Seven rules, one class
Everything below uses the same ten scores, which are also the example built into the calculator:
41, 48, 53, 57, 60, 63, 68, 71, 79, 80 — average 62, lowest 41, highest 80, standard deviation 12.16.
Watch what the choice of rule does to the same ten students.
| Rule | New average | 41 becomes | 80 becomes |
|---|---|---|---|
| Flat bonus of 8 | 70 | 49 | 88 |
| Lift top to 100 | 82 | 61 | 100 |
| Scale to top score | 77.5 | 51.25 | 100 |
| Square root curve | 78.35 | 64.03 | 89.44 |
| Bell curve to 75 ± 10 | 75 | 57.73 | 89.8 |
The bottom student ends on anything from 49 to 64 depending purely on which rule the instructor happened to prefer. That spread is the argument for showing the method, and it is why this page will not name one of them as the standard.
Flat bonus
Add the same number of points to everybody. The simplest rule and the one with the clearest justification: if the paper was uniformly harder than intended, everybody lost the same marks, so everybody gets them back.
It shifts the average by exactly the bonus, leaves the spread untouched and leaves the ranking untouched. A bonus of 8 takes the average from 62 to 70 and turns 41 into 49.
Two variants are worth separating. Lift the top to full marks sets the bonus at 100 minus the highest score — here 20, taking the average to 82, which is a very generous curve that people reach for without noticing how generous it is. Move the average to a target sets the bonus at the gap between the current and desired average, which is 13 to get from 62 to 75.
The one thing to watch: if the bonus pushes anyone past the maximum, they get capped, the promised shift in the average does not actually happen, and students who were not tied become tied. The calculator counts those and says so.
Linear scaling
Multiply every score by the same factor, chosen so the top score lands on the maximum. With a top score of 80 the factor is 100 ÷ 80 = 1.25.
This is not a flat bonus in disguise. Scaling gives more to students who already had more: 80 gains 20 points, while 41 gains only 10.25. If the intention was to compensate for a hard paper, scaling compensates the strongest students twice as much as the weakest, which may or may not be what was meant.
Stretching onto a new range is the variant that also fixes the bottom: map the lowest score to a floor you choose and the highest to the maximum. It is the only rule here that can raise a failing score to a pass by construction, which is either the point or the objection depending on your view.
The square root curve
On a 100-point scale: new score = √score × 10.
- 25 → 50
- 49 → 70
- 64 → 80
- 81 → 90
- 100 → 100
The shape is the appeal. It is enormously generous at the bottom — 41 becomes 64.03, a gain of 23 points — and almost neutral at the top, where 80 becomes 89.44, a gain of 9.4. A student who was already doing well gains little; a student who was struggling gains a lot. Whether that is fair or perverse depends on what you think the paper was measuring.
The √score × 10 shorthand only works out of 100. The general form is √(score ÷ max) × max, which is what this calculator uses, so a test out of 50 curves correctly instead of producing nonsense.
The bell curve
The most misunderstood term on this page. Curving "on a bell curve" usually means rescaling by z-score: how many standard deviations from the class average each student sat, held constant while the average and the spread are moved to targets.
z = (score − mean) ÷ standard deviation
new score = target mean + z × target spread
With this class — mean 62, standard deviation 12.16 — rescaled to an average of 75 with a spread of 10, the student who scored 41 was 1.728 standard deviations below average, so they land at 75 − 17.28 = 57.73. The student on 80 was 1.48 above, so they land at 89.8.
Two things to know before using it. First, it is the only rule here that can lower a score: rescale to an average below the class average and the top of the class comes down. Second, it needs spread to work at all — a class where everyone scored the same has a standard deviation of zero and no z-scores, and the calculator refuses rather than dividing by zero.
Note also what this is not: it is not assigning fixed percentages of A, B and C by rank. That is a different practice, also called curving, which forces a distribution regardless of how well anyone did. This calculator does not implement it, because it depends on a quota table that is entirely institutional.
Population or sample standard deviation
A small distinction that changes the numbers in a small class. The population formula divides by n; the sample formula divides by n − 1 and gives a slightly larger figure.
For a whole class that sat the exam, the population figure is the right one — the class is the population, not a sample drawn from some larger group of hypothetical students. This calculator uses it by default, shows the sample figure beside it, and lets you switch if your department specifies otherwise. In this ten-student class the two are 12.16 and 12.82, which is enough to move a borderline student across a letter boundary.
Letter grades
This page ships no default grading scale. A starting at 90 is a widely used American convention and not a fact — plenty of systems put A at 70, or use 1–5, or First/2:1/2:2, and a calculator that assumes one of them is quietly wrong for everyone else.
