List of 3 integers that gives 'simple' standard deviation

Question

Does anyone know a list of 3 integers that gives a 'simple' answer when you calculate the standard deviation? Ideally it would be an integer that is not too great. For example when I have the vector $$\vec x=(1,2,3)^T$$ The standard deviation is $$\sigma_x=\sqrt{\frac{1}{n}\sum_{i=1}^n(x_i-\langle x\rangle)^2}=\sqrt{2/3}$$ This has a square root and so it is not a simple answer. Ideally it should be easy to calculate so the coefficients should not be too high.

Answer 1

The only triples of rationals $(a,b,c)$ such that the variance $(2/9) (a^2 + b^2 + c^2 - ab - ac - bc)$ is the square of a rational are the trivial cases $a=b=c$.

Since variance is invariant under translation, we may assume wlog $c = 0$, and since variance is homogeneous of degree $2$ we can multiply by a common denominator to assume $a$ and $b$ are integers such that $2 (a^2 + b^2 - ab)$ is a square. Since this is even, $a^2 + b^2 - ab$ is even, but that requires $a$ and $b$ to be both even. Then the same condition is true for $a/2$ and $b/2$, and we get an infinite descent.