Consider $5$ observations $0 \leq a \leq b \leq c \leq d \leq e \leq 100$. If $\sigma$ is the standard deviation then it is always less than or equal to
(a) $20$ $\qquad$ (b) $30$ $\qquad$ (c) $40$ $\qquad$ (d) $50$
Now I know that the formula for standard deviation(as taught to us, I do not know if there are any other formulae) is
= √(∑(ₖ)²-²)
where is the mean of given data
On simplifying using given data,
25²=(∑(-)²)
I do not know how to proceed after this. My intuition says that if the last two numbers be 100 and the others be 0 standard deviation should be maximum(as it's 'spaced out' more, if that makes any sense). Then
² = 2400
but the answer is given as (d)50
Is the answer given wrong? Is there any inequality that would help in solving this question that I am not aware of? I tried using root mean square ≥ mean but that gives a lower bound not an upper bound.
Any help/hint would be appreciated