lets say the first quartile of a random variable (continuous one) has a CDF function F such that $F(x) = .25$... e.g. the random variable is $3\times$ as likely to be larger than the first quartile than it is to be smaller?
What would the first quartile of $X$ be for $X\sim N(μ,σ^2)$?
Would it simply be $.25$ by symmetry of normal distributions?
For $X\sim N(\mu,\sigma^2)$ the first quartile is
$$q_{0.25}=\sigma\cdot \Phi^{-1}(0.25)+\mu$$
where $\Phi$ is the CDF of the standard normal r.v. because $q_{0.25}$ is the solution of
$$P\{X\le q_{0.25}\}=P\left\{\frac{X-\mu}{\sigma}\le \frac{q_{0.25}-\mu}{\sigma}\right\}=\Phi\left(\frac{q_{0.25}-\mu}{\sigma}\right)=0.25$$