How to Curve a Test: Square Root, Linear and Flat Curves
The papers are marked, the class average is in the 50s, and you suspect the test was harder than the lessons. Curving is one way to respond, but there are several ways to do it, and they treat different students very differently. This guide gives the formula for each method, the same student run through all of them, and what to watch for before you change a single grade.
Quick answer: a curve is a rule that raises students' scores. For a student with 64% when the top score was 92%:
- Flat curve: add a fixed number of points. 64 + 5 = 69%.
- Shift to the top score: add 100 − top to everyone. 64 + 8 = 72%.
- Scale to the top score: score ÷ top × 100. 64 ÷ 92 × 100 = 69.57%.
- Square-root curve: 10 × √score. 10 × √64 = 80%.
Curving is a teacher's decision, and some schools restrict it, so check your policy first. Last reviewed: October 2026.
Four ways to curve test scores
Every method below works on a percentage. If your test isn't out of 100, convert first (see the FAQ). To turn raw points into a percentage, divide points earned by points possible, as in how to grade a test.
1. Flat curve (add points)
Curved score = score + bonus (capped at 100)
64 + 5 = 69%
The simplest and easiest to explain. Everyone gains the same amount, so the gaps between students stay the same. The cap at 100 means the highest scorers can gain less than the bonus.
2. Linear curve, version A: shift to the top score
Curved score = score + (100 − top score)
64 + (100 − 92) = 72%
This is a flat curve whose size is chosen for you: the amount that lifts the best paper to exactly 100%. Gaps between students stay the same.
3. Linear curve, version B: scale to the top score
Curved score = score ÷ top score × 100
64 ÷ 92 × 100 = 69.57%
Every score is multiplied by 100 ÷ 92 = 1.087, so the top paper becomes 100% and a score of zero stays zero. Gaps widen slightly in proportion, and students near the top gain more than students at the bottom.
4. Square-root curve
Curved score = 10 × √score
10 × √64 = 10 × 8 = 80%
This one gives the biggest lift to the lowest scores and the smallest to the highest. It doesn't use the class results at all, so it is easy to state in advance, but it won't set the best paper to 100%. A 92% becomes 95.92%.
Comparing the methods on the same scores
Six students from a class where the top score was 92% and the flat bonus is 5 points. Letters use the standard plus/minus scale.
| Original | Flat +5 | Shift (+8) | Scale (×1.087) | Square root |
|---|---|---|---|---|
| 45% (F) | 50% (F) | 53% (F) | 48.91% (F) | 67.08% (D+) |
| 55% (F) | 60% (D-) | 63% (D) | 59.78% (F) | 74.16% (C) |
| 65% (D) | 70% (C-) | 73% (C) | 70.65% (C-) | 80.62% (B-) |
| 75% (C) | 80% (B-) | 83% (B) | 81.52% (B-) | 86.6% (B) |
| 85% (B) | 90% (A-) | 93% (A) | 92.39% (A-) | 92.2% (A-) |
| 92% (A-) | 97% (A+) | 100% (A+) | 100% (A+) | 95.92% (A) |
The letter changes can differ a lot between methods. The same 55% is a D- with a flat curve and a C with a square-root curve.
Test curve calculator
Enter one student's percentage, the top score in the class, and a flat bonus to see all four results. To get the percentage from raw points, use the EZ grader calculator.
The percentage before the curve.
The highest percentage anyone earned.
Points added by the flat method.
| Method | Formula | Curved score | Gain |
|---|---|---|---|
| No curve | — | 64% (D) | — |
| Flat bonus | score + 5 | 69% (D+) | +5 |
| Shift to top score | score + (100 − 92) | 72% (C-) | +8 |
| Scale to top score | score ÷ 92 × 100 | 69.57% (D+) | +5.57 |
| Square root | 10 × √score | 80% (B-) | +16 |
The math runs in your browser, so the scores you type aren't sent anywhere. Letters use the standard plus/minus scale, and no method is allowed to lower a score.
Who each curve helps most
The size of the gain is the real difference between methods. Points gained, with a top score of 92%:
| Original | Flat +5 | Shift | Scale | Square root |
|---|---|---|---|---|
| 50% | +5 | +8 | +4.35 | +20.71 |
| 70% | +5 | +8 | +6.09 | +13.67 |
| 90% | +5 | +8 | +7.83 | +4.87 |
- Flat and shift treat every student alike, so they suit a test that was uniformly too hard.
- Scale gives more to students who already scored well, so it rewards the top of the class.
- Square root gives the most to students who struggled most, which is useful when the test was far too hard but can feel generous to those who did badly because they didn't study.
None of these is more correct than the others. Choose the one that matches what went wrong, tell students which one you used, and use the same method for the whole class.
What about grading on a bell curve?
A bell curve (a mean and standard-deviation curve) re-centers the whole class: it keeps each student's distance from the class average, measured in standard deviations, and moves the average to a target.
Curved = target mean + (score − class mean) × (target SD ÷ class SD)
Class mean 62, SD 12; target mean 75, SD 10
50: 75 + (50 − 62) × (10 ÷ 12) = 65%
74: 75 + (74 − 62) × (10 ÷ 12) = 85%
86: 75 + (86 − 62) × (10 ÷ 12) = 95%
This needs the class mean and standard deviation (a spreadsheet gives both), and it only works well with a large class. It can lower strong students' scores if the target is less generous than the original, so keep the higher of the original and curved score. For most classrooms the four methods above are simpler and easier to explain.
