Grading on a curve: how the mean and standard deviation set grades

Why a teacher curves at all

A test can turn out harder than intended. If a well-prepared class averages 66 on an exam the teacher expected to average 75, the problem is more likely the exam than the students. A curve corrects for that by treating the class's own results as the reference point. The assumption behind it is that this group's ability is about the same as that of the groups the grade scale was designed for, so any shift in the scores reflects the test.

That assumption is also the main argument against curving. In a small class, or a class that really did prepare less, the scores may be telling the truth.

The worked class

Take a hypothetical class of ten with these raw scores out of 100: 48, 55, 58, 62, 64, 67, 70, 73, 78, 85. They sum to 660, so the mean is 66. The class is the whole group being graded, so the population standard deviation is the right one:

μ = 660 / 10 = 66 σ = √[Σ(x − μ)² / n] = √(1100 / 10) = 10.49

If you want the difference between the two formulas explained, see sample vs population standard deviation; the test score standard deviation calculator gives both from a pasted list of marks. On a 90/80/70/60 scale this class earns no A, one B, three Cs, three Ds and three Fs.

Try it: test score standard deviation calculator

The calculator below opens with this class's ten scores in population mode; paste your own class's marks to get their mean and standard deviation, the two numbers every curve here starts from.

Open the full test score standard deviation calculator for example classes and a per-student z-score breakdown.

Method 1: the z-score curve

The standard statistical curve converts each score to a z-score, which is its distance from the mean in standard deviations, then maps that z-score onto a target mean and SD the teacher chooses:

z = (x − μ) / σ curved = target mean + z × target SD

With a target mean of 75 and a target SD of 10, the student who scored 85 has z = (85 − 66) / 10.49 = 1.81, and a curved score of 75 + 1.81 × 10 = 93.1. The full class:

The z-score curve keeps every student's rank and every student's relative gap. Its useful property is that the teacher controls the spread as well as the centre. Choose a target SD smaller than the class's actual SD and the grades bunch together; choose a larger one and they fan out. Here the target of 10 is close to the actual 10.49, so the curve barely changes the spread, and the result is almost identical to adding 9 points to everyone. That is worth noticing: a z-score curve only does something a flat add cannot when the target SD is meaningfully different from the class SD.

Method 2: flat adds

The simplest curves add the same number of points to every paper. "Add 9" moves the mean from 66 to the target of 75. "Top to 100" adds whatever lifts the highest score to full marks, here 15 points, which is generous when one student did unusually well and does little when someone already scored 98. Adding a constant changes the mean and leaves the standard deviation exactly where it was, 10.49, so the distances between students are unchanged.

The square-root curve in the last column (10 times the square root of the raw score) is a different animal. It lifts low scores far more than high ones: 48 gains 21.3 points, 85 gains 7.2. The class SD falls from 10.49 to 6.48 on this data, so it compresses the grade range as well as raising it. It never pushes anyone above 100, which is one reason some instructors like it.

Method 3: fixed percentages

The strictest form of curving hands out a fixed share of each grade regardless of the scores: say 20% A, 30% B, 30% C and 20% D or below. In the class of ten that means the top two get an A, the next three a B, and so on:

This ignores the mean and SD entirely and uses only rank, which is what a percentile rank measures. The student with 78 gets an A while sitting 7 points below the top score, and a student on 67 gets a B that a single extra point from a classmate could take away. Fixed percentages make grades comparable between sections, but they turn a test into a contest.

Some courses use z-score bands instead: A for z ≥ 1, B for 0 ≤ z < 1, C for −1 ≤ z < 0 and so on. For roughly bell-shaped scores those bands give about 16%, 34%, 34% and 16%, the same proportions the normal distribution puts in each one-SD slice. In a class of ten the actual split rarely matches: here it is two, three, three and two, because two students sit above z = 1 and two below z = −1.

What each method costs

Every one of these is sensitive to small classes. With ten students, one absent strong student moves the mean and SD enough to shift several grades. A curve built on the mean and SD is also pulled around by outliers: if the 85 had been a 100, the mean would rise to 67.5 and the SD to 13.7, and every other student's z-score would fall. The median and interquartile range are steadier reference points for small or lopsided classes.

Reading a curved grade as a student

If your teacher reports the class mean and SD, you can locate yourself without waiting for the letter grade. A raw score of 73 in this class is (73 − 66) / 10.49 = 0.67 SD above the mean; on any z-score curve that places you above the middle and below the A band. Two things to keep in mind: a z-score says where you stand in this class, not how much of the material you know, and a curve that suits one exam says nothing about whether the next one will be curved at all.