# Should grades be assigned to students based on a normal distribution?

Assume you have a class of approximately 800 students and following a set of assessments each student has a raw grade.

• How should these raw grades be converted into a final grade?
• Is it a good idea to scale the raw grades to a normal distribution?
• I usually use some sort of marking scheme, but I guess you could just sample them randomly from a normal distribution and see if anyone can tell the difference. It would certainly save time! (Perhaps this isn't what you meant. But I'm afraid I can't tell what you did mean.) – onestop Apr 14 '11 at 10:19
• @user4167 I've tried to rephrase your question to convey what I think you might mean. Feel free to edit if I have misconstrued what you are asking. – Jeromy Anglim Apr 14 '11 at 10:44
• Without a rubric, almost any grading scheme is acceptable, legally. Did you ever see the movie the "Paper Chase". At the end, John Housemen didn't even read the finals; he assigned grades according to the student names! There is no reason for grades to have a normal distribution, regardless of the CLT. Have you found out how profs of previous year amalgamated grades? – schenectady Apr 14 '11 at 13:13
• I think that if any of those 800 students were to read this question, they might be offended. How well did they perform? How much learning was accomplished? That is what a grade should reflect, not some arbitrary statistical summary of their position in a group. IMHO this question should be recast in terms of teaching objectives, not statistical procedure, such as "what is a good way to convert raw scores to grades in a way that respects student accomplishments and advances the learning objectives of this class?" Statistics can help, but blind statistics--like standardization--will not. – whuber Apr 14 '11 at 14:46
• The consensus here seems to be that norm-referenced assessments are uniformly bad and criterion-referenced ones, uniformly good. I think a look at some basic examples of the vast assessment literature will show that each type has its place. (But, yes, more often it is the latter method that will yield a more valid indicator of what a teacher is trying to accomplish.) – rolando2 Apr 14 '11 at 21:42

Why should grades be normally distributed?

Sometimes they are but if the grades are not normally distributed then the bell curve grading system, where the middle say 70% get C's, is probably not a good one to base grades off of. Although that grading is pretty harsh, few instructors would actually do it.

Use distributions to describe the data, don't transform data to fit a particular distribution (although transformations can be helpful at times).

If you use the bell curve grading system and, extreme case, everyone aces the class. How do you decide grades?

Here is how I would decide final grades:

90-100%: A

80-90%: B

...

1.) Usually, grades are on an ordinal scale. So in a strict statistical sense, an overall grade should not be something like the mean grade, since adding such variables is not defined. The median grade however, does not clearly fit since it conveys the ranking between students in the same subject, not between subjects the same student learnt.

In the end, even in statistics departments students get their overall grade as the mean of the single grades. The reasoning is that in the median grade system (as long as the overall grade gets attention, see below), students would have no incentive to improve in the subjects they know they are weak. These weak subjects are like outliers and the median grade would be robust to them.

Maybe one has to remember that grades serve different purposes:

1. Predict the graduate's performance in his future job.
2. Give a selection criterion on whom to give an opportunity for further studies (scholarship, Ph.D. studies).
3. Help the very dump students identify their weak subjects where they have to work more.
4. Help professors to discipline their students (works of course only if purpose 1 and 2 are met).

An overall grade is only necessary for purpose 2, and there, only because a closer examination of the student's aptitude for the particular further studies is too expensive. Personally, I consider final grades as close to useless.

2.) As long as grades are correctly treated ordinal, there is no problem to rescaling them. But also, there would no use since proper statistical methods for such data are invariant under rescaling. However, if you compute mean grades, your transformation on the single grades will affect overall grade. This might be considered as unfair.

Also, the normal distribution on $\mathbb{R}$ is much finer than the coarse, disctrete grading systems we are used to.