 Relative variance

For a random variable $X$ with finite variance and nonzero mean, its relative variance is $R_X=\mathrm{Var}(X)/(\mathbb{E}X{)}^2$. It equals the square of the coefficient of variation under the usual positive-mean convention. Multiplying $X$ by a nonzero constant leaves $R_X$ unchanged; adding a constant generally does not.

For an average of $n$ independent identically distributed observations with mean $\mu \ne 0$ and variance ${\sigma }^2$, the relative variance is ${\sigma }^2/(n{\mu }^2)$. Chebyshev's inequality gives $\mathbb{P}(|X/\mathbb{E}X-1|\ge a)\le R_X/a^2$, explaining its role in self-averaging. At zero mean the ratio is undefined, so another reference scale is needed.