How are two random variables with same mean and different variances are stochastically ordered?

Question

Suppose that we have a random variable $X$ that is normaly distributed with mean $\mu_x$ and variance $\sigma^2_x<\infty$. Any other normal random variable $Y$ with the same mean as the random variable $X$ and different variance $\sigma_y^2\neq \sigma^2_x$ is stochasticly related (ordered if $\sigma_y^2> \sigma^2_x$ or $\sigma_y^2< \sigma^2_x$) to $X$? Namely, is this some kind of stochastic order dominance? Do we need to assume that X and Y are i.i.d. or follow a bi-variate (or multivariate in case of more thatn two random variables) normal distribution?