Division by Zero

We have all given exams where the grades end up lower than we hoped. A curve is in order. How do we do it?

In this post I share my thoughts on when you should (or should not) curve an exam. I give ten sample curving techniques, including pros and cons of each, I explain how to convert grades into letter grades, and I end with three concrete examples.

To keep things simple, I assume that the raw score of the exam is a percentage—a number between 0 and 100. From that we would like to obtain a curved or scaled grade which is again a score between 0 and 100 (or occasionally a number over 100). I am writing this as if the curve is for an exam, but most of the tips work for curving the grades at the end of the semester too.

To curve or not to curve

When I give an exam to a class, I have an intuitive feeling for how the grade distribution should look. I know, roughly, who the A students are, who the F student’s are, and who the average students are. This comes from their homework, their questions in class, our conversations outside of class, and so forth. Individual students may surprise me and do better or worse than I expected, but as a whole, I know the strength of the class when exam-time rolls around. If the class does significantly lower than I think they should have, I will consider curving the exam.

Also, courses have certain historical distributions. For example, in an entry-level course I may want an average (mean) of 80-82% with several A’s. In classes like that, failing grades are not unusual. For my upper-level (majors) class, on the other hand, I may expect higher grades with failures unlikely. If the scores do not fit the historical template, I will consider curving.

I may also consider a curve if there was one (usually high-point-value) problem on which everyone does poorly. I may want to make up for that with a curve.

On the other hand, if I feel that the exam was fair and the class should have done better, then I do not curve. Similarly, if I feel that the class is “weak”—that is, weaker than other classes who have taken the same course from me in the past—then I do not feel an obligation to bring their grade up to fit the template.

My advice is to use your judgement. You know the class, and you know the material.

What’s the goal of the curve?

Before you do any curving, you must determine what you want the curve to accomplish. Determining this will help you choose which curving technique to use. Here are some questions to ask yourself.

• Do you want a particular average?
• Do you want to give the lower-scoring students more of a curve or the same curve as the higher-scoring students? (Rarely do we want weaker students to get less of a curve than the stronger students.)
• Do you want everyone to get a passing grade on the exam?
• Is it OK to have a big group of A’s?
• Is it OK for some students to have a grade over 100%
• Do you want to protect the class from “curve breakers”—outliers who score much higher than the rest of the class and thereby prevent a large curve?

How do I curve an exam?

Below I present ten techniques for curving an exam score. In most case I describe the curve as a function, . By this I mean the raw score and is the curved score.

For example, suppose the curve is . Then a student with a raw score of 80% would get a curved grade of %. In a spreadsheet if the raw score is in column A and we want the curved score to be in column B, then entry B1 should be =4*A1/5+20.

You can choose any function as long as it satisfies the following two properties.

1. is nondecreasing; that is, when . This prevents leapfrogging of grades (i.e., student A scores higher than student B before the curve, and student B does better afterward).

2.  (at least on the range of grades you gave). This ensures that no one gets a lower grade after the curve than they had before the curve.

Here are a couple of other considerations when defining .

3. You probably want on the range of grades that you gave (if the curve is linear, then this means the slope is less than or equal to 1). This will guarantee that the lower-scoring students will get the same or greater boost in points as the higher-scoring students.

4. If you want the final score to be an integer you need to round (or if you’re feeling generous, round up) the grade after applying the function .

Here are ten curves you may want to consider.

1. Return, rewrite, regrade

What is it? This curve is quite different from the other nine, but is my favorite, so I am presenting it first. I can’t always use it, but I do whenever I can.

How it works:

1. Return the graded exam to the students
2. Have them rewrite the problems that they got wrong (completely re-write, not simply “fix”)
3. Have them turn in the original and the rewritten one
4. Grade the rewrite
5. Give them a percentage (30%, say) of their new points

For example, if the raw score is 76% and the “grade” after the rewrite is 96%, the final grade would be %.

I like this curve because it forces the students to go back and correct their mistakes, thereby learning the material that they did not know when they took the exam. They not only improve their grade, they learn from their mistakes.

There are times when this curve does not make sense. For example, if I wrote the correct answers on the students’ tests while grading them initially, then this would be a useless exercise. However, if I wrote comments such as “you need to justify this” or “use the chain rule here,” then rewriting could still be useful. I often write comments such as these in case I need to curve the exam.

