How this calculator works
Every method here applies one rule to all students, so the ranking never changes unless the cap compresses the top. Adding points shifts the whole class up by the same amount. Shifting the mean computes the gap between the current mean and the target and adds or subtracts that gap from every score. Standardising first divides the distance of each score from the mean by the standard deviation, then multiplies by the target spread and adds the target mean; this changes both the average and the spacing. Multiplying scales the spread and the mean together. The cap protects against scores above the maximum, but it also compresses the top of the class.
Formula
original mean = sum(scores) / n. population standard deviation = sqrt(sum((score - mean)^2) / n). add: new = min(score + points, cap). mean shift: new = min(score + target mean - old mean, cap). standardise: new = clamp(target mean + (score - old mean) / old standard deviation x target standard deviation, 0, cap). multiply: new = min(score x multiplier, cap).
Worked example
Scores of 72, 85, 90, 64 and 78 have a mean of 77.8 and a population standard deviation of about 9.22. Adding 5 points gives 77, 90, 95, 69 and 83, a new mean of 82.8. Shifting the mean to 80 adds 2.2 points instead. Standardising to a mean of 80 and a standard deviation of 10 maps 90 to 80 + (90 - 77.8) / 9.22 x 10 = 93.24, and maps 64 to 65.03.
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Frequently asked questions
Which standard deviation does the calculator use?
It uses the population standard deviation, dividing by the number of scores rather than one less. That is the usual choice when the scores are the whole class rather than a sample meant to represent a larger group. For a sample, dividing by n - 1 gives a slightly larger spread and slightly more extreme curved scores.
Can a curve lower scores?
Yes. If the target mean is below the current mean, the shift method subtracts points from everyone. The standardisation method also lowers scores that are above the target mean when the target is set low. This is why the direction and size of the curve should be agreed before it is applied.
Why does the cap change the shape of the class?
Scores that would land above the highest possible grade are pulled down to the cap, so several students can end with the same maximum score. The mean then rises by less than the raw shift would suggest. Raise the cap or use a smaller shift if the class is bunching at the top.
Is curving fair?
A curve can correct a test that was harder than intended, but it can also reward or penalise students for the performance of others. A fixed points addition is the most transparent because everyone gets the same benefit. Standardising to a target distribution makes scores depend on the class spread, which some schools restrict or forbid.