Curve exam scores with the linear (mean + SD) method or the square-root curve
| Raw Score | z-score | Linear Curve | Square-Root Curve | Letter (linear) |
|---|
Linear curve = target mean + z × target SD. Square-root curve = 10 × √raw, capped at 100. Letter grades use the standard 10-point scale (90 A, 80 B, 70 C, 60 D).
| Method | Formula | Raw 44 → | Raw 64 (mean) → | Raw 80 → | Best When |
|---|---|---|---|---|---|
| Linear (mean + SD) | M + (x − m) × (s ÷ SD) | 58.8 | 75.0 | 88.0 | Distribution is roughly normal; you want a controlled shift and squeeze |
| Square-root | 10 × √x | 66.3 | 80.0 | 89.4 | Everyone struggled; low scores need the biggest lift |
| Flat bonus | x + 10 | 54 | 74 | 90 | One bad question tanked the whole exam; every point counts equally |
| Scale to max | x ÷ max × 100 | 55.0 | 80.0 | 100 | The top score is the "real" 100 and should map there |
Comparison uses the worked example below (mean 64, SD 12.33, high score 80) with a 75 ± 10 linear target and a +10 flat bonus. Scale-to-max divides by the class high, so it guarantees one perfect score.
| Raw Score | 25 | 36 | 49 | 64 | 81 | 100 |
|---|---|---|---|---|---|---|
| Curved (10 × √x) | 50 | 60 | 70 | 80 | 90 | 100 |
| Points Gained | +25 | +24 | +21 | +16 | +9 | 0 |
The pattern: the square-root curve gives back the most where scores hurt the most, and it never touches a perfect paper. Any raw score at or below 100 can only go up.
A curve doesn't add free points at random. It rescales a set of raw exam scores so the class lands where you want it: a chosen average, a chosen spread, or a fixed boost to the bottom. This tool runs two of the most widely used methods side by side so you can see which one fits your class before you commit.
The linear method is a z-score rescale. Compute the class mean m and population standard deviation SD, then curved = M + ((x − m) ÷ SD) × s, where M is your target mean and s your target SD. The square-root method is simply 10 × √x, capped at 100. The calculator also reports each score's z-score, which tells you how many standard deviations a student sits from the average, and a letter grade on the 10-point scale.
Paste your scores in any format, commas or spaces or one per line. Set the target mean and SD for the linear curve. The default 75 ± 10 is a common choice: it centers the class on a solid C+ and keeps a reasonable spread. Every input updates the table instantly, and nothing is sent anywhere since it all runs in your browser.
Say five students scored 58, 72, 66, 44, and 80. The mean is 64 and the population SD is 12.33. With a linear target of 75 ± 10, each score moves by (x − 64) × (10 ÷ 12.33): the 44 becomes 58.8, the mean 64 becomes exactly 75, and the 80 becomes 88.0. The square-root curve is more generous to the bottom: the 44 jumps to 66.3 while the 80 lands at 89.4. Neither method changes the order of the class, which is the point. Curving adjusts the scale, not the ranking.
Convert each score to a z-score by subtracting the class mean and dividing by the class standard deviation, then rescale: curved = target mean + z × target SD. On a class with mean 64 and SD 12.3, a 58 is 0.49 SD below average, so with a target mean of 75 and target SD of 10 it curves to 75 − 4.9, which is about 70.
The square-root curve replaces each score with 10 times its square root, capped at 100. It lifts low scores the most: a 25 becomes 50, a 49 becomes 70, and an 81 becomes 90, while a perfect 100 stays 100. Teachers reach for it when an exam was brutal for everyone and the whole distribution needs a floor.
Yes, with the linear method. If the class mean is above your target mean, everyone below the class average loses points. A class averaging 82 curved to a target of 75 drops an 80 to about 73. The square-root curve never lowers a score of 100 or below, since 10 × √x is at least x for every x from 0 to 100.
Use population SD, dividing by n, when the score list is the entire class, which is the normal case for curving. Sample SD, dividing by n − 1, only matters when the scores are a sample of a larger group. The gap closes fast as classes grow: on five scores with a population SD of 12.33 the sample SD is 13.78, but on 30 scores they differ by under 2%.
In most schools, yes. Curving is a grading-policy decision that belongs to the instructor or department, not a legal question, though some schools and states have rules about changing posted grades after the fact. If you curve, say so in the syllabus up front, and never curve down an individual student who would have scored lower after the adjustment than before.