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Given the mean and standard deviation of a set of marks, what is the greatest number of candidates who could have scored 100%?

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David and Goliath

Does weight confer an advantage to shot putters?

Maximum Scattering

Age 16 to 18 Challenge Level:

Having done the first two parts of the question you can show similarly that, with $n$ numbers, there exists a certain value of the variance and hence that the variance can be at least that large. It is quite a subtle point, but you can't be sure that you have found the largest value of the variance with $n = 4$ or more without assuming that the geometrical reasoning generalises from $3$ dimensions to higher dimensions. The results are in fact valid in $n$-dimensional geometry but you are not asked to prove this.