Preface:
I am currently preparing for a statistics olympiad qualifiers of the one of my country’s universities. Yesterday I came across one task, in which I do not quite understand the intuition behind the answer.
The task itself (translated):
We are given a sample $X$ of one hundred observations $X_i$ such that $\forall i: X_i \in \mathbb{N}[-101, 101]$. The sample structure is unknown, but we were still able to construct the following statistics: $$ M = \bar X,~A = \min{X}, ~B=\max{X} $$ Let us note that $A$ and $B$ are unique in $X$, i.e. there is only one unique minimum and only one unique maximum in the given sample.
Now, suppose that we have modified $X$ by negating $A$ [$A$ has now become $-A$] and $B$ [$-A$ and $-B$ are not necessarily the new maximum and minimum (respectively) of the modified $X$]. After applying the above transform to $X$, we have suddenly observed that the mean value of the modified $X$ is now $-M$, i.e. after the transform the mean value of the sample has just became the additive inverse of its past self.
Now the question is: what is the maximum possible value of $M$ in the original (unmodified) sample of $X$?
Answer: $M = 1$
My thoughts:
What I thought of is trying to artificially create a sample of all observations being equal with the only exception to $A$ and $B$, for example $X=(-99, 1, 1, …, 101)$, as here the true mean value is $1$ but then this fails to comply with the fact that the mean value becomes the negation of itself, if we swap the signs of $A$ and $B$. Then I thought that we probably can divide $X$ into two almost equal halves: $(-100, -1, -1,…, 1, 1, 101)$, and it seems like this structure doesn’t fail to comply with mean-inversion, but I am not sure if this example yields that maximum possible value of $M$ is one…
Any help will be appreciated, thank you in advance!
The sum of all the original terms is $100M$
so the sum of the other $98$ terms is $100M-A-B$.
Changing $A$ to $-A$ and $B$ to $-B$ reverses the sign of the mean and so also the sum of the total so $100M -A+(-A)-B +(-B) = -100M$
which implies that $100M-A-B =0$
and $M=\frac{A+B}{100}$
so some of the other terms must be zero or negative.
Given $A$ is a unique minimum, $A$ must be negative, i.e. $A \le -1$
while $B \le 101$ from the question
so $M=\frac{A+B}{100} \le\frac{-1+101}{100}=1$.
The obvious example is $\{-1,0,0,0,\ldots,0,0,0,101\}$.