Is it possible to have 2 different but equal size real number sets that have the same mean and standard deviation? By inspection I notice that


*

*Shifting does not change the standard deviation but change mean.
{1,3,4} has the same standard deviation as {11,13,14} for example.

*Sets with the same (or reversed) sequence of adjacent difference have the same standard deviation. For example, {1,3,4}, {0,2,3}, {0,1,3} have the same standard deviation. But the means are different.
My conjecture: There are no two distinct sets with the same length, mean and standard deviation. 
Question
Is it possible to have 2 different but equal size real number sets that have the same mean and standard deviation?

A simple way to find a counterexample to the conjecture is to focus on sets whose values are symmetrical about zero.  This ensures that the two sets have the same mean, and also simplifies calculation of their standard deviations.
Let $a,b$ be any real numbers.  Then the set $\{a,b,-a,-b\}$ has mean zero and SD $\sqrt{(2a^2+2b^2)/4}$.  Now let $c$ be any real number not equal to any member of that set and such that $c^2 < a^2+b^2$.  Let $d$ be given by:
$$d = \sqrt{a^2 + b^2 - c^2}$$
implying $a^2 + b^2 = c^2 + d^2$.  Then the set $\{c,d,-c,-d\}$ has the same mean and SD.