One down-side is that this requires more time grading. However, since I have the original exam with my comments on them, it is much easier and faster to grade the second time through.

Pros: gets the students to learn from their mistakes, lower-scoring students can get larger curves

Cons: more grading for you, a little complicated to explain to the class

Use when: whenever you can!

2. Flat scale

What is it? This is the simplest and probably the most common means of curving an exam. Simply add the same amount to every student’s score. The function is

where is some fixed value. This curve is like the “flat tax” (or maybe flat tax refund!). Everyone gets treated the same. While that may be good in certain circumstances, there are times when I want to help the lower-scoring students more than the higher-scoring students. A 5-point curve seems like a lot to a student who got an 89%, but it is a drop in the bucket to a student who got a 49%.

I like to use the flat scale when my exam has one unfairly difficult problem that no one can solve.

Often professors do not want anyone to score over 100% on an exam. In this case a “curve breaker” can limit the professor’s ability to apply a curve. If the highest grade is a 97%, then a 3-point curve is all that is allowed, even if the mean is 60%.

Pros: easy to explain to students, easy to implement

Cons: doesn’t significantly help the students who did poorly, can have grades over 100%

Use when: to make small global adjustments, to make up for a single very hard problem

3. High grade to 100%

What is it? In this curve, the professor scales the grades so that the student with the highest grade in the class (call it ) gets 100%; the other students’ grades are compute as the percentage of they scored:

The major problem with this method is that it gives the stronger students a better curve than the weaker students. For example, suppose . Then the student with raw score 90%, gets a 10-point curve, but a student with a raw score of 60% gets a 7-point curve.

A modification of this method is to compute the percentage of some other score (presumably );

Pros: I can’t think of one

Cons: high-scoring students get a larger curve

Use when: maybe useful if there is one question that everyone, or nearly everyone, missed (see “remove question curve” below for another option).

4. Linear scale

What is it? Both of the two previous techniques are specific cases of a linear scale of the form

I use linear scales for my curves all the time, but I view them in a slightly different way. I pick two raw scores ( and ) and decide what grade I want them to become after the curve ( and ). These two points,  and  determine the linear scale:

For example, I often want the grades to have a specific average, say 80%. So, if the mean of the raw scores is 76%, then (76,80) is one point. Then I may take the low score (or high score) and force it to go somewhere. Say the low score is 58% and I want it to be 64%. Then the second point is (58,64). So the function becomes

I always check rules (1) and (2) for defining : that the slope is less than or equal to 1 and that everyone gets a positive curve (it suffices to check the high and low scores).

The one possible down-side of using this method is that different students gets different curves. I’ve never received a complaint about this, but I can imagine it.

Pros: very versatile, can be used to give an extra boost to the weakest students, can adjust the mean to be a target value.

Cons: a little complicated to set up, different students get different curves

Use when: you are willing to finesse the scores to fit the distribution you want

5. Remove a question from the grading

What is it? All of the students, even the A-students, bombed one question. Afterward I realize that it was not appropriate for the exam. I want to excise it from the exam completely. The function becomes

where is the student’s grade on all questions except difficult question and is the point value of the question.

(Of course I would not want to use this curve if the question was fair. There is nothing wrong with putting challenging questions on an exam.)

Pros: students relieved that this question is gone!

Cons: makes the other problems worth more, there may be a handful of students who did well on this problem—they’ll feel cheated

Use when: there is one bad question on the exam

6. Root functions

What is it? I have heard some people suggest the following curve: ”take the square root of the score.” By this they mean treat the raw score as a value between 0 and 1, then take the square root. For scores between 0 and 100 this becomes

.

I propose the following generalization of this curve:

for some chosen value of ().

This curve has the property that students whose raw score is 0 or 100 get no curve, and the lower scores (except for very low scores) get a larger boost than higher scores. To be precise, the largest curve will be for the student who got a grade of and they will receive extra points (this is a good Calc I optimization problem!).

Here are a couple of examples.

First, the square root example: ().

• raw score=25%, curved score=50% (this is the maximum possible curve)
• raw score=50%, curved score=63%
• raw score=75%, curved score=87%
• raw score=90%, curved score=95%

Next, consider ().

• raw score=30%, curved score=45% (this is the maximum possible curve)
• raw score=50%, curved score=58%
• raw score=75%, curved score=82%
• raw score=90%, curved score=93%

This seems like a fine curve. I’ve never used it. It seems unnecessarily complicated and the linear curve is flexible enough that this curve is unnecessary.

