigned to getting a B right now, so I don't have to work very hard."

I try not to think less of students for asking the question, but I don't always succeed.

Yay, Flags of Many Lands™! Yay, Gambia, or as Gambians call it The Gambia! The Gambia has a coastline, but otherwise it is entirely surrounded by Senegal. From what I can tell on the map the westernmost tip of Africa is in The Gambia.

Why does someone from The Gambia come to Lotsa 'Splainin'? The Melissa Theuriau.

Good choice, The Gambia!

dguzman said...

I used to get students who'd ask if I curved grades on essays. ESSAYS! I definitely thought less of them after that; I admit it.

Distributorcap said...

are you one of those non-normal statistics guys...

8-)

Lockwood said...

In some of the sciences- particularly physics and chemistry, but even, I'm sad to say, math, there is what seems like a competition at the college level to create the most impossible test ever. Profs apparently see it as a mark of pride to say "The average score was 12 percent." What they're actually saying is either "I'm such a lousy teacher the students only got 12% of what I thought was important," or "I have such a lousy comprehension of what assessment is that only 12% of my test items had any bearing on what happened in the classroom." Sad. A question that came up in one of my ed classes: If you had a choice between going to two doctors, one of whom was graded strictly on whether s/he met certain criteria and had a B+ GPA, and another who was graded on a "curve" in each class, with an A- GPA, which would you choose?

I just started a blog today at http://outsidetheinterzone.blogspot.com/

The proximal reason was a fun trick (I decided to call the label "stupid math tricks") that I think determines the divisibility of a number by eleven. I describe the chain of thought that leads to it as best as I can. If you have the time and interest, I'd really appreciate your feedback on that particular post (Is that divisible by 11?)