# How do professors interpret normally distributed midterm or final exam scores?

I had recently taken a class where things are poorly taught and much of the studying was self study. This then turned the course into a function of time and workload, and the resulting midterm mark is normally or bell shape distributed with a center of 50.

Needless to say, people who had less workload earned higher scores, people who had more workload earned less scores. After the exam, no one felt that the exam was a good reflection of individual capabilities but it is uncertain if the professor had gotten the hint.

How do professors interpret this outcome? Specifically, are there actions taken to some how adjust this (because most courses have an average passing around a C or 70 instead of 50)?

• You may find the following questions helpful: 1. How to Scale the Grades for an Examination? and 2. Common grade curving techniques Mar 14, 2015 at 1:32
• I'd be astonished if you got a normal distribution for a class of fewer than a couple of hundred students. My own experience is a bimodal distribution, with one hump of the distribution being the people who get it and the other hump being the people who don't. What are the mean, median, and standard deviation of these scores? Mar 14, 2015 at 1:51
• Oops. I've read your question more carefully, and you were a member of the class, so you (probably) don't know the measures of central tendency of the grades. Sorry. I can pretty much promise you it's not a normal distribution. Mar 14, 2015 at 2:01
• Ok it is not perfect normal, out of 100 percent, the 50 percent people were slightly less than the people who got 30-40 and 50 - 60 but not by much Mar 14, 2015 at 2:08
• I once gave a midterm to 150 students where the grade distribution was uniform. With less than ten exceptions, for every possible score between the minimum and the maximum, the number of exams with that score was either 1 or 2. Needless to say, I found this much harder to interpret than the usual normal disitrbution. Mar 15, 2015 at 12:12

Unless you have access to every student's score, you cannot know that the distribution is a normal distribution. It isn't clear that you do have access to the scores, but in a comment you say, "the 50 percent people were slightly less than the people who got 30-40 and 50-60 but not by much." You have described a bimodal distribution -- one with two humps at the ends and a dip in the middle. Your "slightly less" says the dip in the middle is a small one.

My own experience as a college teacher is that a bimodal distribution is a usual and expected result for a fair exam. At the "good" end, you find the people who either worked hard or found the material to be familiar. At the "bad" end, you find the people who did not work hard. That is exactly what you've described, although you equate "worked hard" with "had more time to work."

As far as interpreting the scores, you don't say whether the scores you mention are percentages, i.e. out of 100, or have some other base. If they're grades out of 100, then that is probably, but not certainly, a poor exam. (The other choice is that there's an entire class of poor students, and yes, that does happen.) If I, as a teacher, got a result like that, I'd use the grades as they were, but work much harder on the questions for the next exams.

Finally, you question whether the exam "was a good reflection of individual capabilities." That's not what en exam is supposed to measure. An exam is supposed to measure your knowledge of the course material. A very capable person who doesn't study can and should earn an unsatisfactory grade.

Here is how I normally assign grades in a class of 30-50 students, given a distribution of scores:

1. Declare the average to be either a B or B+, depending on how satisfied I am with the class's performance as a whole.

2. To find the A/B cutoff, look above the average for obvious clumps. If, say, the average is 75% and there are ten scores of 85% or higher, and no scores between 80 and 85, then the scores higher than 85 are the As.

3. There will not always be obvious clumps, and in that case I put the cutoff somewhere between the average and the maximum in a semi-arbitrary way. I may look back at some exams to see if particular students did what I consider A work.

4. Determine the B/C and C/D cutoffs, then pluses and minuses, using a similar process.

5. To distinguish Ds from Fs, I don't make a numerical cutoff. Fs are reserved for students who have not shown me that they're actually trying.

The benefit of this method is that if I accidentally make the exam too hard or too long, the students don't pay the price. But I always try to make my exams straightforward, with at most one challenging problem, because experience shows that asking students to regurgitate what I've told them already provides a good enough range of scores to separate students from each other. If I've done things right, the average is usually in the low 80s.

Now, your professor may or may not use a system like mine. He may simply enter the scores into the grade book and use rigid, pre-determined cutoffs, regardless of the class's performance. The only way to know is to ask him.

By the way, you've observed that the people who spent more time studying got higher scores, and this strikes me as completely fair. If the class could be aced without studying, it wouldn't be a very useful class.

• This page explains why clumps may not be the best way to go. (See "The Distribution Gap Method" section). Jul 15, 2015 at 18:02
• I don't agree with that reasoning. The locations of the gaps may be random, but so are the scores themselves. If the test were given again, the ordering of the students would change, but we have to grade them based on the exam they took. The presence of a gap means that two students on either side of the gap would be less likely to exchange places if the test were given again, so it is more reasonable to assign them different grades than if their scores were very close. It's not perfect, but for a small class, using standard deviations feels at least as arbitrary.
– user37208
Jul 15, 2015 at 18:16
• I have to admit that when I read procedures like the one above, I'm glad of leaving in a country where grading is typically not done on a curve... less headaches! :-) Jul 15, 2015 at 18:37