## Wednesday, September 29, 2010

### PH253: Exam 1 scaling

Long story short, it didn't really work out to simply add points to everyone's score or take the best 4 of 5. I suppose this is a common theme if you read the news, but either of those benefited the grade-rich quite nicely, but did little for those of you who really needed a boost. Luckily, our problem has a nice solution.

Our exam had an average of A=72.46% and a standard deviation of s=22.30%. The former is too low, while the latter is too high - the average should be a bit higher, but more importantly the spread was way too large. What I wanted to do was map your grades to a more reasonable distribution, such that most of you ended up in the middle or above, and not too many were well below 60% or so. There is a mathematical procedure for mapping one distribution to another, probably familiar to those of you who have taken statistics. To start, I calculated what is called a "z score", or "standard score," which is basically how far from the average you were relative to the standard deviation:

z = (your score - A)/s

This tells you how you did relative to your classmates, essentially. If you were 1 standard deviation below the mean (so a score of 72.46-22.30=50.16), your z score is -1. The higher the z score, the more above or below class average you were. I used this score to calculate a new distribution based on a desired class average of A'=78.5% and standard deviation s'=18%, making your new score

(your scaled score) = z*s' + B

So, if your original score was 50.16, your scaled score is (-1)*(18)+78.5=60.5. Basically, the effect is to move scores well below the mean toward it, while scores well above the mean change little. Another way of thinking of it is that your score is tied more to how you did relative to your classmates than to a particular average score. In the parlance of our system, you would call this 'curving' the test.

Everyone gains points in this scheme, but those farthest below the mean gain a bit more. The resulting distribution looks like this (click for larger):
The number of you with A's stayed about the same, but there are more B's and C's, and (importantly) the long tail of lower grades has moved up. It is still a slightly wacky distribution, but quite a bit better, and I think more representative of how you really did.

Let me know if you have questions. The Exam 1 grade you see on moodle (see previous post) is this scaled one, and it should be much happier than your raw score. Again, it might not be as good as taking the best 4/5 questions for some of you, but this was I felt the way to be most fair to the class as a whole.