Grade curve calculator: rescale a class's scores to a target mean and standard deviation, or add flat points, with z-scores and percentile ranks.
By AdminDrafted with AI writing toolsUpdated Checked examples: 4
The entered score 68 becomes 71.2 after the curve. Its percentile rank in the original class is 37.5. Its original z-score is −0.38, relative to the class mean of 71.75. Before capping, the transformation preserves the order of students; a cap can create new ties.
| # | Score | z | Curved | Percentile rank |
|---|---|---|---|---|
| 1 | 52 | −1.9993 | 55.0074 | 2.5 |
| 2 | 58 | −1.3919 | 61.0811 | 7.5 |
| 3 | 61 | −1.0882 | 64.1179 | 12.5 |
| 4 | 63 | −0.8857 | 66.1425 | 17.5 |
| 5 | 64 | −0.7845 | 67.1548 | 22.5 |
| 6 | 66 | −0.5821 | 69.1794 | 27.5 |
| 7 | 67 | −0.4808 | 70.1916 | 32.5 |
| 8 | 68 | −0.3796 | 71.2039 | 37.5 |
| 9 | 70 | −0.1771 | 73.2285 | 42.5 |
| 10 | 71 | −0.0759 | 74.2408 | 47.5 |
| 11 | 72 | 0.0253 | 75.2531 | 55 |
| 12 | 72 | 0.0253 | 75.2531 | 55 |
Before capping, a positive linear rescale preserves z-scores and ranks.
B counts class scores below this score and E counts scores equal to it. Ranks come from the original scores.
A z-score curve rescales every score linearly. Each student's z-score, z = (x − class mean)/SD, is kept and mapped onto the target: curved = target mean + target SD × z. A flat curve instead adds the same number of points to everyone so that the class mean reaches the target. Before capping, both preserve the order and percentile ranks. Scores above the highest possible score are capped; capping can create new ties. Percentile ranks shown here always refer to the original class scores. In the formula y is the curved score, μₜ and σₜ are the target mean and SD, and B and E count original scores below and equal to x.
The default class of 20 has a mean of 71.75 and a population SD of 9.88. Curving it to a mean of 75 and an SD of 10 turns a 68 (z = −0.38) into 71.20, and its percentile rank stays 37.5.
A z-score curve does not make scores normally distributed: a skewed class stays skewed. When scores are capped, the curved mean falls below the target, and a warning shows the actual value.
To curve grades, convert each score to a z-score, z = (x − mean) ÷ SD, then compute curved = target mean + target SD × z. In a class with mean 71.75 and SD 9.88, a 68 has z = −0.38; curving to mean 75 and SD 10 gives 75 + 10 × (−0.38) = 71.20.
A flat curve is simpler: add the gap between the target mean and the class mean to every score. A class averaging 65 reaches 70 when everyone gets 5 points, so a 60 becomes 65.
Take these 20 scores: 52, 58, 61, 63, 64, 66, 67, 68, 70, 71, 72, 72, 74, 75, 77, 79, 81, 84, 88, 93. The goal is a mean of 75 and a standard deviation of 10.
The same rule sends the lowest score, 52, to 55.01 and the top score, 93, to 96.51. The curved class has a mean of exactly 75 and an SD of exactly 10.
A flat curve moves every score by the same amount. The standard deviation stays as it was, so the gaps between students do not change.
A z-score curve, often called a bell curve, also resets the spread. If the target SD is larger than the class SD, the lowest scores can fall even when the mean rises.
Curving the class above to an SD of 15 instead of 10 drops 52 to 45.01 and 64 to 63.23, while 66 rises to 66.27.
Neither method makes scores normally distributed. A linear rescale keeps the shape of the class, so a long tail of low scores is still there after the curve.
Population SD divides the squared deviations by n and treats the class as the whole group being graded, the usual choice for curving. Sample SD divides by n − 1 and estimates the spread of a larger group the class was drawn from.
For 20 students the two differ by a factor of √(20/19) ≈ 1.026. With the sample SD of 10.1353, the 68 curves to 71.30 instead of 71.20, and the 93 to 95.97 instead of 96.51. The NIST/SEMATECH e-Handbook defines both measures.
A z-score curve can push the best scores past the maximum mark. Capping them at 100 removes points from the class total, so the curved mean ends up below the target.
For scores 40, 55, 60, 70 and 95 curved to a mean of 80 and an SD of 15, the 95 would become 105.44. Capped at 100, the class mean is 78.91 rather than 80, and any two students who pass the cap end up tied.
A percentile rank is the share of the class scoring below a score, with ties counted as half: PR = 100 × (B + E/2) ÷ n. B counts scores below and E counts equal scores.
In the class of 20, seven scores are below 68 and one equals it, so PR = 100 × 7.5 ÷ 20 = 37.5. Both kinds of curve preserve the order of students, so ranks only change when a cap creates ties. Ranks are always computed from the original scores.
| Scenario | Original selected score | Curved score |
|---|---|---|
| Default class and targets | 68 | 71.20 |
| Add five points | 60 | 65.00 |
| Top score capped at 100 | 95 | 100.00 |
| Equal scores shifted to 75 | 60 | 75.00 |
The first row is the class of 20 curved to 75 and 10. The second adds 5 points to 55, 60, 65, 70 and 75, and the third is the capped class above.
The last row curves three scores of 60 to a mean of 75. Only a flat curve can do that: identical scores have an SD of 0, so their z-scores are undefined.
To check a class's mean, spread and skew before choosing a curve, use the descriptive statistics calculator. To read a z-score as a percentile under a normal model, use the normal distribution calculator.
Checked against: curved = 75 + 10·(68 − 71.75)/σ in decimal; PR = (7 + ½·1)/20 × 100
Checked against: Hand computation: mean 65, so +5 points; PR of 60 = (1 + ½·1)/5 × 100; population SD √50 is unchanged by a shift
Checked against: σ = 18.2757; 95 → 80 + 15 × 1.69624 = 105.44, capped to 100; mean of the capped curved list
Checked against: median 74 → 75 + 10·(74 − 75)/√160
Find the class mean and standard deviation, turn each score into a z-score, then map it onto the target: curved = target mean + target SD × z. In the default class (mean 71.75, SD 9.88) with a target of 75 and 10, a score of 68 has z = −0.38 and becomes 75 + 10 × (−0.38) = 71.20. Without capping, every student keeps the same rank; a cap can create new ties.
A flat curve adds the same points to every score, so the spread stays as it was: raising a class mean from 65 to 70 adds 5 points to everyone. A z-score curve also sets the spread. With the defaults the SD goes from 9.88 to 10, so 52 rises to 55.01 and 93 to 96.51. If the target SD is well above the class SD, low scores can even fall.
The percentage of the class scoring below a score, with ties counted as half: PR = (B + ½E)/n × 100, the definition used in classical test theory (Crocker & Algina, 1986). In the default class of 20, 7 scores are below 68 and 1 equals it, so PR = (7 + 0.5)/20 × 100 = 37.5. A positive linear transformation preserves percentile ranks before capping. Capping can create new ties; this calculator reports ranks from the original scores.
Use the population SD (divide by n) when the class is the whole group being graded, which is the default. The sample SD (divide by n − 1) estimates the spread of a larger group the class was drawn from. For 20 students the two differ by a factor of √(20/19) = 1.026; switching to the sample SD moves the default 68 from 71.20 to 71.30 and the top score of 93 from 96.51 to 95.97.
This calculator includes 4 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.