There is no single official curve formula
Curves are teacher or institution choices, and different named methods applied to the same raw scores give different curved scores — sometimes different letter grades. Any page that implies “this is how your teacher will curve” is making a false claim.
The University of Alberta states that grades in any course “shall not be mandated on the basis of a curve or historic distribution of student grades” and that the distribution of grades “shall not be predetermined by any system of quotas.” The University of Illinois CS 277 course policy says the course “is not designed to operate on a curve,” and any curve is done by lowering grade cutoffs only, never raising them — scores themselves are not even changed. An Ohio State Astronomy 161 instructor sets the mean to a C+, makes each letter bin about as wide as the standard deviation, and then adjusts by judgment, historical question data, and question review.
The five named methods
Flat / additive. Add the same number of points to every score:
Pure shift. Every score gains the same absolute amount, gaps between students are unchanged, and rank order is preserved. It helps low and high scorers equally in absolute points. It does nothing about spread, skew, or clustering. Capping at the scale maximum is a teacher choice — this page makes it a toggle.
Top-score. Add the difference between the scale maximum and the highest score:
A special case of flat with a derived number of points: the highest score becomes the maximum by construction. It is not the same method as teacher-chosen flat — in flat, the top score need not reach the maximum, and scores can exceed the maximum if uncapped.
Scale-to-max. Multiply every score by the ratio of the scale maximum to the highest score:
Multiplicative, not additive. Higher scores gain more absolute points than lower scores — a 90 gains 10 while a 40 gains 4.44 when the top score is 90. Gaps stretch proportionally. Do not describe scaling as “everyone gets the same boost.” If the highest score is already the maximum, nothing changes. If every score is 0, the factor is undefined — the calculator shows an error, not an invented value.
Square-root. On a 0–100 scale:
General form for other scales: Curved = Scale maximum × √(Raw ÷ Scale maximum). The general form is this page’s stated generalisation of the 0–100 form — no source was found prescribing it for non-100 scales, but the two are algebraically equivalent. The curve is concave: the lowest scores gain most and the highest gain least. Anchor examples: 25 → 50.00, 36 → 60.00, 49 → 70.00, 64 → 80.00, 81 → 90.00, 96 → 97.98. It never lowers a score (for raw scores between 0 and the maximum, the curved score is at least the raw score), and 0 stays 0 and the maximum stays the maximum. It is sometimes called the “Texas curve” — that name’s origin is unverified, so this page does not assert it. Claims that it is the “fairest” method are value judgments, not facts, and do not appear here as fact.
Z-rescale. Convert each score to a z-score from the class mean and SD, then rescale to a target mean and SD:
Unlike the other four methods, this one can lower a score: if the target mean or SD is below the class’s actual mean or SD, above-average scores map downward. Target mean and SD are teacher choices — this page’s 75 / 10 defaults are editable examples, not rules. The SD convention matters: population (÷N) versus sample (÷n−1) changes every z-score. Full treatment: Bell Curve Grade Calculator.
Proof: one raw score, different curved scores
Same class set [40, 55, 65, 75, 90], mean 65.0. Focus on the student with raw 65:
| Method | Formula applied | Curved score for raw 65 |
|---|---|---|
| Flat +10 | 65 + 10 | 75.00 |
| Top-score (top 90) | 65 + (100 − 90) | 75.00 |
| Scale-to-max (top 90) | 65 × (100 ÷ 90) | 72.22 |
| Square-root | 10 × √65 | 80.62 |
| Z-rescale, target 75/10 (population SD) | 75 + ((65 − 65) ÷ 17.03) × 10 | 75.00 |
| Z-rescale, target 80/10 (population SD) | 80 + 0 × 10 | 80.00 |
Same student, same raw score: 72.22 to 80.62 across methods — an 8.4-point spread. The flat and bell (75/10) methods coincide at 75.00 here purely by arithmetic accident: 65 is the mean, and 65 + 10 = 75 = the target mean. A second student, raw 40: flat 50.00, scale 44.44, square-root 63.25, bell (75/10, population SD) 60.32 — an 18.81-point spread. No method can be inferred from the raw score alone; the method must be known.
The ordering also differs by method: square-root lifts the bottom most; scale-to-max lifts the top most.
A curve that changes no score: cutoff adjustment
Some teachers curve by moving the grade cutoffs, not the scores. Illinois CS 277 adjusts cutoffs lower and never raises them. A student who only sees score-transformation methods may wrongly conclude a curve must change their number. If your teacher curved this way, your raw score stays the same and the boundary for each letter moves.
Assumptions on this page
- Cap at the scale maximum: toggle, default on. A page default, not a rule.
- Rounding: display rounding, default 2 decimals. No source sets a standard rounding rule.
- Target mean / SD (75 / 10): editable example defaults, not rules.
- SD convention: population by default; both offered; the choice is stated in the result.
- General square-root form: the page’s stated generalisation for non-100 scales.
Common mistakes
- Assuming one standard curve. There is none. Ask your teacher which method, if any, applies.
- Assuming curves only raise scores. False for z-rescale and quota methods.
- Treating flat and top-score as the same method. Flat points are teacher-chosen; top-score points are derived from the highest score.
- Applying the 0–100 square-root form to another scale. Convert to a percentage first, or use the general form.
- Treating example defaults as rules. Target mean/SD, cutoffs, cap, and rounding on this page are editable defaults.
- Treating the computed curve as the recorded grade. Your teacher’s actual method, if any, governs.
Full walkthrough: How to Curve Grades. Bell-curve grading: Bell Curve Grade Calculator. Test scores: Test Grade Calculator.