Enter your own cut-offs and the calculator shows the letter before and after for every student, and counts how many crossed a boundary. That count is usually a better measure of whether a curve achieved anything than the change in the average: an eight-point bonus that moves the average by eight points but only moves two students across a boundary has not done much.
Choosing a rule honestly
A short version of the reasoning, offered as reasoning rather than as a rule:
- The paper was uniformly harder than intended. A flat bonus matches the diagnosis: everyone lost the same marks, everyone gets them back.
- One question was impossible and nobody got it. Drop the question and rescale, which is linear scaling with a known factor rather than a curve at all.
- The scores are too low across the board and you want a specific average. Move the average to a target, and look at how many scores get capped.
- You want the relative standing preserved but the numbers normalised across sections or years. A bell curve by z-score does exactly that and nothing else.
- You want to lift the bottom of the class in particular. The square root curve, with the caveat that it is a large intervention and it is worth checking whether a 64 for a paper that scored 41 is a number you can defend.
Whichever you choose, the case for telling students which rule you used is that they can check it. Every figure on this page is reproducible from the scores and the rule, which is not true of a curve that is simply announced.
Telling students what you did
A curve that is announced as a number — "I added eight points" — can be checked. A curve that is announced as a result — "grades have been adjusted" — cannot, and the difference matters more than it looks.
Every figure this page produces is reproducible from two things: the raw scores and the rule. A student who knows both can verify their own grade, which removes an entire category of anxious email. It also removes the suspicion that the curve was fitted to the outcome rather than to the paper — which is what people assume when a method is not stated, whether or not it is true.
Frequently asked questions
What does it mean to curve a grade?
To adjust a set of scores by a rule applied to all of them, usually because the test turned out harder than intended. The rule can be additive (everyone gets eight more points), multiplicative (everyone is scaled so the top mark becomes 100), or statistical (scores are rescaled to a target mean and spread). Which one your instructor uses is their choice — there is no standard.
What is the square root curve?
Take the square root of the score and multiply by ten, on a 100-point scale: 64 becomes 80, 49 becomes 70, 25 becomes 50. It is generous to low scores and barely touches high ones, and 100 stays 100 because the square root of 1 is 1. Written properly it is √(score ÷ max) × max, which is the same thing on a 100-point test and still correct on a test out of 50.
How do I calculate a bell curve for grades?
Work out the class mean and standard deviation, convert each score to a z-score — (score − mean) ÷ standard deviation — and then rebuild it at a target mean and spread: new score = target mean + z × target spread. A student one standard deviation above the class average stays one target spread above the new average, whatever the raw numbers were.
Which curving method is the standard one?
There is not one. Flat bonuses, square root curves, linear scaling and z-score rescaling are all in regular use, and departments differ, instructors within a department differ, and the same instructor may curve one paper and not another. This calculator applies whichever rule you pick and shows all seven on the same scores so you can see how much the choice matters.
Does curving ever lower a grade?
It can. A flat bonus and the square root curve never lower anything. Linear scaling to the top score never lowers anything either. But a bell curve rescaling to a target mean below the class average will pull high scores down, and linear-range scaling onto a narrower band can too. The calculator reports the change for every student, including negative ones.
Should I use the population or sample standard deviation?
For a whole class that sat the exam, the population figure — the class is the population, not a sample drawn from one. This calculator uses population by default, shows the sample figure alongside, and lets you switch, because some departments specify the sample version and the two differ noticeably in a small class.
What happens to scores that go above 100?
They are capped at the maximum, and the calculator tells you how many were capped. That matters: capping quietly changes the shape of the result, can tie students who were not tied before, and means a rule that promised to move the mean by eight points will have moved it by less. You can also turn the cap off if your department allows scores above the maximum.
Can I curve a single score without the whole class?
Partly. The square root curve and a flat bonus need only your own score. Linear scaling also needs the highest score in the class. A bell curve needs the class mean and standard deviation. Anything else — scaling the observed range, hitting a target mean — needs the full list, and the calculator says so rather than guessing at the rest of the class.
Why did my letter grade not change after a curve?
Because letter bands are wide and curves are often narrow. An eight-point bonus moves a 71 to a 79, which is still a C on a 90/80/70/60 scale. The calculator counts how many students cross a boundary, which is usually a better measure of whether a curve did anything than the change in the average.
Does this page set the A/B/C cut-offs?
No. It ships no default scale at all, because A starting at 90 is an American convention rather than a fact and plenty of systems do it differently. Enter the cut-offs your course uses and the calculator will show letters before and after; leave them alone and it works purely in scores.
Are the scores I paste in sent anywhere?
No. Parsing and every calculation run in JavaScript in your own browser. A pasted column of student marks is not transmitted or stored.