You may also like

problem icon

Spread

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

problem icon

David and Goliath

Does weight confer an advantage to shot putters?

Maximum Scattering

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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.