Partial credit: grade in points, then convert
Partial credit works best when you score in points, not questions. Give each question a point value, award points in whole, half or quarter steps, then divide the points earned by the points possible.
Percentage = total points earned ÷ total points possible × 100
A worked example
| Question | Points possible | Points earned |
|---|---|---|
| Questions 1–5 (multiple choice, 2 points each) | 10 | 8 |
| Question 6 (short answer) | 3 | 2.5 |
| Question 7 (short answer) | 3 | 2.25 |
| Question 8 (problem) | 4 | 3 |
| Question 9 (essay) | 10 | 7.5 |
| Total | 30 | 23.25 |
23.25 ÷ 30 × 100 = 77.5%, C+. In the EZ grader, enter 30 as the test size and the points lost (6.75) as the number wrong. The box accepts decimals, so quarter points work.
A partial credit conversion table
To keep partial credit consistent from paper to paper, decide in advance what each level is worth, as a fraction of the question's points:
| Question worth | Full credit (100%) | Minor error (75%) | Half credit (50%) | Started (25%) | No credit (0%) |
|---|---|---|---|---|---|
| 2 points | 2 | 1.5 | 1 | 0.5 | 0 |
| 3 points | 3 | 2.25 | 1.5 | 0.75 | 0 |
| 4 points | 4 | 3 | 2 | 1 | 0 |
| 6 points | 6 | 4.5 | 3 | 1.5 | 0 |
| 10 points | 10 | 7.5 | 5 | 2.5 | 0 |
Write your rubric before you start marking so the first and last papers are treated alike. For the whole marking routine, see how to grade a test.
Extra credit and curves together
Extra credit is added to the points earned, and a curve is applied to percentages, so the order matters. With 42 of 50 points and 3 bonus points, the score is (42 + 3) ÷ 50 = 90%. If you also curve, add the extra credit first, convert to a percentage, then curve once. Never stack two curves, and decide in advance whether scores can exceed 100%.
A simple workflow: mark, check, curve, record
- Mark every paper and total the points. For same-length tests, the printable EZ grader charts (for example the 20-question chart) turn wrong answers into percentages with one lookup.
- List the percentages and find the highest score, the average and the spread.
- Check for a bad question first. If one item caused the problem, fixing that item may be fairer than a curve.
- Choose one method and write down the rule before you apply it.
- Apply it to every student, and keep each student's higher score.
- Convert to letters using your school's scale (see the grading scale chart).
- Record the final scores in your gradebook, and tell students how the curve worked.
To see how a curved test score changes a course grade, use the average grade calculator, or check what a student needs on the final with the final grade calculator.
Mistakes to avoid when curving
- Curving before checking the marking. A scoring error on one question can look like a hard test.
- Curving points instead of percentages. Convert first, so a 40-point test and a 100-point test are handled the same way.
- Letting one outlier set the scale. If one student scored far above everyone, scaling to the top score barely helps. Use a typical high score or a different method.
- Stacking curves. Applying a flat bonus and then a scale is hard to explain and easy to get wrong.
- Rounding early. Round once, at the end.
- Lowering a grade. Keep the higher of the original and curved score.
Frequently asked questions
How do you curve a test?
Pick one method, apply it to every student's score, and make sure no grade goes down. The four common methods are adding a flat number of points, shifting scores so the top score becomes 100, scaling scores so the top score becomes 100, and taking the square root. For a student with 64% when the top score was 92%, those give 69% (flat +5), 72% (shift), 69.57% (scale) and 80% (square root).
What is a square root curve and how do you calculate it?
A square root curve turns a score out of 100 into 10 times its square root. A 64% becomes 10 × √64 = 80%. It helps low scores most: 50% becomes 70.71% (+20.71 points) while 90% becomes 94.87% (+4.87). It doesn't set the best score to 100, so a 92% becomes 95.92%.
How do you curve a test if nobody got 100%?
Use the top score. Shifting adds the gap to every score, so with a top score of 92% you add 8 points. Scaling multiplies every score by 100 ÷ 92 = 1.087. A 64% becomes 72% by shifting or 69.57% by scaling, and the top student ends up at 100% either way.
How do you grade a test with partial credit?
Give each question a point value, award points in steps such as halves or quarters, add up the points earned, and divide by the points possible. For example, 23.25 points out of 30 is 77.5%, C+ on the standard plus/minus scale.
Does curving ever lower a student's grade?
A curve should not. Flat, shift, scale and square-root curves all raise or keep every score, as long as the curve is applied to scores that are not above the top score. Methods based on the class mean and standard deviation can lower the scores of strong students, so keep the higher of the original and curved score for each student.
How do you add extra credit to a test?
The simplest method adds the bonus points to the points earned and leaves the total unchanged. With 42 of 50 points and 3 bonus points, the score is 45 ÷ 50 = 90%. Decide beforehand whether scores are capped at 100% and apply the rule to everyone.
Can I curve a test that isn't scored out of 100?
Yes. Convert each score to a percentage first, curve the percentage, then convert back if you need points. For example, 32 out of 40 is 80%. A square root curve gives 89.44%, which is 35.78 out of 40.