Question

The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75. What are the quartiles of the standard Normal distribution N(0,1)?

Answer 1

$\mathbb P(X\le Q_1)=F_X(Q_1)=0.25\implies Q_1={F_X}^{-1}(0.25)\approx-0.6745$

where $F_X(x)$ is the CDF for $X\sim\mathcal N(0,1)$.

By symmetry of the distribution, you then have

$Q_3=-Q_1\approx0.6745$