Pros: can be used to give an extra boost to the weakest students and a smaller boost to the strongest students

Cons: complicated, hard to explain to students

Use when: you really want to test your skill with the spreadsheet

7. Bell curve

What is it? Here’s the way I understand the “bell curve”: make the mean a C, then the mean plus/minus a half standard deviation would be the C-/C/C+ scores, one more standard deviation out would give the B’s and D’s, and the tails would give the A’s and F’s. This could be tweaked in any number of ways—change the mean, fatten or slim the distribution.

I don’t know if this is used by any professors anymore (in small classes, at least).

Pros: grades end up with a very predictable distribution

Cons: ruthless, students competing against classmates

Use when: for standardized tests in which only a certain number of students can pass, for large classes or multiple sections when there must be a fixed distribution

8. Extra credit problems

What is it? Give the class some challenging question to solve. If they get it right, they get extra points on their exam.

Don’t do it! Extra credit problems typically benefit the stronger students (who do not need the points). The weaker students do not try or cannot solve the extra credit problems. If a weak student in my class is going to spend extra time working on my class, then I would like it to be on the core material, not on extra credit problems.

9. Grading by gravity

What is it? Toss the exams down the stairs—the farther they fly, the higher the grade (or lower, if you want).

10. “I don’t believe in grades”/”I’m a grouch waiting for retirement” grading

What is it? Give everyone an A or everyone an F.

How to assign letter grades

I don’t like letter grades. I only use them at the end of the semester when I have to submit my final grades. What good are they in the middle of the semester? How do you average a B-, an A, and a B+?

This is the procedure I use at the end of the semester.

1. Decide on a fixed scale—i.e., how to translate percentage grades to letter grades. There does not appear to be a standard for how to do this. Here are two examples—one for straight letter grades and one including +/- grades (my college does not have an A+, but I included it because some schools do).

Percent (min)	Grade		Percent (min)	Grade
0	F		0	F
60	D		60	D-
70	C		63.3	D
80	B		66.7	D+
90	A		70	C-
			73.3	C
			76.7	C+
			80	B-
			83.3	B
			86.7	B+
			90	A-
			93.3	A
			96.7	A+

2. Quickly go through and assign letter grades using this scale.

If you are using Excel you can use this function to assign the grades automatically (if the percent grade is in column A):

=LOOKUP(A1,{0,”F”;60,”D”;70,”C”;80,”B”;90,”A”})
=LOOKUP(A1,{0,”F”;60,”D-”;63.3,”D”;66.7,”D+”;70,”C-”;73.3,”C”;76.7,”C+”;80,”B-”;83.3,”B”;86.7,”B+”;90,”A-”;93.3,”A”;96.7,”A+”})

If you are using Google Docs you can use this combination of functions:

=INDEX(FILTER({“A”;”B”;”C”;”D”;”F”};A1>= {90;80;70;60;0});1;1)
=INDEX(FILTER({“A+”;”A”;”A-”;”B+”;”B”;”B-”;”C+”;”C”;”C-”;”D+”;”D”;”D-”;”F”};A1>= {96.7;93.3;90;86.7;83.3;80;76.7;73.3;70;66.7;63.3;60;0});1;1)

3. I always go in and see if any of the grades need tweaking. I try to put the dividing lines between the grades in the “gaps.” For example, if there are students with grades …87.8, 88, 89.8, 90.0,…, then I will likely bump the 89.7 student up to an A-. I also bump the borderline students up or down depending on class participation, attendance, tardiness, illnesses during the semester, etc. (Except in exceptional circumstances, I still avoid letting students “leapfrog” each other.)

4. I take a close look at the failing students. I don’t like failing them, but it is often the right thing to do. Despite the atmosphere of grade inflation, do not pass a student who should not pass.

Examples

Finally, I am going to end with three examples. I created a spreadsheet using Google Docs and included sample scores of 45 students. The mean of the raw scores was 75.1%. I applied three different different curves all of which raised the mean to approximately 82.1%.

Flat curve:

Linear curve:  (the two points are (75,82) and (99,100))

Root curve:  ()

The histograms are shown below. As you can see, the distributions are quite different.

(See the Google docs spreadsheet.)

I would be happy to hear your thoughts, comments, and ideas!