A score of 68 is 0.38 standard deviations below the class mean of 71.75 and becomes 71.2 after the curve. Its percentile rank stays 37.5: rescaling changes scores, not the order of students.
Original score → curved score
StudentsNo change
Every student (20 rows)
#
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
How it's calculated S
Class mean
xˉ=n∑xi=71.75
Class standard deviation (population)
σ=n∑(xi−xˉ)2=9.878639
Curve rule
xcurved=75+10×9.878639x−71.75
A linear rescale keeps every student's z-score and rank.
Your score
z=9.87863968−71.75=−0.379607,xcurved=71.20393
Percentile rank
PR=nB+21E×100=207+21×1×100=37.5
B counts class scores below this score and E counts scores equal to it. Ranks come from the original scores.
About the grade curve calculator
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. Both keep the order of students, so percentile ranks do not change; only scores pushed above the highest possible score are cut back to it.
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.
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. Every student keeps the same rank.
What is the difference between a flat curve and a bell curve?
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.
What is a percentile rank?
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 linear curve never changes percentile ranks, because it keeps the order of the scores.
Should I use the population or sample standard deviation to curve grades?
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.
How accurate is the grade curve calculator?
Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 4 worked examples whose answers come from independent sources; for example, “Default class, score 68” is checked against Python statistics.mean and pstdev (9.87863857); curved = 75 + 10·(68 − 71.75)/σ in decimal; PR = (7 + ½·1)/20 × 100.
Where does the method come from?
NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.1 (mean) and §1.3.5.6 (standard deviation); Crocker & Algina, Introduction to Classical and Modern Test Theory (1986), ch. 19: percentile ranks and linear standard scores.